Integrand size = 27, antiderivative size = 910 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^5} \, dx=\frac {35 \left (64 c^3 d^3 (e f-2 d g)-b^2 e^3 (b e f-8 b d g+6 a e g)-4 c e^2 \left (2 a^2 e^2 g+a b e (3 e f-16 d g)-2 b^2 d (3 e f-10 d g)\right )-16 c^2 d e (b d (5 e f-12 d g)-2 a e (e f-3 d g))+e \left (b^3 e^3 g+32 c^3 d^2 (e f-2 d g)+6 b c e^2 (b e f-4 b d g+2 a e g)-8 c^2 e (2 b d (2 e f-5 d g)-a e (e f-4 d g))\right ) x\right ) \sqrt {a+b x+c x^2}}{64 e^8 (d+e x)}-\frac {35 \left (16 c^2 d^2 (e f-2 d g)+b e^2 (b e f-6 b d g+4 a e g)-4 c e (b d (3 e f-8 d g)-a e (e f-4 d g))-e \left (b^2 e^2 g-8 c^2 d (e f-2 d g)+4 c e (b e f-3 b d g+a e g)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 e^6 (d+e x)^2}+\frac {7 (4 c d (e f-2 d g)-e (b e f-4 b d g+2 a e g)+e (2 c e f-4 c d g+b e g) x) \left (a+b x+c x^2\right )^{5/2}}{24 e^4 (d+e x)^3}-\frac {(e f-2 d g-e g x) \left (a+b x+c x^2\right )^{7/2}}{4 e^2 (d+e x)^4}+\frac {35 \left (b^4 e^4 g-128 c^4 d^3 (e f-2 d g)+8 b^2 c e^3 (b e f-5 b d g+3 a e g)+16 c^2 e^2 \left (a^2 e^2 g+2 a b e (e f-5 d g)-5 b^2 d (e f-3 d g)\right )+64 c^3 d e (b d (3 e f-7 d g)-a e (e f-3 d g))\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 \sqrt {c} e^9}+\frac {35 \left (128 c^4 d^4 (e f-2 d g)+b^3 e^4 (b e f-9 b d g+8 a e g)+8 b c e^3 \left (4 a^2 e^2 g+a b e (3 e f-17 d g)-b^2 d (4 e f-15 d g)\right )+16 c^2 e^2 \left (b^2 d^2 (10 e f-27 d g)-2 a b d e (4 e f-13 d g)+a^2 e^2 (e f-5 d g)\right )-64 c^3 d^2 e (b d (4 e f-9 d g)-a e (2 e f-5 d g))\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{128 e^9 \sqrt {c d^2-b d e+a e^2}} \] Output:
35/64*(64*c^3*d^3*(-2*d*g+e*f)-b^2*e^3*(6*a*e*g-8*b*d*g+b*e*f)-4*c*e^2*(2* a^2*e^2*g+a*b*e*(-16*d*g+3*e*f)-2*b^2*d*(-10*d*g+3*e*f))-16*c^2*d*e*(b*d*( -12*d*g+5*e*f)-2*a*e*(-3*d*g+e*f))+e*(b^3*e^3*g+32*c^3*d^2*(-2*d*g+e*f)+6* b*c*e^2*(2*a*e*g-4*b*d*g+b*e*f)-8*c^2*e*(2*b*d*(-5*d*g+2*e*f)-a*e*(-4*d*g+ e*f)))*x)*(c*x^2+b*x+a)^(1/2)/e^8/(e*x+d)-35/96*(16*c^2*d^2*(-2*d*g+e*f)+b *e^2*(4*a*e*g-6*b*d*g+b*e*f)-4*c*e*(b*d*(-8*d*g+3*e*f)-a*e*(-4*d*g+e*f))-e *(b^2*e^2*g-8*c^2*d*(-2*d*g+e*f)+4*c*e*(a*e*g-3*b*d*g+b*e*f))*x)*(c*x^2+b* x+a)^(3/2)/e^6/(e*x+d)^2+7/24*(4*c*d*(-2*d*g+e*f)-e*(2*a*e*g-4*b*d*g+b*e*f )+e*(b*e*g-4*c*d*g+2*c*e*f)*x)*(c*x^2+b*x+a)^(5/2)/e^4/(e*x+d)^3-1/4*(-e*g *x-2*d*g+e*f)*(c*x^2+b*x+a)^(7/2)/e^2/(e*x+d)^4+35/128*(b^4*e^4*g-128*c^4* d^3*(-2*d*g+e*f)+8*b^2*c*e^3*(3*a*e*g-5*b*d*g+b*e*f)+16*c^2*e^2*(a^2*e^2*g +2*a*b*e*(-5*d*g+e*f)-5*b^2*d*(-3*d*g+e*f))+64*c^3*d*e*(b*d*(-7*d*g+3*e*f) -a*e*(-3*d*g+e*f)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^( 1/2)/e^9+35/128*(128*c^4*d^4*(-2*d*g+e*f)+b^3*e^4*(8*a*e*g-9*b*d*g+b*e*f)+ 8*b*c*e^3*(4*a^2*e^2*g+a*b*e*(-17*d*g+3*e*f)-b^2*d*(-15*d*g+4*e*f))+16*c^2 *e^2*(b^2*d^2*(-27*d*g+10*e*f)-2*a*b*d*e*(-13*d*g+4*e*f)+a^2*e^2*(-5*d*g+e *f))-64*c^3*d^2*e*(b*d*(-9*d*g+4*e*f)-a*e*(-5*d*g+2*e*f)))*arctanh(1/2*(b* d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/e^9 /(a*e^2-b*d*e+c*d^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(22765\) vs. \(2(910)=1820\).
