\(\int \frac {(f+g x) (a+b x+c x^2)^{5/4}}{(d+e x)^2} \, dx\) [1111]

Optimal result

Integrand size = 27, antiderivative size = 1220 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx =\text {Too large to display} \] Output:

1/6*(b^2*e^2*g-6*c^2*d*(-7*d*g+5*e*f)+2*c*e*(6*a*e*g-17*b*d*g+10*b*e*f)+2* 
c*e*(b*e*g-7*c*d*g+5*c*e*f)*x)*(c*x^2+b*x+a)^(1/4)/c/e^4-1/5*(c*d*(-7*d*g+ 
5*e*f)-e*(2*a*e*g-7*b*d*g+5*b*e*f)-5*c*e*(-d*g+e*f)*x)*(c*x^2+b*x+a)^(5/4) 
/e^2/(a*e^2-b*d*e+c*d^2)-(-d*g+e*f)*(c*x^2+b*x+a)^(9/4)/(a*e^2-b*d*e+c*d^2 
)/(e*x+d)+1/4*(4*a*c-b^2)^(3/4)*(a*e^2-b*d*e+c*d^2)^(1/4)*(2*c*d*(-7*d*g+5 
*e*f)-e*(4*a*e*g-9*b*d*g+5*b*e*f))*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*a 
rctan(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 
)+1/4*(4*a*c-b^2)^(3/4)*(a*e^2-b*d*e+c*d^2)^(1/4)*(2*c*d*(-7*d*g+5*e*f)-e* 
(4*a*e*g-9*b*d*g+5*b*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)-1/12* 
(4*a*c-b^2)^(1/2)*(b^3*e^3*g-12*c^3*d^2*(-7*d*g+5*e*f)-4*c^2*e*(a*e*(-13*d 
*g+5*e*f)-3*b*d*(-8*d*g+5*e*f))-2*b*c*e^2*(8*a*e*g+5*b*(-2*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^2/e^5/(c*x^2+b*x+a)^(3/4)+1/8*(-4*a*c+b 
^2)*(-b*e+2*c*d)*(2*c*d*(-7*d*g+5*e*f)-e*(4*a*e*g-9*b*d*g+5*b*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)*EllipticP 
i((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/...
 

Mathematica [A] (warning: unable to verify)

Time = 17.29 (sec) , antiderivative size = 2107, normalized size of antiderivative = 1.73 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\text {Result too large to show} \] Input:

Integrate[((f + g*x)*(a + b*x + c*x^2)^(5/4))/(d + e*x)^2,x]
 

Output:

((e*f - d*g)*(a + b*x + c*x^2)*(a + x*(b + c*x))^(5/4))/((-(c*d^2) + b*d*e 
 - a*e^2)*(d + e*x)) + ((a + x*(b + c*x))^(5/4)*((4*((7*c*((7*c*d)/2 - (5* 
b*e)/4)*(e*f - d*g))/2 + (7*c*e*(-4*c*d*f - 5*b*e*f + 9*b*d*g - 4*a*e*g))/ 
8 - (35*c^2*e*(e*f - d*g)*x)/4)*(a + b*x + c*x^2)^(5/4))/(35*c*e^2) - ((4* 
((21*c^2*e*(c*d^2 - e*(b*d - a*e))*(5*b*e*f - 7*b*d*g + 4*a*e*g))/8 - (7*c 
*((3*c*d)/2 - (b*e)/4)*(c*d^2 - e*(b*d - a*e))*(5*c*e*f - 7*c*d*g + b*e*g) 
)/2 + (7*c^2*e*(c*d^2 - e*(b*d - a*e))*(5*c*e*f - 7*c*d*g + b*e*g)*x)/4)*( 
a + b*x + c*x^2)^(1/4))/(3*c*e^2) - ((-7*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)*(c*d^ 
2 - e*(b*d - a*e))*(3*c*e*(2*c*d - b*e)*(5*b*e*f - 7*b*d*g + 4*a*e*g) - (1 
2*c^2*d^2 - 6*b*c*d*e - b^2*e^2 + 4*a*c*e^2)*(5*c*e*f - 7*c*d*g + b*e*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])/(2* 
Sqrt[2]*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)*((7*c*e*(c*d^2 - 
 e*(b*d - a*e))*(3*c*e*(b*d - 2*a*e)*(5*b*e*f - 7*b*d*g + 4*a*e*g) - d*(6* 
b*c*d - b^2*e - 8*a*c*e)*(5*c*e*f - 7*c*d*g + b*e*g)))/8 - (7*c*d*(c*d^2 - 
 e*(b*d - a*e))*(3*c*e*(2*c*d - b*e)*(5*b*e*f - 7*b*d*g + 4*a*e*g) - (12*c 
^2*d^2 - 6*b*c*d*e - b^2*e^2 + 4*a*c*e^2)*(5*c*e*f - 7*c*d*g + b*e*g)))/8) 
*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*(-(e*((Sqrt[2]*c^(1/4)*...
 

