Integrand size = 27, antiderivative size = 1151 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx =\text {Too large to display} \] Output:
-1/84*(5*b^3*e^3*g-168*c^3*d^2*(-d*g+e*f)+28*c^2*e*(-6*a*e+7*b*d)*(-d*g+e* f)-2*b*c*e^2*(10*a*e*g-7*b*d*g+7*b*e*f)+2*c*e*(14*c*e*(-b*e+2*c*d)*f+5*(-4 *a*c+b^2)*e^2*g-14*c*d*(-b*e+2*c*d)*g)*x)*(c*x^2+b*x+a)^(1/4)/c^2/e^4+1/35 *(10*c*e*g*x+5*b*e*g-14*c*d*g+14*c*e*f)*(c*x^2+b*x+a)^(5/4)/c/e^2-(4*a*c-b ^2)^(3/4)*(a*e^2-b*d*e+c*d^2)^(5/4)*(-d*g+e*f)*(-c*(c*x^2+b*x+a)/(-4*a*c+b ^2))^(3/4)*arctan(1/2*(4*a*c-b^2)^(1/4)*e^(1/2)*(1-(2*c*x+b)^2/(-4*a*c+b^2 ))^(1/4)*2^(1/2)/c^(1/4)/(a*e^2-b*d*e+c*d^2)^(1/4))/c^(3/4)/e^(9/2)/(c*x^2 +b*x+a)^(3/4)-(4*a*c-b^2)^(3/4)*(a*e^2-b*d*e+c*d^2)^(5/4)*(-d*g+e*f)*(-c*( c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*arctanh(1/2*(4*a*c-b^2)^(1/4)*e^(1/2)*(1- (2*c*x+b)^2/(-4*a*c+b^2))^(1/4)*2^(1/2)/c^(1/4)/(a*e^2-b*d*e+c*d^2)^(1/4)) /c^(3/4)/e^(9/2)/(c*x^2+b*x+a)^(3/4)+1/168*(4*a*c-b^2)^(1/2)*(5*b^4*e^4*g- 336*c^4*d^3*(-d*g+e*f)+56*c^3*d*e*(-8*a*e+9*b*d)*(-d*g+e*f)-2*b^2*c*e^3*(2 0*a*e*g-7*b*d*g+7*b*e*f)+4*c^2*e^2*(20*a^2*e^2*g-35*b^2*d*(-d*g+e*f)+56*a* b*e*(-d*g+e*f)))*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*InverseJacobiAM(1/2 *arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)),2^(1/2))*2^(1/2)/c^3/e^5/(c*x^2+b*x+a )^(3/4)-1/2*(-4*a*c+b^2)*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(-d*g+e*f)*((2*c *x+b)^2/(-4*a*c+b^2))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*Elliptic Pi((1-(2*c*x+b)^2/(-4*a*c+b^2))^(1/4),-1/2*(4*a*c-b^2)^(1/2)*e/c^(1/2)/(a* e^2-b*d*e+c*d^2)^(1/2),I)*2^(1/2)/c/e^5/(2*c*x+b)/(c*x^2+b*x+a)^(3/4)-1/2* (-4*a*c+b^2)*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(-d*g+e*f)*((2*c*x+b)^2/(...