Time = 17.59 (sec) , antiderivative size = 22765, normalized size of antiderivative = 25.02 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^5} \, dx=\text {Result too large to show} \] Input:
Integrate[((f + g*x)*(a + b*x + c*x^2)^(7/2))/(d + e*x)^5,x]
Output:
Result too large to show
Time = 3.60 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {1230, 27, 1230, 27, 1230, 27, 25, 1230, 25, 1269, 1092, 219, 1154, 219}
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.
| \(\displaystyle \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^5} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {7 \int -\frac {4 (b e f-2 b d g+2 a e g+(2 c e f-4 c d g+b e g) x) \left (c x^2+b x+a\right )^{5/2}}{(d+e x)^4}dx}{32 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-2 d g+e f-e g x)}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 \int \frac {(b e f-2 b d g+2 a e g+(2 c e f-4 c d g+b e g) x) \left (c x^2+b x+a\right )^{5/2}}{(d+e x)^4}dx}{8 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-2 d g+e f-e g x)}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {7 \left (\frac {\left (a+b x+c x^2\right )^{5/2} (-e (2 a e g-4 b d g+b e f)+e x (b e g-4 c d g+2 c e f)+4 c d (e f-2 d g))}{3 e^2 (d+e x)^3}-\frac {5 \int -\frac {3 \left (b e (b e f-2 b d g+2 a e g)-2 (b d-a e) (2 c e f-4 c d g+b e g)+\left (-8 d (e f-2 d g) c^2+4 e (b e f-3 b d g+a e g) c+b^2 e^2 g\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^3}dx}{18 e^2}\right )}{8 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-2 d g+e f-e g x)}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 \left (\frac {5 \int \frac {\left (b e (b e f-2 b d g+2 a e g)-2 (b d-a e) (2 c e f-4 c d g+b e g)+\left (-8 d (e f-2 d g) c^2+4 e (b e f-3 b d g+a e g) c+b^2 e^2 g\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^3}dx}{6 e^2}+\frac {\left (a+b x+c x^2\right )^{5/2} (-e (2 a e g-4 b d g+b e f)+e x (b e g-4 c d g+2 c e f)+4 c d (e f-2 d g))}{3 e^2 (d+e x)^3}\right )}{8 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-2 d g+e f-e g x)}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (-\frac {3 \int \frac {2 \left (\frac {1}{2} (4 b d-4 a e) \left (-8 d (e f-2 d g) c^2+4 e (b e f-3 b d g+a e g) c+b^2 e^2 g\right )-b e (b e (b e f-2 b d g+2 a e g)-2 (b d-a e) (2 c e f-4 c d g+b e g))-\left (32 d^2 (e f-2 d g) c^3-8 e (2 b d (2 e f-5 d g)-a e (e f-4 d g)) c^2+6 b e^2 (b e f-4 b d g+2 a e g) c+b^3 e^3 g\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{8 e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x \left (4 c e (a e g-3 b d g+b e f)+b^2 e^2 g-8 c^2 d (e f-2 d g)\right )-4 c e (b d (3 e f-8 d g)-a e (e f-4 d g))+b e^2 (4 a e g-6 b d g+b e f)+16 