Rubi [A] (warning: unable to verify)

Time = 3.79 (sec) , antiderivative size = 1226, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.815, Rules used = {1230, 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.

\(\displaystyle \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx\)

\(\Big \downarrow \) 1230

\(\displaystyle -\frac {\int -\frac {(5 b e f-7 b d g+4 a e g+2 (5 c e f-7 c d g+b e g) x) \sqrt [4]{c x^2+b x+a}}{2 (d+e x)}dx}{2 e^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (-7 d g+5 e f-2 e g x)}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(5 b e f-7 b d g+4 a e g+2 (5 c e f-7 c d g+b e g) x) \sqrt [4]{c x^2+b x+a}}{d+e x}dx}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (-7 d g+5 e f-2 e g x)}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\frac {2 \sqrt [4]{a+b x+c x^2} \left (2 c e (6 a e g-17 b d g+10 b e f)+b^2 e^2 g+2 c e x (b e g-7 c d g+5 c e f)-6 c^2 d (5 e f-7 d g)\right )}{3 c e^2}-\frac {\int \frac {3 c e (b d-2 a e) (5 b e f-7 b d g+4 a e g)-d \left (-e b^2+6 c d b-8 a c e\right ) (5 c e f-7 c d g+b e g)+\left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e g)\right ) x}{2 (d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{3 c e^2}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (-7 d g+5 e f-2 e g x)}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \sqrt [4]{a+b x+c x^2} \left (2 c e (6 a e g-17 b d g+10 b e f)+b^2 e^2 g+2 c e x (b e g-7 c d g+5 c e f)-6 c^2 d (5 e f-7 d g)\right )}{3 c e^2}-\frac {\int \frac {3 c e (b d-2 a e) (5 b e f-7 b d g+4 a e g)-d \left (-e b^2+6 c d b-8 a c e\right ) (5 c e f-7 c d g+b e g)+\left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e g)\right ) x}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{6 c e^2}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (-7 d g+5 e f-2 e g x)}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {2 \sqrt [4]{a+b x+c x^2} \left (2 c e (6 a e g-17 b d g+10 b e f)+b^2 e^2 g+2 c e x (b e g-7 c d g+5 c e f)-6 c^2 d (5 e f-7 d g)\right )}{3 c e^2}-\frac {\frac {\left (3 c e (2 c d-b e) (4 a e g-7 b d g+5 b e f)-\left (4 a c e^2-b^2 e^2-6 b c d e+12 c^2 d^2\right ) (b e g-7 c d g+5 c e f)\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/4}}dx}{e}+\frac {6 c \left (a e^2-b d e+c d^2\right ) (2 c d (5 e f-7 d g)-e (4 a e g-9 b d g+5 b e f)) \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}}{6 c e^2}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (-7 d g+5 e f-2 e g x)}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {\frac {2 \sqrt [4]{a+b x+c x^2} \left (2 c e (6 a e g-17 b d g+10 b e f)+b^2 e^2 g+2 c e x (b e g-7 c d g+5 c e f)-6 c^2 d (5 e f-7 d g)\right )}{3 c e^2}-\frac {\frac {4 \sqrt {(b+2 c x)^2} \left (3 c e (2 c d-b e) (4 a e g-7 b d g+5 b e f)-\left (4 a c e^2-b^2 e^2-6 b c d e+12 c^2 d^2\right ) (b e g-7 c d g+5 c e f)\right ) \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{e (b+2 c x)}+\frac {6 c \left (a e^2-b d e+c d^2\right ) (2 c d (5 e f-7 d g)-e (4 a e g-9 b d g+5 b e f)) \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}}{6 c e^2}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (-7 d g+5 e f-2 e g x)}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {2 \sqrt [4]{a+b x+c x^2} \left (2 c e (6 a e g-17 b d g+10 b e f)+b^2 e^2 g+2 c e x (b e g-7 c d g+5 c e f)-6 c^2 d (5 e f-7 d g)\right )}{3 c e^2}-\frac {\frac {6 c \left (a e^2-b d e+c d^2\right ) (2 c d (5 e f-7 d g)-e (4 a e g-9 b d g+5 b e f)) \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right ) \left (3 c e (2 c d-b e) (4 a e g-7 b d g+5 b e f)-\left (4 a c e^2-b^2 e^2-6 b c d e+12 c^2 d^2\right ) (b e g-7 c d g+5 c e f)\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{6 c e^2}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (-7 d g+5 e f-2 e g x)}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 1174

\(\displaystyle \frac {\frac {2 \sqrt [4]{a+b x+c x^2} \left (2 c e (6 a e g-17 b d g+10 b e f)+b^2 e^2 g+2 c e x (b e g-7 c d g+5 c e f)-6 c^2 d (5 e f-7 d g)\right )}{3 c e^2}-\frac {\frac {6 c \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (a e^2-b d e+c d^2\right ) (2 c d (5 e f-7 d g)-e (4 a e g-9 b d g+5 b e f)) \int \frac {1}{(d+e x) \left (-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}\right )^{3/4}}dx}{e \left (a+b x+c x^2\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right ) \left (3 c e (2 c d-b e) (4 a e g-7 b d g+5 b e f)-\left (4 a c e^2-b^2 e^2-6 b c d e+12 c^2 d^2\right ) (b e g-7 c d g+5 c e f)\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{6 c e^2}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (-7 d g+5 e f-2 e g x)}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 1173

\(\displaystyle \frac {\frac {2 \sqrt [4]{a+b x+c x^2} \left (2 c e (6 a e g-17 b d g+10 b e f)+b^2 e^2 g+2 c e x (b e g-7 c d g+5 c e f)-6 c^2 d (5 e f-7 d g)\right )}{3 c e^2}-\frac {\frac {12 \sqrt {2} c \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (a e^2-b d e+c d^2\right ) (2 c d (5 e f-7 d g)-e (4 a e g-9 b d g+5 b e f)) \int -\frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{e \left (a+b x+c x^2\right )^{3/4}}+\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right ) \left (3 c e (2 c d-b e) (4 a e g-7 b d g+5 b e f)-\left (4 a c e^2-b^2 e^2-6 b c d e+12 c^2 d^2\right ) (b e g-7 c d g+5 c e f)\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{6 c e^2}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (-7 d g+5 e f-2 e g x)}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \sqrt [4]{a+b x+c x^2} \left (2 c e (6 a e g-17 b d g+10 b e f)+b^2 e^2 g+2 c e x (b e g-7 c d g+5 c e f)-6 c^2 d (5 e f-7 d g)\right )}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right ) \left (3 c e (2 c d-b e) (4 a e g-7 b d g+5 b e f)-\left (4 a c e^2-b^2 e^2-6 b c d e+12 c^2 d^2\right ) (b e g-7 c d g+5 c e f)\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {12 \sqrt {2} c \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (a e^2-b d e+c d^2\right ) (2 c d (5 e f-7 d g)-e (4 a e g-9 b d g+5 b e f)) \int \frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{e \left (a+b x+c x^2\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{5/4} (-7 d g+5 e f-2 e g x)}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 504

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e g)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e g)) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (-\frac {c (2 c d-b e) \int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{b^2-4 a c}-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 312

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e 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 )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e 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 )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 118

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e 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 )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e 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 )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e 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 )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e 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 )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e 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 )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e 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 )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e 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 )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e 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 )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e 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 )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e 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 )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e 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 )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e 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 )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 925

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e 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 )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e 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 )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e 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 )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e 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 )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 1537

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e 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 )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e 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 )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {\frac {2 \left (-6 d (5 e f-7 d g) c^2+2 e (10 b e f-17 b d g+6 a e g) c+2 e (5 c e f-7 c d g+b e g) x c+b^2 e^2 g\right ) \sqrt [4]{c x^2+b x+a}}{3 c e^2}-\frac {\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (3 c e (2 c d-b e) (5 b e f-7 b d g+4 a e g)-\left (12 c^2 d^2-6 b c e d-b^2 e^2+4 a c e^2\right ) (5 c e f-7 c d g+b e 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 )}}+\frac {12 \sqrt {2} c \left (c d^2-b e d+a e^2\right ) (2 c d (5 e f-7 d g)-e (5 b e f-9 b d g+4 a e 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 )}{e \left (c x^2+b x+a\right )^{3/4}}}{6 c e^2}}{4 e^2}-\frac {(5 e f-7 d g-2 e g x) \left (c x^2+b x+a\right )^{5/4}}{5 e^2 (d+e x)}\)

Input:

Int[((f + g*x)*(a + b*x + c*x^2)^(5/4))/(d + e*x)^2,x]
 

Output:

-1/5*((5*e*f - 7*d*g - 2*e*g*x)*(a + b*x + c*x^2)^(5/4))/(e^2*(d + e*x)) + 
 ((2*(b^2*e^2*g - 6*c^2*d*(5*e*f - 7*d*g) + 2*c*e*(10*b*e*f - 17*b*d*g + 6 
*a*e*g) + 2*c*e*(5*c*e*f - 7*c*d*g + b*e*g)*x)*(a + b*x + c*x^2)^(1/4))/(3 
*c*e^2) - ((Sqrt[2]*(b^2 - 4*a*c)^(1/4)*(3*c*e*(2*c*d - b*e)*(5*b*e*f - 7* 
b*d*g + 4*a*e*g) - (12*c^2*d^2 - 6*b*c*d*e - b^2*e^2 + 4*a*c*e^2)*(5*c*e*f 
 - 7*c*d*g + b*e*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)]*Ell 
ipticF[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 
)]) + (12*Sqrt[2]*c*(c*d^2 - b*d*e + a*e^2)*(2*c*d*(5*e*f - 7*d*g) - e*(5* 
b*e*f - 9*b*d*g + 4*a*e*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*e^2)^(1/4))])/(4*Sqrt[2]*c^(11/4)*(-b^2 + 4*a*c)^(1/4)*Sq 
rt[e]*(c*d^2 - b*d*e + a*e^2)^(3/4))))/(b^2 - 4*a*c) + (2*c*(2*c*d - b*...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

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]
 

rule 118

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]
 

rule 218

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]
 

rule 221

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]
 

rule 312

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]
 

rule 353

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]
 

rule 412

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])
 

rule 504

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]
 

rule 756

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]
 

rule 761

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]
 

rule 925

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]
 

rule 1094

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]
 

rule 1173

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]
 

rule 1174

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]
 

rule 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1231

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])
 

rule 1269

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]
 

rule 1537

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]
 

Maple [F]

\[\int \frac {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}}{\left (e x +d \right )^{2}}d x\]

Input:

int((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x)
 

Output:

int((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\int \frac {\left (f + g x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{4}}}{\left (d + e x\right )^{2}}\, dx \] Input:

integrate((g*x+f)*(c*x**2+b*x+a)**(5/4)/(e*x+d)**2,x)
 

Output:

Integral((f + g*x)*(a + b*x + c*x**2)**(5/4)/(d + e*x)**2, x)
 

Maxima [F]

\[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(5/4)*(g*x + f)/(e*x + d)^2, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{1,[0,1,1,2,0,0,0]%%%}+%%%{-1,[0,1,0,1,1,1,0]%%%}+%%%{1,[0, 
1,0,0,2,0
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/4}}{{\left (d+e\,x\right )}^2} \,d x \] Input:

int(((f + g*x)*(a + b*x + c*x^2)^(5/4))/(d + e*x)^2,x)
 

Output:

int(((f + g*x)*(a + b*x + c*x^2)^(5/4))/(d + e*x)^2, x)
 

Reduce [F]

\[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\text {too large to display} \] Input:

int((g*x+f)*(c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x)
 

Output:

( - 480*(a + b*x + c*x**2)**(1/4)*a**2*c*e**3*g + 2144*(a + b*x + c*x**2)* 
*(1/4)*a*b*c*d*e**2*g - 960*(a + b*x + c*x**2)**(1/4)*a*b*c*e**3*f + 864*( 
a + b*x + c*x**2)**(1/4)*a*b*c*e**3*g*x - 896*(a + b*x + c*x**2)**(1/4)*a* 
c**2*d**2*e*g + 640*(a + b*x + c*x**2)**(1/4)*a*c**2*d*e**2*f - 576*(a + b 
*x + c*x**2)**(1/4)*a*c**2*d*e**2*g*x + 100*(a + b*x + c*x**2)**(1/4)*b**3 
*d*e**2*g + 60*(a + b*x + c*x**2)**(1/4)*b**3*e**3*g*x - 2360*(a + b*x + c 
*x**2)**(1/4)*b**2*c*d**2*e*g + 1400*(a + b*x + c*x**2)**(1/4)*b**2*c*d*e* 
*2*f - 1456*(a + b*x + c*x**2)**(1/4)*b**2*c*d*e**2*g*x + 840*(a + b*x + c 
*x**2)**(1/4)*b**2*c*e**3*f*x + 264*(a + b*x + c*x**2)**(1/4)*b**2*c*e**3* 
g*x**2 + 2800*(a + b*x + c*x**2)**(1/4)*b*c**2*d**3*g - 2000*(a + b*x + c* 
x**2)**(1/4)*b*c**2*d**2*e*f + 2624*(a + b*x + c*x**2)**(1/4)*b*c**2*d**2* 
e*g*x - 1760*(a + b*x + c*x**2)**(1/4)*b*c**2*d*e**2*f*x - 512*(a + b*x + 
c*x**2)**(1/4)*b*c**2*d*e**2*g*x**2 + 240*(a + b*x + c*x**2)**(1/4)*b*c**2 
*e**3*f*x**2 + 144*(a + b*x + c*x**2)**(1/4)*b*c**2*e**3*g*x**3 - 1120*(a 
+ b*x + c*x**2)**(1/4)*c**3*d**3*g*x + 800*(a + b*x + c*x**2)**(1/4)*c**3* 
d**2*e*f*x + 224*(a + b*x + c*x**2)**(1/4)*c**3*d**2*e*g*x**2 - 160*(a + b 
*x + c*x**2)**(1/4)*c**3*d*e**2*f*x**2 - 96*(a + b*x + c*x**2)**(1/4)*c**3 
*d*e**2*g*x**3 - 1440*int((a + b*x + c*x**2)**(1/4)/(3*a*b*d**2*e + 6*a*b* 
d*e**2*x + 3*a*b*e**3*x**2 - 2*a*c*d**3 - 4*a*c*d**2*e*x - 2*a*c*d*e**2*x* 
*2 + 3*b**2*d**2*e*x + 6*b**2*d*e**2*x**2 + 3*b**2*e**3*x**3 - 2*b*c*d*...