Time = 16.93 (sec) , antiderivative size = 2095, normalized size of antiderivative = 1.82 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\text {Result too large to show} \] Input:
Integrate[((f + g*x)*(a + b*x + c*x^2)^(5/4))/(d + e*x),x]
Output:
(4*((7*c*e*f)/2 - ((7*c*d)/2 - (5*b*e)/4)*g + (5*c*e*g*x)/2)*(a + x*(b + c *x))^(5/4))/(35*c*e^2) - ((a + x*(b + c*x))^(5/4)*((4*((3*c*e*(5*b^2*d*e*g - 4*a*c*e*(7*e*f - 2*d*g) + 14*b*c*d*(e*f - d*g)))/8 - (((3*c*d)/2 - (b*e )/4)*(5*b^2*e^2*g + 28*c^2*d*(e*f - d*g) - 2*c*e*(7*b*e*f - 7*b*d*g + 10*a *e*g)))/4 + (c*e*(5*b^2*e^2*g + 28*c^2*d*(e*f - d*g) - 2*c*e*(7*b*e*f - 7* b*d*g + 10*a*e*g))*x)/8)*(a + b*x + c*x^2)^(1/4))/(3*c*e^2) - (-1/4*(Sqrt[ b^2 - 4*a*c]*(c^2/((b^2 - 4*a*c)*((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b ^2 - 4*a*c)^2)))^(3/4)*(5*b^4*e^4*g - 336*c^4*d^3*(e*f - d*g) + 56*c^3*d*e *(9*b*d - 8*a*e)*(e*f - d*g) - 2*b^2*c*e^3*(7*b*e*f - 7*b*d*g + 20*a*e*g) + 4*c^2*e^2*(20*a^2*e^2*g - 35*b^2*d*(e*f - d*g) + 56*a*b*e*(e*f - d*g)))* (-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*EllipticF[ArcSin[(Sqrt[b^2 - 4*a*c]*(-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c)))/c]/2, 2])/(Sq rt[2]*c*e*(a + b*x + c*x^2)^(3/4)) + (2*Sqrt[2]*(c^2/((b^2 - 4*a*c)*((b^2* c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2)))^(3/4)*(-1/16*(d*(5*b^4 *e^4*g - 336*c^4*d^3*(e*f - d*g) + 56*c^3*d*e*(9*b*d - 8*a*e)*(e*f - d*g) - 2*b^2*c*e^3*(7*b*e*f - 7*b*d*g + 20*a*e*g) + 4*c^2*e^2*(20*a^2*e^2*g - 3 5*b^2*d*(e*f - d*g) + 56*a*b*e*(e*f - d*g)))) + (e*(6*c*e*(b*d - 2*a*e)*(5 *b^2*d*e*g - 4*a*c*e*(7*e*f - 2*d*g) + 14*b*c*d*(e*f - d*g)) - d*(6*b*c*d - b^2*e - 8*a*c*e)*(5*b^2*e^2*g + 28*c^2*d*(e*f - d*g) - 2*c*e*(7*b*e*f - 7*b*d*g + 10*a*e*g))))/16)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/...
Time = 4.74 (sec) , antiderivative size = 1312, normalized size of antiderivative = 1.14, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.815, Rules used = {1231, 27, 1231, 27, 1269, 1094, 761, 1174, 1173, 25, 504, 312, 118, 353, 73, 756, 218, 221, 925, 27, 1537, 412}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/4} (5 b e g-14 c d g+14 c e f+10 c e g x)}{35 c e^2}-\frac {\int \frac {\left (5 d e g b^2+14 c d (e f-d g) b-4 a c e (7 e f-2 d g)+\left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x\right ) \sqrt [4]{c x^2+b x+a}}{4 (d+e x)}dx}{7 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/4} (5 b e g-14 c d g+14 c e f+10 c e g x)}{35 c e^2}-\frac {\int \frac {\left (5 d e g b^2+14 c d (e f-d g) b-4 a c e (7 e f-2 d g)+\left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x\right ) \sqrt [4]{c x^2+b x+a}}{d+e x}dx}{28 c e^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/4} (5 b e g-14 c d g+14 c e f+10 c e g x)}{35 c e^2}-\frac {\frac {\sqrt [4]{a+b x+c x^2} \left (2 c e x \left (-2 c e (10 a e g-7 b d g+7 b e f)+5 b^2 e^2 g+28 c^2 d (e f-d g)\right )+28 c^2 e (7 b d-6 a e) (e f-d g)-2 b c e^2 (10 a e g-7 b d g+7 b e f)+5 b^3 e^3 g-168 c^3 d^2 (e f-d g)\right )}{3 c e^2}-\frac {\int \frac {6 c e (b d-2 a e) \left (5 d e g b^2+14 c d (e f-d g) b-4 a c e (7 e f-2 d g)\right )-d \left (-e b^2+6 c d b-8 a c e\right ) \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right )+\left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) x}{4 (d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{3 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/4} (5 b e g-14 c d g+14 c e f+10 c e g x)}{35 c e^2}-\frac {\frac {\sqrt [4]{a+b x+c x^2} \left (2 c e x \left (-2 c e (10 a e g-7 b d g+7 b e f)+5 b^2 e^2 