c^2 d^2 (e f-2 d g)\right )}{2 e^2 (d+e x)^2}\right )}{6 e^2}+\frac {\left (a+b x+c x^2\right )^{5/2} (-e (2 a e g-4 b d g+b e f)+e x (b e g-4 c d g+2 c e f)+4 c d (e f-2 d g))}{3 e^2 (d+e x)^3}\right )}{8 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-2 d g+e f-e g x)}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (-\frac {3 \int -\frac {\left (e^2 (e f-6 d g) b^3+2 \left (3 a e^3 g-2 c d e (3 e f-8 d g)\right ) b^2+4 c \left (4 c (e f-2 d g) d^2+a e^2 (3 e f-10 d g)\right ) b+8 a c e \left (a e^2 g-2 c d (e f-2 d g)\right )+\left (32 d^2 (e f-2 d g) c^3-8 e (2 b d (2 e f-5 d g)-a e (e f-4 d g)) c^2+6 b e^2 (b e f-4 b d g+2 a e g) c+b^3 e^3 g\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x \left (4 c e (a e g-3 b d g+b e f)+b^2 e^2 g-8 c^2 d (e f-2 d g)\right )-4 c e (b d (3 e f-8 d g)-a e (e f-4 d g))+b e^2 (4 a e g-6 b d g+b e f)+16 c^2 d^2 (e f-2 d g)\right )}{2 e^2 (d+e x)^2}\right )}{6 e^2}+\frac {\left (a+b x+c x^2\right )^{5/2} (-e (2 a e g-4 b d g+b e f)+e x (b e g-4 c d g+2 c e f)+4 c d (e f-2 d g))}{3 e^2 (d+e x)^3}\right )}{8 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-2 d g+e f-e g x)}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \int \frac {\left (e^2 (e f-6 d g) b^3+2 \left (3 a e^3 g-2 c d e (3 e f-8 d g)\right ) b^2+4 c \left (4 c (e f-2 d g) d^2+a e^2 (3 e f-10 d g)\right ) b+8 a c e \left (a e^2 g-2 c d (e f-2 d g)\right )+\left (32 d^2 (e f-2 d g) c^3-8 e (2 b d (2 e f-5 d g)-a e (e f-4 d g)) c^2+6 b e^2 (b e f-4 b d g+2 a e g) c+b^3 e^3 g\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x \left (4 c e (a e g-3 b d g+b e f)+b^2 e^2 g-8 c^2 d (e f-2 d g)\right )-4 c e (b d (3 e f-8 d g)-a e (e f-4 d g))+b e^2 (4 a e g-6 b d g+b e f)+16 c^2 d^2 (e f-2 d g)\right )}{2 e^2 (d+e x)^2}\right )}{6 e^2}+\frac {\left (a+b x+c x^2\right )^{5/2} (-e (2 a e g-4 b d g+b e f)+e x (b e g-4 c d g+2 c e f)+4 c d (e f-2 d g))}{3 e^2 (d+e x)^3}\right )}{8 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-2 d g+e f-e g x)}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {7 \left (\frac {(4 c d (e f-2 d g)-e (b e f-4 b d g+2 a e g)+e (2 c e f-4 c d g+b e g) x) \left (c x^2+b x+a\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac {5 \left (\frac {3 \left (\frac {\left (64 c^3 (e f-2 d g) d^3-16 c^2 e (b d (5 e f-12 d g)-2 a e (e f-3 d g)) d-b^2 e^3 (b e f-8 b d g+6 a e g)-4 c e^2 \left (-2 d (3 e f-10 d g) b^2+a e (3 e f-16 d g) b+2 a^2 e^2 g\right )+e \left (32 d^2 (e f-2 d g) c^3-8 e (2 b d (2 e f-5 d g)-a e (e f-4 d g)) c^2+6 b e^2 (b e f-4 b d g+2 a e g) c+b^3 e^3 g\right ) x\right ) \sqrt {c x^2+b x+a}}{e^2 (d+e x)}-\frac {\int -\frac {e^3 (e f-8 d g) b^4+8 \left (a e^4 g-c d e^2 (3 e f-10 d g)\right ) b^3+8 c e \left (2 c (5 e f-12 d g) d^2+a e^2 (3 e f-14 d g)\right ) b^2+32 c \left (a^2 g e^4-a c d (3 e f-8 d g) e^2-2 c^2 d^3 (e f-2 d g)\right ) b+16 a c^2 e \left (4 c (e f-2 d g) d^2+a e^2 (e f-4 d g)\right )+\left (-128 d^3 (e f-2 d g) c^4+64 d e (b d (3 e f-7 d g)-a e (e f-3 d g)) c^3+16 e^2 \left (-5 d (e f-3 d g) b^2+2 a e (e f-5 d g) b+a^2 e^2 g\right ) c^2+8 b^2 e^3 (b e f-5 b d g+3 a e g) c+b^4 e^4 g\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}\right )}{4 e^2}-\frac {\left (16 c^2 (e f-2 d g) d^2+b e^2 (b e f-6 b d g+4 a e g)-4 c e (b d (3 e f-8 d g)-a e (e f-4 d g))-e \left (-8 d (e f-2 d g) c^2+4 e (b e f-3 b d g+a e g) c+b^2 e^2 g\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 e^2 (d+e x)^2}\right )}{6 e^2}\right )}{8 e^2}-\frac {(e f-2 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {7 \left (\frac {(4 c d (e f-2 d g)-e (b e f-4 b d g+2 a e g)+e (2 c e f-4 c d g+b e g) x) \left (c x^2+b x+a\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac {5 \left (\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (e f-2 d g) d^3-16 c^2 e (b d (5 e f-12 d g)-2 a e (e f-3 d g)) d-b^2 e^3 (b e f-8 b d g+6 a e g)-4 c e^2 \left (-2 d (3 e f-10 d g) b^2+a e (3 e f-16 d g) b+2 a^2 e^2 g\right )+e \left (32 d^2 (e f-2 d g) c^3-8 e (2 b d (2 e f-5 d g)-a e (e f-4 d g)) c^2+6 b e^2 (b e f-4 b d g+2 a e g) c+b^3 e^3 g\right ) x\right )}{e^2 (d+e x)}+\frac {\int \frac {e^3 (e f-8 d g) b^4+8 \left (a e^4 g-c d e^2 (3 e f-10 d g)\right ) b^3+8 c e \left (2 c (5 e f-12 d g) d^2+a e^2 (3 e f-14 d g)\right ) b^2+32 c \left (a^2 g e^4-a c d (3 e f-8 d g) e^2-2 c^2 d^3 (e f-2 d g)\right ) b+16 a c^2 e \left (4 c (e f-2 d g) d^2+a e^2 (e f-4 d g)\right )+\left (-128 d^3 (e f-2 d g) c^4+64 d e (b d (3 e f-7 d g)-a e (e f-3 d g)) c^3+16 e^2 \left (-5 d (e f-3 d g) b^2+2 a e (e f-5 d g) b+a^2 e^2 g\right ) c^2+8 b^2 e^3 (b e f-5 b d g+3 a e g) c+b^4 e^4 g\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}\right )}{4 e^2}-\frac {\left (16 c^2 (e f-2 d g) d^2+b e^2 (b e f-6 b d g+4 a e g)-4 c e (b d (3 e f-8 d g)-a e (e f-4 d g))-e \left (-8 d (e f-2 d g) c^2+4 e (b e f-3 b d g+a e g) c+b^2 e^2 g\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 e^2 (d+e x)^2}\right )}{6 e^2}\right )}{8 e^2}-\frac {(e f-2 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {7 \left (\frac {(4 c d (e f-2 d g)-e (b e f-4 b d g+2 a e g)+e (2 c e f-4 c d g+b e g) x) \left (c x^2+b x+a\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac {5 \left (\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (e f-2 d g) d^3-16 c^2 e (b d (5 e f-12 d g)-2 a e (e f-3 d g)) d-b^2 e^3 (b e f-8 b d g+6 a e g)-4 c e^2 \left (-2 d (3 e f-10 d g) b^2+a e (3 e f-16 d g) b+2 a^2 e^2 g\right )+e \left (32 d^2 (e f-2 d g) c^3-8 e (2 b d (2 e f-5 d g)-a e (e f-4 d g)) c^2+6 b e^2 (b e f-4 b d g+2 a e g) c+b^3 e^3 g\right ) x\right )}{e^2 (d+e x)}+\frac {\frac {\left (-128 d^3 (e f-2 d g) c^4+64 d e (b d (3 e f-7 d g)-a e (e f-3 d g)) c^3+16 e^2 \left (-5 d (e f-3 d g) b^2+2 a e (e f-5 d g) b+a^2 