g+28 c^2 d (e f-d g)\right )+28 c^2 e (7 b d-6 a e) (e f-d g)-2 b c e^2 (10 a e g-7 b d g+7 b e f)+5 b^3 e^3 g-168 c^3 d^2 (e f-d g)\right )}{3 c e^2}-\frac {\int \frac {6 c e (b d-2 a e) \left (5 d e g b^2+14 c d (e f-d g) b-4 a c e (7 e f-2 d g)\right )-d \left (-e b^2+6 c d b-8 a c e\right ) \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right )+\left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) x}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/4} (5 b e g-14 c d g+14 c e f+10 c e g x)}{35 c e^2}-\frac {\frac {\sqrt [4]{a+b x+c x^2} \left (2 c e x \left (-2 c e (10 a e g-7 b d g+7 b e f)+5 b^2 e^2 g+28 c^2 d (e f-d g)\right )+28 c^2 e (7 b d-6 a e) (e f-d g)-2 b c e^2 (10 a e g-7 b d g+7 b e f)+5 b^3 e^3 g-168 c^3 d^2 (e f-d g)\right )}{3 c e^2}-\frac {\frac {\left (4 c^2 e^2 \left (20 a^2 e^2 g+56 a b e (e f-d g)-35 b^2 d (e f-d g)\right )-2 b^2 c e^3 (20 a e g-7 b d g+7 b e f)+56 c^3 d e (9 b d-8 a e) (e f-d g)+5 b^4 e^4 g-336 c^4 d^3 (e f-d g)\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/4}}dx}{e}+\frac {336 c^2 (e f-d g) \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 1094 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/4} (5 b e g-14 c d g+14 c e f+10 c e g x)}{35 c e^2}-\frac {\frac {\sqrt [4]{a+b x+c x^2} \left (2 c e x \left (-2 c e (10 a e g-7 b d g+7 b e f)+5 b^2 e^2 g+28 c^2 d (e f-d g)\right )+28 c^2 e (7 b d-6 a e) (e f-d g)-2 b c e^2 (10 a e g-7 b d g+7 b e f)+5 b^3 e^3 g-168 c^3 d^2 (e f-d g)\right )}{3 c e^2}-\frac {\frac {4 \sqrt {(b+2 c x)^2} \left (4 c^2 e^2 \left (20 a^2 e^2 g+56 a b e (e f-d g)-35 b^2 d (e f-d g)\right )-2 b^2 c e^3 (20 a e g-7 b d g+7 b e f)+56 c^3 d e (9 b d-8 a e) (e f-d g)+5 b^4 e^4 g-336 c^4 d^3 (e f-d g)\right ) \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{e (b+2 c x)}+\frac {336 c^2 (e f-d g) \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/4} (5 b e g-14 c d g+14 c e f+10 c e g x)}{35 c e^2}-\frac {\frac {\sqrt [4]{a+b x+c x^2} \left (2 c e x \left (-2 c e (10 a e g-7 b d g+7 b e f)+5 b^2 e^2 g+28 c^2 d (e f-d g)\right )+28 c^2 e (7 b d-6 a e) (e f-d g)-2 b c e^2 (10 a e g-7 b d g+7 b e f)+5 b^3 e^3 g-168 c^3 d^2 (e f-d g)\right )}{3 c e^2}-\frac {\frac {336 c^2 (e f-d g) \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right ) \left (4 c^2 e^2 \left (20 a^2 e^2 g+56 a b e (e f-d g)-35 b^2 d (e f-d g)\right )-2 b^2 c e^3 (20 a e g-7 b d g+7 b e f)+56 c^3 d e (9 b d-8 a e) (e f-d g)+5 b^4 e^4 g-336 c^4 d^3 (e f-d g)\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 1174 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/4} (5 b e g-14 c d g+14 c e f+10 c e g x)}{35 c e^2}-\frac {\frac {\sqrt [4]{a+b x+c x^2} \left (2 c e x \left (-2 c e (10 a e g-7 b d g+7 b e f)+5 b^2 e^2 g+28 c^2 d (e f-d g)\right )+28 c^2 e (7 b d-6 a e) (e f-d g)-2 b c e^2 (10 a e g-7 b d g+7 b e f)+5 b^3 e^3 g-168 c^3 d^2 (e f-d g)\right )}{3 c e^2}-\frac {\frac {336 c^2 \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (e f-d g) \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \left (-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}\right )^{3/4}}dx}{e \left (a+b x+c x^2\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right ) \left (4 c^2 e^2 \left (20 a^2 e^2 g+56 a b e (e f-d g)-35 b^2 d (e f-d g)\right )-2 b^2 c e^3 (20 a e g-7 b d g+7 b e f)+56 c^3 d e (9 b d-8 a e) (e f-d g)+5 b^4 e^4 g-336 c^4 d^3 (e f-d g)\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 1173 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/4} (5 b e g-14 c d g+14 c e f+10 c e g x)}{35 c e^2}-\frac {\frac {\sqrt [4]{a+b x+c x^2} \left (2 c e x \left (-2 c e (10 a e g-7 b d g+7 b e f)+5 b^2 e^2 g+28 c^2 d (e f-d g)\right )+28 c^2 e (7 b d-6 a e) (e f-d g)-2 b c e^2 (10 a e g-7 b d g+7 b e f)+5 b^3 e^3 g-168 c^3 d^2 (e f-d g)\right )}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (e f-d g) \left (a e^2-b d e+c d^2\right )^2 \int -\frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{e \left (a+b x+c x^2\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right ) \left (4 c^2 e^2 \left (20 a^2 e^2 g+56 a b e (e f-d g)-35 b^2 d (e f-d g)\right )-2 b^2 c e^3 (20 a e g-7 b d g+7 b e f)+56 c^3 d e (9 b d-8 a e) (e f-d g)+5 b^4 e^4 g-336 c^4 d^3 (e f-d g)\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/4} (5 b e g-14 c d g+14 c e f+10 c e g x)}{35 c e^2}-\frac {\frac {\sqrt [4]{a+b x+c x^2} \left (2 c e x \left (-2 c e (10 a e g-7 b d g+7 b e f)+5 b^2 e^2 g+28 c^2 d (e f-d g)\right )+28 c^2 e (7 b d-6 a e) (e f-d g)-2 b c e^2 (10 a e g-7 b d g+7 b e f)+5 b^3 e^3 g-168 c^3 d^2 (e f-d g)\right )}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right ) \left (4 c^2 e^2 \left (20 a^2 e^2 g+56 a b e (e f-d g)-35 b^2 d (e f-d g)\right )-2 b^2 c e^3 (20 a e g-7 b d g+7 b e f)+56 c^3 d e (9 b d-8 a e) (e f-d g)+5 b^4 e^4 g-336 c^4 d^3 (e f-d g)\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {672 \sqrt {2} c^2 \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (e f-d g) \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{e \left (a+b x+c x^2\right )^{3/4}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 504 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (-\frac {c (2 c d-b e) \int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{b^2-4 a c}-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 312 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )-\frac {c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{2 \left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 118 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}-\frac {1}{2} e \int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \int \frac {1}{\frac {c^2 e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}{b^2-4 a c}+\frac {4 c^3 \left (c d^2-b e d+a e^2\right )}{\left (b^2-4 a c\right )^2}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\int \frac {1}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}-\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (b^2-4 a c\right )^2}{4 c^{5/2} \sqrt {c d^2-b e d+a e^2}}+\frac {\int \frac {1}{\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4+2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (b^2-4 a c\right )^2}{4 c^{5/2} \sqrt {c d^2-b e d+a e^2}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\int \frac {1}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}-\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (b^2-4 a c\right )^2}{4 c^{5/2} \sqrt {c d^2-b e d+a e^2}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {\left (b^2-4 a c\right ) \int \frac {2 \sqrt {c}}{\left (2 \sqrt {c}-\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}\right ) \sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{8 c \left (c d^2-b e d+a e^2\right )}+\frac {\left (b^2-4 a c\right ) \int \frac {2 \sqrt {c}}{\left (\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}+2 \sqrt {c}\right ) \sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{8 c \left (c d^2-b e d+a e^2\right )}\right ) c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (2 \sqrt {c}-\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}\right ) \sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{4 \sqrt {c} \left (c d^2-b e d+a e^2\right )}+\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}+2 \sqrt {c}\right ) \sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{4 \sqrt {c} \left (c d^2-b e d+a e^2\right )}\right ) c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 1537 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4} \sqrt {\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4+1} \left (2 \sqrt {c}-\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{4 \sqrt {c} \left (c d^2-b e d+a e^2\right )}+\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4} \sqrt {\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4+1} \left (\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}+2 \sqrt {c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{4 \sqrt {c} \left (c d^2-b e d+a e^2\right )}\right ) c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {(14 c e f-14 c d g+5 b e g+10 c e g x) \left (c x^2+b x+a\right )^{5/4}}{35 c e^2}-\frac {\frac {\left (-168 d^2 (e f-d g) c^3+28 e (7 b d-6 a e) (e f-d g) c^2-2 b e^2 (7 b e f-7 b d g+10 a e g) c+2 e \left (28 d (e f-d g) c^2-2 e (7 b e f-7 b d g+10 a e g) c+5 b^2 e^2 g\right ) x c+5 b^3 e^3 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {672 \sqrt {2} c^2 (e f-d g) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac {\left (b^2-4 a c\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}},\arcsin \left (\frac {2 x c^2}{b^2-4 a c}+\frac {b c}{b^2-4 a c}\right ),-1\right )}{8 c \left (c d^2-b e d+a e^2\right )}-\frac {\left (b^2-4 a c\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}},\arcsin \left (\frac {2 x c^2}{b^2-4 a c}+\frac {b c}{b^2-4 a c}\right ),-1\right )}{8 c \left (c d^2-b e d+a e^2\right )}\right ) c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) \left (c d^2-b e d+a e^2\right )^2}{e \left (c x^2+b x+a\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (-336 d^3 (e f-d g) c^4+56 d e (9 b d-8 a e) (e f-d g) c^3+4 e^2 \left (-35 d (e f-d g) b^2+56 a e (e f-d g) b+20 a^2 e^2 g\right ) c^2-2 b^2 e^3 (7 b e f-7 b d g+20 a e g) c+5 b^4 e^4 g\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{28 c e^2}\) |
Input:
Int[((f + g*x)*(a + b*x + c*x^2)^(5/4))/(d + e*x),x]
Output:
((14*c*e*f - 14*c*d*g + 5*b*e*g + 10*c*e*g*x)*(a + b*x + c*x^2)^(5/4))/(35 *c*e^2) - (((5*b^3*e^3*g - 168*c^3*d^2*(e*f - d*g) + 28*c^2*e*(7*b*d - 6*a *e)*(e*f - d*g) - 2*b*c*e^2*(7*b*e*f - 7*b*d*g + 10*a*e*g) + 2*c*e*(5*b^2* e^2*g + 28*c^2*d*(e*f - d*g) - 2*c*e*(7*b*e*f - 7*b*d*g + 10*a*e*g))*x)*(a + b*x + c*x^2)^(1/4))/(3*c*e^2) - ((Sqrt[2]*(b^2 - 4*a*c)^(1/4)*(5*b^4*e^ 4*g - 336*c^4*d^3*(e*f - d*g) + 56*c^3*d*e*(9*b*d - 8*a*e)*(e*f - d*g) - 2 *b^2*c*e^3*(7*b*e*f - 7*b*d*g + 20*a*e*g) + 4*c^2*e^2*(20*a^2*e^2*g - 35*b ^2*d*(e*f - d*g) + 56*a*b*e*(e*f - d*g)))*Sqrt[(b + 2*c*x)^2]*(1 + (2*Sqrt [c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*Sqrt[(b^2 - 4*a*c + 4*c*(a + b*x + c*x^2))/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[ b^2 - 4*a*c])^2)]*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1 /4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(c^(1/4)*e*(b + 2*c*x)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)]) + (672*Sqrt[2]*c^2*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*((2*c^2*e*(((b^2 - 4*a*c)^2*ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - ((b^2 - 4*a*c)*(-((b*c) /(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c))^2)/c^2)^(1/4))/(Sqrt[2]*c^(1/4) *(c*d^2 - b*d*e + a*e^2)^(1/4))])/(4*Sqrt[2]*c^(11/4)*(-b^2 + 