e^2 g\right ) c^2+8 b^2 e^3 (b e f-5 b d g+3 a e g) c+b^4 e^4 g\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}+\frac {\left (128 c^4 (e f-2 d g) d^4-64 c^3 e (b d (4 e f-9 d g)-a e (2 e f-5 d g)) d^2+b^3 e^4 (b e f-9 b d g+8 a e g)+8 b c e^3 \left (-d (4 e f-15 d g) b^2+a e (3 e f-17 d g) b+4 a^2 e^2 g\right )+16 c^2 e^2 \left (b^2 (10 e f-27 d g) d^2-2 a b e (4 e f-13 d g) d+a^2 e^2 (e f-5 d g)\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}\right )}{4 e^2}-\frac {\left (16 c^2 (e f-2 d g) d^2+b e^2 (b e f-6 b d g+4 a e g)-4 c e (b d (3 e f-8 d g)-a e (e f-4 d g))-e \left (-8 d (e f-2 d g) c^2+4 e (b e f-3 b d g+a e g) c+b^2 e^2 g\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 e^2 (d+e x)^2}\right )}{6 e^2}\right )}{8 e^2}-\frac {(e f-2 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {7 \left (\frac {(4 c d (e f-2 d g)-e (b e f-4 b d g+2 a e g)+e (2 c e f-4 c d g+b e g) x) \left (c x^2+b x+a\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac {5 \left (\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (e f-2 d g) d^3-16 c^2 e (b d (5 e f-12 d g)-2 a e (e f-3 d g)) d-b^2 e^3 (b e f-8 b d g+6 a e g)-4 c e^2 \left (-2 d (3 e f-10 d g) b^2+a e (3 e f-16 d g) b+2 a^2 e^2 g\right )+e \left (32 d^2 (e f-2 d g) c^3-8 e (2 b d (2 e f-5 d g)-a e (e f-4 d g)) c^2+6 b e^2 (b e f-4 b d g+2 a e g) c+b^3 e^3 g\right ) x\right )}{e^2 (d+e x)}+\frac {\frac {\left (128 c^4 (e f-2 d g) d^4-64 c^3 e (b d (4 e f-9 d g)-a e (2 e f-5 d g)) d^2+b^3 e^4 (b e f-9 b d g+8 a e g)+8 b c e^3 \left (-d (4 e f-15 d g) b^2+a e (3 e f-17 d g) b+4 a^2 e^2 g\right )+16 c^2 e^2 \left (b^2 (10 e f-27 d g) d^2-2 a b e (4 e f-13 d g) d+a^2 e^2 (e f-5 d g)\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}+\frac {2 \left (-128 d^3 (e f-2 d g) c^4+64 d e (b d (3 e f-7 d g)-a e (e f-3 d g)) c^3+16 e^2 \left (-5 d (e f-3 d g) b^2+2 a e (e f-5 d g) b+a^2 e^2 g\right ) c^2+8 b^2 e^3 (b e f-5 b d g+3 a e g) c+b^4 e^4 g\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{2 e^2}\right )}{4 e^2}-\frac {\left (16 c^2 (e f-2 d g) d^2+b e^2 (b e f-6 b d g+4 a e g)-4 c e (b d (3 e f-8 d g)-a e (e f-4 d g))-e \left (-8 d (e f-2 d g) c^2+4 e (b e f-3 b d g+a e g) c+b^2 e^2 g\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 e^2 (d+e x)^2}\right )}{6 e^2}\right )}{8 e^2}-\frac {(e f-2 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7 \left (\frac {(4 c d (e f-2 d g)-e (b e f-4 b d g+2 a e g)+e (2 c e f-4 c d g+b e g) x) \left (c x^2+b x+a\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac {5 \left (\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (e f-2 d g) d^3-16 c^2 e (b d (5 e f-12 d g)-2 a e (e f-3 d g)) d-b^2 e^3 (b e f-8 b d g+6 a e g)-4 c e^2 \left (-2 d (3 e f-10 d g) b^2+a e (3 e f-16 d g) b+2 a^2 e^2 g\right )+e \left (32 d^2 (e f-2 d g) c^3-8 e (2 b d (2 e f-5 d g)-a e (e f-4 d g)) c^2+6 b e^2 (b e f-4 b d g+2 a e g) c+b^3 e^3 g\right ) x\right )}{e^2 (d+e x)}+\frac {\frac {\left (-128 d^3 (e f-2 d g) c^4+64 d e (b d (3 e f-7 d g)-a e (e f-3 d g)) c^3+16 e^2 \left (-5 d (e f-3 d g) b^2+2 a e (e f-5 d g) b+a^2 e^2 g\right ) c^2+8 b^2 e^3 (b e f-5 b d g+3 a e g) c+b^4 e^4 g\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{\sqrt {c} e}+\frac {\left (128 c^4 (e f-2 d g) d^4-64 c^3 e (b d (4 e f-9 d g)-a e (2 e f-5 d g)) d^2+b^3 e^4 (b e f-9 b d g+8 a e g)+8 b c e^3 \left (-d (4 e f-15 d g) b^2+a e (3 e f-17 d g) b+4 a^2 e^2 g\right )+16 c^2 e^2 \left (b^2 (10 e f-27 d g) d^2-2 a b e (4 e f-13 d g) d+a^2 e^2 (e f-5 d g)\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}\right )}{4 e^2}-\frac {\left (16 c^2 (e f-2 d g) d^2+b e^2 (b e f-6 b d g+4 a e g)-4 c e (b d (3 e f-8 d g)-a e (e f-4 d g))-e \left (-8 d (e f-2 d g) c^2+4 e (b e f-3 b d g+a e g) c+b^2 e^2 g\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 e^2 (d+e x)^2}\right )}{6 e^2}\right )}{8 e^2}-\frac {(e f-2 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {7 \left (\frac {(4 c d (e f-2 d g)-e (b e f-4 b d g+2 a e g)+e (2 c e f-4 c d g+b e g) x) \left (c x^2+b x+a\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac {5 \left (\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (e f-2 d g) d^3-16 c^2 e (b d (5 e f-12 d g)-2 a e (e f-3 d g)) d-b^2 e^3 (b e f-8 b d g+6 a e g)-4 c e^2 \left (-2 d (3 e f-10 d g) b^2+a e (3 e f-16 d g) b+2 a^2 e^2 g\right )+e \left (32 d^2 (e f-2 d g) c^3-8 e (2 b d (2 e f-5 d g)-a e (e f-4 d g)) c^2+6 b e^2 (b e f-4 b d g+2 a e g) c+b^3 e^3 g\right ) x\right )}{e^2 (d+e x)}+\frac {\frac {\left (-128 d^3 (e f-2 d g) c^4+64 d e (b d (3 e f-7 d g)-a e (e f-3 d g)) c^3+16 e^2 \left (-5 d (e f-3 d g) b^2+2 a e (e f-5 d g) b+a^2 e^2 g\right ) c^2+8 b^2 e^3 (b e f-5 b d g+3 a e g) c+b^4 e^4 g\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{\sqrt {c} e}-\frac {2 \left (128 c^4 (e f-2 d g) d^4-64 c^3 e (b d (4 e f-9 d g)-a e (2 e f-5 d g)) d^2+b^3 e^4 (b e f-9 b d g+8 a e g)+8 b c e^3 \left (-d (4 e f-15 d g) b^2+a e (3 e f-17 d g) b+4 a^2 e^2 g\right )+16 c^2 e^2 \left (b^2 (10 e f-27 d g) d^2-2 a b e (4 e f-13 d g) d+a^2 e^2 (e f-5 d g)\right )\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}}{2 e^2}\right )}{4 e^2}-\frac {\left (16 c^2 (e f-2 d g) d^2+b e^2 (b e f-6 b d g+4 a e g)-4 c e (b d (3 e f-8 d g)-a e (e f-4 d g))-e \left (-8 d (e f-2 d g) c^2+4 e (b e f-3 b d g+a e g) c+b^2 e^2 g\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 e^2 (d+e x)^2}\right )}{6 e^2}\right )}{8 e^2}-\frac {(e f-2 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7 \left (\frac {(4 c d (e f-2 d g)-e (b e f-4 b d g+2 a e g)+e (2 c e f-4 c d g+b e g) x) \left (c x^2+b x+a\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac {5 \left (\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (64 c^3 (e f-2 d g) d^3-16 