4*a*c)^(1/4) *Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(3/4)) + ((b^2 - 4*a*c)^2*ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - ((b^2 - 4*a*c)*(-((b*c)/(b^2 - 4*a*c)) - (2*c^ 2*x)/(b^2 - 4*a*c))^2)/c^2)^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 3/4)), x_] :> Simp[-4 Subst[Int[1/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) + d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e, f}, x] & & GtQ[-f/(d*e - c*f), 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim p[Sqrt[(-b)*(x^2/a)]/(2*x) Subst[Int[1/(Sqrt[(-b)*(x/a)]*(a + b*x)^(3/4)* (c + d*x)), x], x, x^2], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[((a_) + (b_.)*(x_)^2)^(p_)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[c I nt[(a + b*x^2)^p/(c^2 - d^2*x^2), x], x] - Simp[d Int[x*((a + b*x^2)^p/(c ^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, p}, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 1/(2*c) Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 *c) Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[4*(Sqrt[(b + 2*c*x)^2]/(b + 2*c*x)) Subst[Int[x^(4*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 *c*x^4], x], x, (a + b*x + c*x^2)^(1/4)], x] /; FreeQ[{a, b, c}, x] && Inte gerQ[4*p]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/((d_.) + (e_.)*(x_)), x_Symbol ] :> Simp[1/(-4*(c/(b^2 - 4*a*c)))^p Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c) , x]^p/Simp[2*c*d - b*e + e*x, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, d, e, p}, x] && GtQ[4*a - b^2/c, 0] && IntegerQ[4*p]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/((d_.) + (e_.)*(x_)), x_Symbol ] :> Simp[(a + b*x + c*x^2)^p/((-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c)))^p Int[((-a)*(c/(b^2 - 4*a*c)) - b*c*(x/(b^2 - 4*a*c)) - c^2*(x^2/(b^2 - 4*a*c )))^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && !GtQ[4*a - b^2/ c, 0] && IntegerQ[4*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[(-a)*c, 2]}, Simp[Sqrt[-c] Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqr t[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] & & GtQ[a, 0] && LtQ[c, 0]
\[\int \frac {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}}{e x +d}d x\]
Input:
int((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d),x)
Output:
int((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d),x)
Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\int \frac {\left (f + g x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{4}}}{d + e x}\, dx \] Input:
integrate((g*x+f)*(c*x**2+b*x+a)**(5/4)/(e*x+d),x)
Output:
Integral((f + g*x)*(a + b*x + c*x**2)**(5/4)/(d + e*x), x)
\[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (g x + f\right )}}{e x + d} \,d x } \] Input:
integrate((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x + a)^(5/4)*(g*x + f)/(e*x + d), x)
\[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (g x + f\right )}}{e x + d} \,d x } \] Input:
integrate((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d),x, algorithm="giac")
Output:
integrate((c*x^2 + b*x + a)^(5/4)*(g*x + f)/(e*x + d), x)
Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/4}}{d+e\,x} \,d x \] Input:
int(((f + g*x)*(a + b*x + c*x^2)^(5/4))/(d + e*x),x)
Output:
int(((f + g*x)*(a + b*x + c*x^2)^(5/4))/(d + e*x), x)
\[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\int \frac {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}}{e x +d}d x \] Input:
int((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d),x)
Output:
int((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d),x)