c^2 e (b d (5 e f-12 d g)-2 a e (e f-3 d g)) d-b^2 e^3 (b e f-8 b d g+6 a e g)-4 c e^2 \left (-2 d (3 e f-10 d g) b^2+a e (3 e f-16 d g) b+2 a^2 e^2 g\right )+e \left (32 d^2 (e f-2 d g) c^3-8 e (2 b d (2 e f-5 d g)-a e (e f-4 d g)) c^2+6 b e^2 (b e f-4 b d g+2 a e g) c+b^3 e^3 g\right ) x\right )}{e^2 (d+e x)}+\frac {\frac {\left (-128 d^3 (e f-2 d g) c^4+64 d e (b d (3 e f-7 d g)-a e (e f-3 d g)) c^3+16 e^2 \left (-5 d (e f-3 d g) b^2+2 a e (e f-5 d g) b+a^2 e^2 g\right ) c^2+8 b^2 e^3 (b e f-5 b d g+3 a e g) c+b^4 e^4 g\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{\sqrt {c} e}+\frac {\left (128 c^4 (e f-2 d g) d^4-64 c^3 e (b d (4 e f-9 d g)-a e (2 e f-5 d g)) d^2+b^3 e^4 (b e f-9 b d g+8 a e g)+8 b c e^3 \left (-d (4 e f-15 d g) b^2+a e (3 e f-17 d g) b+4 a^2 e^2 g\right )+16 c^2 e^2 \left (b^2 (10 e f-27 d g) d^2-2 a b e (4 e f-13 d g) d+a^2 e^2 (e f-5 d g)\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b e d+a e^2} \sqrt {c x^2+b x+a}}\right )}{e \sqrt {c d^2-b e d+a e^2}}}{2 e^2}\right )}{4 e^2}-\frac {\left (16 c^2 (e f-2 d g) d^2+b e^2 (b e f-6 b d g+4 a e g)-4 c e (b d (3 e f-8 d g)-a e (e f-4 d g))-e \left (-8 d (e f-2 d g) c^2+4 e (b e f-3 b d g+a e g) c+b^2 e^2 g\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 e^2 (d+e x)^2}\right )}{6 e^2}\right )}{8 e^2}-\frac {(e f-2 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{4 e^2 (d+e x)^4}\) |
Input:
Int[((f + g*x)*(a + b*x + c*x^2)^(7/2))/(d + e*x)^5,x]
Output:
-1/4*((e*f - 2*d*g - e*g*x)*(a + b*x + c*x^2)^(7/2))/(e^2*(d + e*x)^4) + ( 7*(((4*c*d*(e*f - 2*d*g) - e*(b*e*f - 4*b*d*g + 2*a*e*g) + e*(2*c*e*f - 4* c*d*g + b*e*g)*x)*(a + b*x + c*x^2)^(5/2))/(3*e^2*(d + e*x)^3) + (5*(-1/2* ((16*c^2*d^2*(e*f - 2*d*g) + b*e^2*(b*e*f - 6*b*d*g + 4*a*e*g) - 4*c*e*(b* d*(3*e*f - 8*d*g) - a*e*(e*f - 4*d*g)) - e*(b^2*e^2*g - 8*c^2*d*(e*f - 2*d *g) + 4*c*e*(b*e*f - 3*b*d*g + a*e*g))*x)*(a + b*x + c*x^2)^(3/2))/(e^2*(d + e*x)^2) + (3*(((64*c^3*d^3*(e*f - 2*d*g) - b^2*e^3*(b*e*f - 8*b*d*g + 6 *a*e*g) - 4*c*e^2*(2*a^2*e^2*g + a*b*e*(3*e*f - 16*d*g) - 2*b^2*d*(3*e*f - 10*d*g)) - 16*c^2*d*e*(b*d*(5*e*f - 12*d*g) - 2*a*e*(e*f - 3*d*g)) + e*(b ^3*e^3*g + 32*c^3*d^2*(e*f - 2*d*g) + 6*b*c*e^2*(b*e*f - 4*b*d*g + 2*a*e*g ) - 8*c^2*e*(2*b*d*(2*e*f - 5*d*g) - a*e*(e*f - 4*d*g)))*x)*Sqrt[a + b*x + c*x^2])/(e^2*(d + e*x)) + (((b^4*e^4*g - 128*c^4*d^3*(e*f - 2*d*g) + 8*b^ 2*c*e^3*(b*e*f - 5*b*d*g + 3*a*e*g) + 16*c^2*e^2*(a^2*e^2*g + 2*a*b*e*(e*f - 5*d*g) - 5*b^2*d*(e*f - 3*d*g)) + 64*c^3*d*e*(b*d*(3*e*f - 7*d*g) - a*e *(e*f - 3*d*g)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/( Sqrt[c]*e) + ((128*c^4*d^4*(e*f - 2*d*g) + b^3*e^4*(b*e*f - 9*b*d*g + 8*a* e*g) + 8*b*c*e^3*(4*a^2*e^2*g + a*b*e*(3*e*f - 17*d*g) - b^2*d*(4*e*f - 15 *d*g)) + 16*c^2*e^2*(b^2*d^2*(10*e*f - 27*d*g) - 2*a*b*d*e*(4*e*f - 13*d*g ) + a^2*e^2*(e*f - 5*d*g)) - 64*c^3*d^2*e*(b*d*(4*e*f - 9*d*g) - a*e*(2*e* f - 5*d*g)))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*...
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
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])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(5694\) vs. \(2(874)=1748\).
Time = 3.84 (sec) , antiderivative size = 5695, normalized size of antiderivative = 6.26
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(5695\) |
| default | \(\text {Expression too large to display}\) | \(17155\) |
Input:
int((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^5,x,method=_RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^5} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^5,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^5} \, dx=\int \frac {\left (f + g x\right ) \left (a + b x + c x^{2}\right )^{\frac {7}{2}}}{\left (d + e x\right )^{5}}\, dx \] Input:
integrate((g*x+f)*(c*x**2+b*x+a)**(7/2)/(e*x+d)**5,x)
Output:
Integral((f + g*x)*(a + b*x + c*x**2)**(7/2)/(d + e*x)**5, x)
Exception generated. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^5,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?` f or more de
Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^5} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^5,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^5} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{{\left (d+e\,x\right )}^5} \,d x \] Input:
int(((f + g*x)*(a + b*x + c*x^2)^(7/2))/(d + e*x)^5,x)
Output:
int(((f + g*x)*(a + b*x + c*x^2)^(7/2))/(d + e*x)^5, x)
Time = 13.17 (sec) , antiderivative size = 21669, normalized size of antiderivative = 23.81 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:
int((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^5,x)
Output:
( - 3360*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqr t(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b*c**2*d* *4*e**5*g - 13440*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c* x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2* b*c**2*d**3*e**6*g*x - 20160*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c *d*x)*a**2*b*c**2*d**2*e**7*g*x**2 - 13440*sqrt(a*e**2 - b*d*e + c*d**2)*l og( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b*c**2*d*e**8*g*x**3 - 3360*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2 *a*e + b*d - b*e*x + 2*c*d*x)*a**2*b*c**2*e**9*g*x**4 + 8400*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d **2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**3*d**5*e**4*g - 1680*sqrt(a* e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**3*d**4*e**5*f + 33600* sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**3*d**4*e**5*g*x - 6720*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt (a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**3*d**3* e**6*f*x + 50400*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + ...