\(\int \frac {(f+g x) (a+b x+c x^2)^{7/2}}{(d+e x)^2} \, dx\) [903]

Optimal result

Integrand size = 27, antiderivative size = 1306 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^2} \, dx =\text {Too large to display} \] Output:

1/3072*(15*b^6*e^6*g-3072*c^6*d^5*(-8*d*g+7*e*f)+48*c^3*e^3*(64*a^3*e^3*g- 
2*a*b^2*d*e*(-451*d*g+322*e*f)+b^3*d^2*(-436*d*g+343*e*f)+2*a^2*b*e^2*(-26 
2*d*g+147*e*f))+768*c^5*d^3*e*(b*d*(-90*d*g+77*e*f)-4*a*e*(-17*d*g+14*e*f) 
)+192*c^4*d*e^2*(a*b*d*e*(-492*d*g+385*e*f)-2*b^2*d^2*(-169*d*g+140*e*f)-1 
6*a^2*e^2*(-10*d*g+7*e*f))-6*b^4*c*e^5*(32*a*e*g+7*b*(-2*d*g+e*f))+24*b^2* 
c^2*e^4*(38*a^2*e^2*g-7*b^2*d*(-3*d*g+2*e*f)+28*a*b*e*(-2*d*g+e*f))+2*c*e* 
(8*c*e*(-b*e+2*c*d)*(12*c*e*(-2*a*e+b*d)*(2*a*e*g-8*b*d*g+7*b*e*f)-d*(-4*a 
*c*e-5*b^2*e+12*b*c*d)*(b*e*g-16*c*d*g+14*c*e*f))+2*(3/2*(-4*a*c+b^2)*e^2- 
4*c*d*(-b*e+2*c*d))*(12*c*e*(-b*e+2*c*d)*(2*a*e*g-8*b*d*g+7*b*e*f)+2*(5/2* 
(-4*a*c+b^2)*e^2-6*c*d*(-b*e+2*c*d))*(b*e*g+2*c*(-8*d*g+7*e*f))))*x)*(c*x^ 
2+b*x+a)^(1/2)/c^3/e^8-1/384*(5*b^4*e^4*g+128*c^4*d^3*(-8*d*g+7*e*f)-8*c^2 
*e^2*(16*a^2*e^2*g+a*b*e*(-134*d*g+91*e*f)-b^2*d*(-123*d*g+98*e*f))-16*c^3 
*d*e*(b*d*(-124*d*g+105*e*f)-8*a*e*(-9*d*g+7*e*f))-2*b^2*c*e^3*(22*a*e*g+7 
*b*(-2*d*g+e*f))+2*c*e*(12*c*e*(-b*e+2*c*d)*(2*a*e*g-8*b*d*g+7*b*e*f)+2*(5 
/2*(-4*a*c+b^2)*e^2-6*c*d*(-b*e+2*c*d))*(b*e*g+2*c*(-8*d*g+7*e*f)))*x)*(c* 
x^2+b*x+a)^(3/2)/c^2/e^6+1/120*(5*b^2*e^2*g-24*c^2*d*(-8*d*g+7*e*f)+2*c*e* 
(12*a*e*g-94*b*d*g+77*b*e*f)+10*c*e*(b*e*g-16*c*d*g+14*c*e*f)*x)*(c*x^2+b* 
x+a)^(5/2)/c/e^4-1/7*(-e*g*x-8*d*g+7*e*f)*(c*x^2+b*x+a)^(7/2)/e^2/(e*x+d)- 
1/2048*(5*b^7*e^7*g-2048*c^7*d^6*(-8*d*g+7*e*f)-14*b^5*c*e^6*(6*a*e*g-2*b* 
d*g+b*e*f)+7168*c^6*d^4*e*(b*d*(-7*d*g+6*e*f)-a*e*(-6*d*g+5*e*f))-8960*...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(3123\) vs. \(2(1306)=2612\).

Time = 16.51 (sec) , antiderivative size = 3123, normalized size of antiderivative = 2.39 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^2} \, dx=\text {Result too large to show} \] Input:

Integrate[((f + g*x)*(a + b*x + c*x^2)^(7/2))/(d + e*x)^2,x]
 

Output:

((e*f - d*g)*(a + b*x + c*x^2)*(a + x*(b + c*x))^(7/2))/((-(c*d^2) + b*d*e 
 - a*e^2)*(d + e*x)) + ((a + x*(b + c*x))^(7/2)*(((8*c*(8*c*d - (7*b*e)/2) 
*(e*f - d*g) + 4*c*e*(-2*c*d*f - 7*b*e*f + 9*b*d*g - 2*a*e*g) - 56*c^2*e*( 
e*f - d*g)*x)*(a + b*x + c*x^2)^(7/2))/(56*c*e^2) - (((48*c^2*e*(c*d^2 - e 
*(b*d - a*e))*(7*b*e*f - 8*b*d*g + 2*a*e*g) - 8*c*(6*c*d - (5*b*e)/2)*(c*d 
^2 - e*(b*d - a*e))*(14*c*e*f - 16*c*d*g + b*e*g) + 40*c^2*e*(c*d^2 - e*(b 
*d - a*e))*(14*c*e*f - 16*c*d*g + b*e*g)*x)*(a + b*x + c*x^2)^(5/2))/(30*c 
*e^2) - (((-4*c*(4*c*d - (3*b*e)/2)*(c*d^2 - e*(b*d - a*e))*(12*c*e*(2*c*d 
 - b*e)*(7*b*e*f - 8*b*d*g + 2*a*e*g) - 2*(12*c^2*d^2 - (5*b^2*e^2)/2 - 2* 
c*e*(3*b*d - 5*a*e))*(b*e*g + 2*c*(7*e*f - 8*d*g))) + 16*c^2*e*(c*d^2 - e* 
(b*d - a*e))*(12*c*e*(b*d - 2*a*e)*(7*b*e*f - 8*b*d*g + 2*a*e*g) + 2*(2*a* 
c*d*e - b*d*(6*c*d - (5*b*e)/2))*(b*e*g + 2*c*(7*e*f - 8*d*g))) + 12*c^2*e 
*(c*d^2 - e*(b*d - a*e))*(12*c*e*(2*c*d - b*e)*(7*b*e*f - 8*b*d*g + 2*a*e* 
g) - 2*(12*c^2*d^2 - (5*b^2*e^2)/2 - 2*c*e*(3*b*d - 5*a*e))*(b*e*g + 2*c*( 
7*e*f - 8*d*g)))*x)*(a + b*x + c*x^2)^(3/2))/(12*c*e^2) - (((2*c*e*(4*c*(c 
*d^2 - e*(b*d - a*e))*(2*a*c*d*e + b*d*(-4*c*d + (3*b*e)/2))*(12*c*e*(2*c* 
d - b*e)*(7*b*e*f - 8*b*d*g + 2*a*e*g) - 2*(12*c^2*d^2 - (5*b^2*e^2)/2 - 2 
*c*e*(3*b*d - 5*a*e))*(b*e*g + 2*c*(7*e*f - 8*d*g))) + 16*c^2*e*(b*d - 2*a 
*e)*(c*d^2 - e*(b*d - a*e))*(12*c*e*(b*d - 2*a*e)*(7*b*e*f - 8*b*d*g + 2*a 
*e*g) + 2*(2*a*c*d*e - b*d*(6*c*d - (5*b*e)/2))*(b*e*g + 2*c*(7*e*f - 8...
 

Rubi [A] (verified)

Time = 6.38 (sec) , antiderivative size = 1350, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1230, 25, 1231, 27, 1231, 27, 1231, 27, 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.

\(\displaystyle \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^2} \, dx\)

\(\Big \downarrow \) 1230

\(\displaystyle -\frac {\int -\frac {(7 b e f-8 b d g+2 a e g+(14 c e f-16 c d g+b e g) x) \left (c x^2+b x+a\right )^{5/2}}{d+e x}dx}{2 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-8 d g+7 e f-e g x)}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(7 b e f-8 b d g+2 a e g+(14 c e f-16 c d g+b e g) x) \left (c x^2+b x+a\right )^{5/2}}{d+e x}dx}{2 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-8 d g+7 e f-e g x)}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\frac {\left (a+b x+c x^2\right )^{5/2} \left (2 c e (12 a e g-94 b d g+77 b e f)+5 b^2 e^2 g+10 c e x (b e g-16 c d g+14 c e f)-24 c^2 d (7 e f-8 d g)\right )}{60 c e^2}-\frac {\int \frac {\left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (b e g+2 c (7 e f-8 d g))+\left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 (d+e x)}dx}{12 c e^2}}{2 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-8 d g+7 e f-e g x)}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\left (a+b x+c x^2\right )^{5/2} \left (2 c e (12 a e g-94 b d g+77 b e f)+5 b^2 e^2 g+10 c e x (b e g-16 c d g+14 c e f)-24 c^2 d (7 e f-8 d g)\right )}{60 c e^2}-\frac {\int \frac {\left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)+\left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{d+e x}dx}{24 c e^2}}{2 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-8 d g+7 e f-e g x)}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\frac {\left (-24 d (7 e f-8 d g) c^2+2 e (77 b e f-94 b d g+12 a e g) c+10 e (14 c e f-16 c d g+b e g) x c+5 b^2 e^2 g\right ) \left (c x^2+b x+a\right )^{5/2}}{60 c e^2}-\frac {\frac {\left (128 d^3 (7 e f-8 d g) c^4-16 d e (b d (105 e f-124 d g)-8 a e (7 e f-9 d g)) c^3-8 e^2 \left (-d (98 e f-123 d g) b^2+a e (91 e f-134 d g) b+16 a^2 e^2 g\right ) c^2-2 b^2 e^3 (22 a e g+7 b (e f-2 d g)) c+2 e \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right ) x c+5 b^4 e^4 g\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\int \frac {\left (8 c e (b d-2 a e) \left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right )+\left (8 c e (2 c d-b e) \left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)}dx}{8 c e^2}}{24 c e^2}}{2 e^2}-\frac {(7 e f-8 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\left (-24 d (7 e f-8 d g) c^2+2 e (77 b e f-94 b d g+12 a e g) c+10 e (14 c e f-16 c d g+b e g) x c+5 b^2 e^2 g\right ) \left (c x^2+b x+a\right )^{5/2}}{60 c e^2}-\frac {\frac {\left (128 d^3 (7 e f-8 d g) c^4-16 d e (b d (105 e f-124 d g)-8 a e (7 e f-9 d g)) c^3-8 e^2 \left (-d (98 e f-123 d g) b^2+a e (91 e f-134 d g) b+16 a^2 e^2 g\right ) c^2-2 b^2 e^3 (22 a e g+7 b (e f-2 d g)) c+2 e \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right ) x c+5 b^4 e^4 g\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\int \frac {\left (8 c e (b d-2 a e) \left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right )+\left (8 c e (2 c d-b e) \left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{16 c e^2}}{24 c e^2}}{2 e^2}-\frac {(7 e f-8 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\frac {\left (-24 d (7 e f-8 d g) c^2+2 e (77 b e f-94 b d g+12 a e g) c+10 e (14 c e f-16 c d g+b e g) x c+5 b^2 e^2 g\right ) \left (c x^2+b x+a\right )^{5/2}}{60 c e^2}-\frac {\frac {\left (128 d^3 (7 e f-8 d g) c^4-16 d e (b d (105 e f-124 d g)-8 a e (7 e f-9 d g)) c^3-8 e^2 \left (-d (98 e f-123 d g) b^2+a e (91 e f-134 d g) b+16 a^2 e^2 g\right ) c^2-2 b^2 e^3 (22 a e g+7 b (e f-2 d g)) c+2 e \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right ) x c+5 b^4 e^4 g\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-1024 d^5 (7 e f-8 d g) c^6+256 d^3 e (b d (77 e f-90 d g)-4 a e (14 e f-17 d g)) c^5+64 d e^2 \left (-2 b^2 (140 e f-169 d g) d^2+a b e (385 e f-492 d g) d-16 a^2 e^2 (7 e f-10 d g)\right ) c^4+16 e^3 \left (d^2 (343 e f-436 d g) b^3-2 a d e (322 e f-451 d g) b^2+2 a^2 e^2 (147 e f-262 d g) b+64 a^3 e^3 g\right ) c^3+8 b^2 e^4 \left (-7 d (2 e f-3 d g) b^2+28 a e (e f-2 d g) b+38 a^2 e^2 g\right ) c^2-2 b^4 e^5 (32 a e g+7 b (e f-2 d g)) c+5 b^6 e^6 g\right )+2 c e \left (8 c e (2 c d-b e) \left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {\int \frac {3 \left (5 d e^6 g b^7-14 c d e^5 (e f-2 d g) b^6-28 c d e^4 \left (3 a g e^2+2 c d (2 e f-3 d g)\right ) b^5+8 c^2 d e^3 \left (2 c (343 e f-436 d g) d^2+35 a e^2 (e f-2 d g)\right ) b^4+16 c^2 d e^2 \left (35 a^2 g e^4-4 a c d (266 e f-359 d g) e^2-8 c^2 d^3 (140 e f-169 d g)\right ) b^3+32 c^3 d e \left (8 c^2 (77 e f-90 d g) d^4+12 a c e^2 (119 e f-148 d g) d^2+9 a^2 e^4 (63 e f-94 d g)\right ) b^2-64 c^3 \left (16 c^3 (7 e f-8 d g) d^6+8 a c^2 e^2 (70 e f-83 d g) d^4+6 a^2 c e^4 (98 e f-127 d g) d^2+a^3 e^6 (112 e f-205 d g)\right ) b-128 a c^3 e \left (16 a^3 g e^6-a^2 c d (77 e f-106 d g) e^4-2 a c^2 d^3 (63 e f-76 d g) e^2-8 c^3 d^5 (7 e f-8 d g)\right )+\left (-2048 d^6 (7 e f-8 d g) c^7+7168 d^4 e (b d (6 e f-7 d g)-a e (5 e f-6 d g)) c^6-8960 d^2 e^2 (b d-a e) (b d (5 e f-6 d g)-a e (3 e f-4 d g)) c^5+4480 e^3 (b d-a e)^2 (b d (4 e f-5 d g)-a e (e f-2 d g)) c^4-560 b e^4 \left (d^2 (3 e f-4 d g) b^3-4 a d e (2 e f-3 d g) b^2+6 a^2 e^2 (e f-2 d g) b+4 a^3 e^3 g\right ) c^3+56 b^3 e^5 \left (-d (2 e f-3 d g) b^2+5 a e (e f-2 d g) b+10 a^2 e^2 g\right ) c^2-14 b^5 e^6 (b e f-2 b d g+6 a e g) c+5 b^7 e^7 g\right ) x\right )}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}}{16 c e^2}}{24 c e^2}}{2 e^2}-\frac {(7 e f-8 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\left (-24 d (7 e f-8 d g) c^2+2 e (77 b e f-94 b d g+12 a e g) c+10 e (14 c e f-16 c d g+b e g) x c+5 b^2 e^2 g\right ) \left (c x^2+b x+a\right )^{5/2}}{60 c e^2}-\frac {\frac {\left (128 d^3 (7 e f-8 d g) c^4-16 d e (b d (105 e f-124 d g)-8 a e (7 e f-9 d g)) c^3-8 e^2 \left (-d (98 e f-123 d g) b^2+a e (91 e f-134 d g) b+16 a^2 e^2 g\right ) c^2-2 b^2 e^3 (22 a e g+7 b (e f-2 d g)) c+2 e \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right ) x c+5 b^4 e^4 g\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-1024 d^5 (7 e f-8 d g) c^6+256 d^3 e (b d (77 e f-90 d g)-4 a e (14 e f-17 d g)) c^5+64 d e^2 \left (-2 b^2 (140 e f-169 d g) d^2+a b e (385 e f-492 d g) d-16 a^2 e^2 (7 e f-10 d g)\right ) c^4+16 e^3 \left (d^2 (343 e f-436 d g) b^3-2 a d e (322 e f-451 d g) b^2+2 a^2 e^2 (147 e f-262 d g) b+64 a^3 e^3 g\right ) c^3+8 b^2 e^4 \left (-7 d (2 e f-3 d g) b^2+28 a e (e f-2 d g) b+38 a^2 e^2 g\right ) c^2-2 b^4 e^5 (32 a e g+7 b (e f-2 d g)) c+5 b^6 e^6 g\right )+2 c e \left (8 c e (2 c d-b e) \left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {3 \int \frac {5 d e^6 g b^7-14 c d e^5 (e f-2 d g) b^6-28 c d e^4 \left (3 a g e^2+2 c d (2 e f-3 d g)\right ) b^5+8 c^2 d e^3 \left (2 c (343 e f-436 d g) d^2+35 a e^2 (e f-2 d g)\right ) b^4+16 c^2 d e^2 \left (35 a^2 g e^4-4 a c d (266 e f-359 d g) e^2-8 c^2 d^3 (140 e f-169 d g)\right ) b^3+32 c^3 d e \left (8 c^2 (77 e f-90 d g) d^4+12 a c e^2 (119 e f-148 d g) d^2+9 a^2 e^4 (63 e f-94 d g)\right ) b^2-64 c^3 \left (16 c^3 (7 e f-8 d g) d^6+8 a c^2 e^2 (70 e f-83 d g) d^4+6 a^2 c e^4 (98 e f-127 d g) d^2+a^3 e^6 (112 e f-205 d g)\right ) b-128 a c^3 e \left (16 a^3 g e^6-a^2 c d (77 e f-106 d g) e^4-2 a c^2 d^3 (63 e f-76 d g) e^2-8 c^3 d^5 (7 e f-8 d g)\right )+\left (-2048 d^6 (7 e f-8 d g) c^7+7168 d^4 e (b d (6 e f-7 d g)-a e (5 e f-6 d g)) c^6-8960 d^2 e^2 (b d-a e) (b d (5 e f-6 d g)-a e (3 e f-4 d g)) c^5+4480 e^3 (b d-a e)^2 (b d (4 e f-5 d g)-a e (e f-2 d g)) c^4-560 b e^4 \left (d^2 (3 e f-4 d g) b^3-4 a d e (2 e f-3 d g) b^2+6 a^2 e^2 (e f-2 d g) b+4 a^3 e^3 g\right ) c^3+56 b^3 e^5 \left (-d (2 e f-3 d g) b^2+5 a e (e f-2 d g) b+10 a^2 e^2 g\right ) c^2-14 b^5 e^6 (b e f-2 b d g+6 a e g) c+5 b^7 e^7 g\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}}{16 c e^2}}{24 c e^2}}{2 e^2}-\frac {(7 e f-8 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\left (-24 d (7 e f-8 d g) c^2+2 e (77 b e f-94 b d g+12 a e g) c+10 e (14 c e f-16 c d g+b e g) x c+5 b^2 e^2 g\right ) \left (c x^2+b x+a\right )^{5/2}}{60 c e^2}-\frac {\frac {\left (128 d^3 (7 e f-8 d g) c^4-16 d e (b d (105 e f-124 d g)-8 a e (7 e f-9 d g)) c^3-8 e^2 \left (-d (98 e f-123 d g) b^2+a e (91 e f-134 d g) b+16 a^2 e^2 g\right ) c^2-2 b^2 e^3 (22 a e g+7 b (e f-2 d g)) c+2 e \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right ) x c+5 b^4 e^4 g\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-1024 d^5 (7 e f-8 d g) c^6+256 d^3 e (b d (77 e f-90 d g)-4 a e (14 e f-17 d g)) c^5+64 d e^2 \left (-2 b^2 (140 e f-169 d g) d^2+a b e (385 e f-492 d g) d-16 a^2 e^2 (7 e f-10 d g)\right ) c^4+16 e^3 \left (d^2 (343 e f-436 d g) b^3-2 a d e (322 e f-451 d g) b^2+2 a^2 e^2 (147 e f-262 d g) b+64 a^3 e^3 g\right ) c^3+8 b^2 e^4 \left (-7 d (2 e f-3 d g) b^2+28 a e (e f-2 d g) b+38 a^2 e^2 g\right ) c^2-2 b^4 e^5 (32 a e g+7 b (e f-2 d g)) c+5 b^6 e^6 g\right )+2 c e \left (8 c e (2 c d-b e) \left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {3 \left (\frac {1024 c^3 (2 c d (7 e f-8 d g)-e (7 b e f-9 b d g+2 a e g)) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx \left (c d^2-b e d+a e^2\right )^3}{e}+\frac {\left (-2048 d^6 (7 e f-8 d g) c^7+7168 d^4 e (b d (6 e f-7 d g)-a e (5 e f-6 d g)) c^6-8960 d^2 e^2 (b d-a e) (b d (5 e f-6 d g)-a e (3 e f-4 d g)) c^5+4480 e^3 (b d-a e)^2 (b d (4 e f-5 d g)-a e (e f-2 d g)) c^4-560 b e^4 \left (d^2 (3 e f-4 d g) b^3-4 a d e (2 e f-3 d g) b^2+6 a^2 e^2 (e f-2 d g) b+4 a^3 e^3 g\right ) c^3+56 b^3 e^5 \left (-d (2 e f-3 d g) b^2+5 a e (e f-2 d g) b+10 a^2 e^2 g\right ) c^2-14 b^5 e^6 (b e f-2 b d g+6 a e g) c+5 b^7 e^7 g\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}\right )}{8 c e^2}}{16 c e^2}}{24 c e^2}}{2 e^2}-\frac {(7 e f-8 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\left (-24 d (7 e f-8 d g) c^2+2 e (77 b e f-94 b d g+12 a e g) c+10 e (14 c e f-16 c d g+b e g) x c+5 b^2 e^2 g\right ) \left (c x^2+b x+a\right )^{5/2}}{60 c e^2}-\frac {\frac {\left (128 d^3 (7 e f-8 d g) c^4-16 d e (b d (105 e f-124 d g)-8 a e (7 e f-9 d g)) c^3-8 e^2 \left (-d (98 e f-123 d g) b^2+a e (91 e f-134 d g) b+16 a^2 e^2 g\right ) c^2-2 b^2 e^3 (22 a e g+7 b (e f-2 d g)) c+2 e \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right ) x c+5 b^4 e^4 g\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-1024 d^5 (7 e f-8 d g) c^6+256 d^3 e (b d (77 e f-90 d g)-4 a e (14 e f-17 d g)) c^5+64 d e^2 \left (-2 b^2 (140 e f-169 d g) d^2+a b e (385 e f-492 d g) d-16 a^2 e^2 (7 e f-10 d g)\right ) c^4+16 e^3 \left (d^2 (343 e f-436 d g) b^3-2 a d e (322 e f-451 d g) b^2+2 a^2 e^2 (147 e f-262 d g) b+64 a^3 e^3 g\right ) c^3+8 b^2 e^4 \left (-7 d (2 e f-3 d g) b^2+28 a e (e f-2 d g) b+38 a^2 e^2 g\right ) c^2-2 b^4 e^5 (32 a e g+7 b (e f-2 d g)) c+5 b^6 e^6 g\right )+2 c e \left (8 c e (2 c d-b e) \left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {3 \left (\frac {1024 c^3 (2 c d (7 e f-8 d g)-e (7 b e f-9 b d g+2 a e g)) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx \left (c d^2-b e d+a e^2\right )^3}{e}+\frac {2 \left (-2048 d^6 (7 e f-8 d g) c^7+7168 d^4 e (b d (6 e f-7 d g)-a e (5 e f-6 d g)) c^6-8960 d^2 e^2 (b d-a e) (b d (5 e f-6 d g)-a e (3 e f-4 d g)) c^5+4480 e^3 (b d-a e)^2 (b d (4 e f-5 d g)-a e (e f-2 d g)) c^4-560 b e^4 \left (d^2 (3 e f-4 d g) b^3-4 a d e (2 e f-3 d g) b^2+6 a^2 e^2 (e f-2 d g) b+4 a^3 e^3 g\right ) c^3+56 b^3 e^5 \left (-d (2 e f-3 d g) b^2+5 a e (e f-2 d g) b+10 a^2 e^2 g\right ) c^2-14 b^5 e^6 (b e f-2 b d g+6 a e g) c+5 b^7 e^7 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}\right )}{8 c e^2}}{16 c e^2}}{24 c e^2}}{2 e^2}-\frac {(7 e f-8 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (-24 d (7 e f-8 d g) c^2+2 e (77 b e f-94 b d g+12 a e g) c+10 e (14 c e f-16 c d g+b e g) x c+5 b^2 e^2 g\right ) \left (c x^2+b x+a\right )^{5/2}}{60 c e^2}-\frac {\frac {\left (128 d^3 (7 e f-8 d g) c^4-16 d e (b d (105 e f-124 d g)-8 a e (7 e f-9 d g)) c^3-8 e^2 \left (-d (98 e f-123 d g) b^2+a e (91 e f-134 d g) b+16 a^2 e^2 g\right ) c^2-2 b^2 e^3 (22 a e g+7 b (e f-2 d g)) c+2 e \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right ) x c+5 b^4 e^4 g\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-1024 d^5 (7 e f-8 d g) c^6+256 d^3 e (b d (77 e f-90 d g)-4 a e (14 e f-17 d g)) c^5+64 d e^2 \left (-2 b^2 (140 e f-169 d g) d^2+a b e (385 e f-492 d g) d-16 a^2 e^2 (7 e f-10 d g)\right ) c^4+16 e^3 \left (d^2 (343 e f-436 d g) b^3-2 a d e (322 e f-451 d g) b^2+2 a^2 e^2 (147 e f-262 d g) b+64 a^3 e^3 g\right ) c^3+8 b^2 e^4 \left (-7 d (2 e f-3 d g) b^2+28 a e (e f-2 d g) b+38 a^2 e^2 g\right ) c^2-2 b^4 e^5 (32 a e g+7 b (e f-2 d g)) c+5 b^6 e^6 g\right )+2 c e \left (8 c e (2 c d-b e) \left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {3 \left (\frac {1024 c^3 (2 c d (7 e f-8 d g)-e (7 b e f-9 b d g+2 a e g)) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx \left (c d^2-b e d+a e^2\right )^3}{e}+\frac {\left (-2048 d^6 (7 e f-8 d g) c^7+7168 d^4 e (b d (6 e f-7 d g)-a e (5 e f-6 d g)) c^6-8960 d^2 e^2 (b d-a e) (b d (5 e f-6 d g)-a e (3 e f-4 d g)) c^5+4480 e^3 (b d-a e)^2 (b d (4 e f-5 d g)-a e (e f-2 d g)) c^4-560 b e^4 \left (d^2 (3 e f-4 d g) b^3-4 a d e (2 e f-3 d g) b^2+6 a^2 e^2 (e f-2 d g) b+4 a^3 e^3 g\right ) c^3+56 b^3 e^5 \left (-d (2 e f-3 d g) b^2+5 a e (e f-2 d g) b+10 a^2 e^2 g\right ) c^2-14 b^5 e^6 (b e f-2 b d g+6 a e g) c+5 b^7 e^7 g\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{\sqrt {c} e}\right )}{8 c e^2}}{16 c e^2}}{24 c e^2}}{2 e^2}-\frac {(7 e f-8 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\left (-24 d (7 e f-8 d g) c^2+2 e (77 b e f-94 b d g+12 a e g) c+10 e (14 c e f-16 c d g+b e g) x c+5 b^2 e^2 g\right ) \left (c x^2+b x+a\right )^{5/2}}{60 c e^2}-\frac {\frac {\left (128 d^3 (7 e f-8 d g) c^4-16 d e (b d (105 e f-124 d g)-8 a e (7 e f-9 d g)) c^3-8 e^2 \left (-d (98 e f-123 d g) b^2+a e (91 e f-134 d g) b+16 a^2 e^2 g\right ) c^2-2 b^2 e^3 (22 a e g+7 b (e f-2 d g)) c+2 e \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right ) x c+5 b^4 e^4 g\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-1024 d^5 (7 e f-8 d g) c^6+256 d^3 e (b d (77 e f-90 d g)-4 a e (14 e f-17 d g)) c^5+64 d e^2 \left (-2 b^2 (140 e f-169 d g) d^2+a b e (385 e f-492 d g) d-16 a^2 e^2 (7 e f-10 d g)\right ) c^4+16 e^3 \left (d^2 (343 e f-436 d g) b^3-2 a d e (322 e f-451 d g) b^2+2 a^2 e^2 (147 e f-262 d g) b+64 a^3 e^3 g\right ) c^3+8 b^2 e^4 \left (-7 d (2 e f-3 d g) b^2+28 a e (e f-2 d g) b+38 a^2 e^2 g\right ) c^2-2 b^4 e^5 (32 a e g+7 b (e f-2 d g)) c+5 b^6 e^6 g\right )+2 c e \left (8 c e (2 c d-b e) \left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {3 \left (\frac {\left (-2048 d^6 (7 e f-8 d g) c^7+7168 d^4 e (b d (6 e f-7 d g)-a e (5 e f-6 d g)) c^6-8960 d^2 e^2 (b d-a e) (b d (5 e f-6 d g)-a e (3 e f-4 d g)) c^5+4480 e^3 (b d-a e)^2 (b d (4 e f-5 d g)-a e (e f-2 d g)) c^4-560 b e^4 \left (d^2 (3 e f-4 d g) b^3-4 a d e (2 e f-3 d g) b^2+6 a^2 e^2 (e f-2 d g) b+4 a^3 e^3 g\right ) c^3+56 b^3 e^5 \left (-d (2 e f-3 d g) b^2+5 a e (e f-2 d g) b+10 a^2 e^2 g\right ) c^2-14 b^5 e^6 (b e f-2 b d g+6 a e g) c+5 b^7 e^7 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 {2048 c^3 \left (c d^2-b e d+a e^2\right )^3 (2 c d (7 e f-8 d g)-e (7 b e f-9 b d g+2 a e g)) \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}\right )}{8 c e^2}}{16 c e^2}}{24 c e^2}}{2 e^2}-\frac {(7 e f-8 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{7 e^2 (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (-24 d (7 e f-8 d g) c^2+2 e (77 b e f-94 b d g+12 a e g) c+10 e (14 c e f-16 c d g+b e g) x c+5 b^2 e^2 g\right ) \left (c x^2+b x+a\right )^{5/2}}{60 c e^2}-\frac {\frac {\left (128 d^3 (7 e f-8 d g) c^4-16 d e (b d (105 e f-124 d g)-8 a e (7 e f-9 d g)) c^3-8 e^2 \left (-d (98 e f-123 d g) b^2+a e (91 e f-134 d g) b+16 a^2 e^2 g\right ) c^2-2 b^2 e^3 (22 a e g+7 b (e f-2 d g)) c+2 e \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right ) x c+5 b^4 e^4 g\right ) \left (c x^2+b x+a\right )^{3/2}}{8 c e^2}-\frac {\frac {\left (3 \left (-1024 d^5 (7 e f-8 d g) c^6+256 d^3 e (b d (77 e f-90 d g)-4 a e (14 e f-17 d g)) c^5+64 d e^2 \left (-2 b^2 (140 e f-169 d g) d^2+a b e (385 e f-492 d g) d-16 a^2 e^2 (7 e f-10 d g)\right ) c^4+16 e^3 \left (d^2 (343 e f-436 d g) b^3-2 a d e (322 e f-451 d g) b^2+2 a^2 e^2 (147 e f-262 d g) b+64 a^3 e^3 g\right ) c^3+8 b^2 e^4 \left (-7 d (2 e f-3 d g) b^2+28 a e (e f-2 d g) b+38 a^2 e^2 g\right ) c^2-2 b^4 e^5 (32 a e g+7 b (e f-2 d g)) c+5 b^6 e^6 g\right )+2 c e \left (8 c e (2 c d-b e) \left (12 c e (b d-2 a e) (7 b e f-8 b d g+2 a e g)-d \left (-5 e b^2+12 c d b-4 a c e\right ) (14 c e f-16 c d g+b e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) \left (12 c e (2 c d-b e) (7 b e f-8 b d g+2 a e g)-\left (24 c^2 d^2-5 b^2 e^2-4 c e (3 b d-5 a e)\right ) (b e g+2 c (7 e f-8 d g))\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{4 c e^2}-\frac {3 \left (\frac {1024 \left (c d^2-b e d+a e^2\right )^{5/2} (2 c d (7 e f-8 d g)-e (7 b e f-9 b d g+2 a e g)) \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 ) c^3}{e}+\frac {\left (-2048 d^6 (7 e f-8 d g) c^7+7168 d^4 e (b d (6 e f-7 d g)-a e (5 e f-6 d g)) c^6-8960 d^2 e^2 (b d-a e) (b d (5 e f-6 d g)-a e (3 e f-4 d g)) c^5+4480 e^3 (b d-a e)^2 (b d (4 e f-5 d g)-a e (e f-2 d g)) c^4-560 b e^4 \left (d^2 (3 e f-4 d g) b^3-4 a d e (2 e f-3 d g) b^2+6 a^2 e^2 (e f-2 d g) b+4 a^3 e^3 g\right ) c^3+56 b^3 e^5 \left (-d (2 e f-3 d g) b^2+5 a e (e f-2 d g) b+10 a^2 e^2 g\right ) c^2-14 b^5 e^6 (b e f-2 b d g+6 a e g) c+5 b^7 e^7 g\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{e \sqrt {c}}\right )}{8 c e^2}}{16 c e^2}}{24 c e^2}}{2 e^2}-\frac {(7 e f-8 d g-e g x) \left (c x^2+b x+a\right )^{7/2}}{7 e^2 (d+e x)}\)

Input:

Int[((f + g*x)*(a + b*x + c*x^2)^(7/2))/(d + e*x)^2,x]
 

Output:

-1/7*((7*e*f - 8*d*g - e*g*x)*(a + b*x + c*x^2)^(7/2))/(e^2*(d + e*x)) + ( 
((5*b^2*e^2*g - 24*c^2*d*(7*e*f - 8*d*g) + 2*c*e*(77*b*e*f - 94*b*d*g + 12 
*a*e*g) + 10*c*e*(14*c*e*f - 16*c*d*g + b*e*g)*x)*(a + b*x + c*x^2)^(5/2)) 
/(60*c*e^2) - (((5*b^4*e^4*g + 128*c^4*d^3*(7*e*f - 8*d*g) - 8*c^2*e^2*(16 
*a^2*e^2*g + a*b*e*(91*e*f - 134*d*g) - b^2*d*(98*e*f - 123*d*g)) - 16*c^3 
*d*e*(b*d*(105*e*f - 124*d*g) - 8*a*e*(7*e*f - 9*d*g)) - 2*b^2*c*e^3*(22*a 
*e*g + 7*b*(e*f - 2*d*g)) + 2*c*e*(12*c*e*(2*c*d - b*e)*(7*b*e*f - 8*b*d*g 
 + 2*a*e*g) - (24*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(3*b*d - 5*a*e))*(b*e*g + 2* 
c*(7*e*f - 8*d*g)))*x)*(a + b*x + c*x^2)^(3/2))/(8*c*e^2) - (((3*(5*b^6*e^ 
6*g - 1024*c^6*d^5*(7*e*f - 8*d*g) + 16*c^3*e^3*(64*a^3*e^3*g - 2*a*b^2*d* 
e*(322*e*f - 451*d*g) + b^3*d^2*(343*e*f - 436*d*g) + 2*a^2*b*e^2*(147*e*f 
 - 262*d*g)) + 256*c^5*d^3*e*(b*d*(77*e*f - 90*d*g) - 4*a*e*(14*e*f - 17*d 
*g)) + 64*c^4*d*e^2*(a*b*d*e*(385*e*f - 492*d*g) - 2*b^2*d^2*(140*e*f - 16 
9*d*g) - 16*a^2*e^2*(7*e*f - 10*d*g)) - 2*b^4*c*e^5*(32*a*e*g + 7*b*(e*f - 
 2*d*g)) + 8*b^2*c^2*e^4*(38*a^2*e^2*g - 7*b^2*d*(2*e*f - 3*d*g) + 28*a*b* 
e*(e*f - 2*d*g))) + 2*c*e*(8*c*e*(2*c*d - b*e)*(12*c*e*(b*d - 2*a*e)*(7*b* 
e*f - 8*b*d*g + 2*a*e*g) - d*(12*b*c*d - 5*b^2*e - 4*a*c*e)*(14*c*e*f - 16 
*c*d*g + b*e*g)) - 2*(8*c^2*d^2 - 4*b*c*d*e - (3*b^2*e^2)/2 + 6*a*c*e^2)*( 
12*c*e*(2*c*d - b*e)*(7*b*e*f - 8*b*d*g + 2*a*e*g) - (24*c^2*d^2 - 5*b^2*e 
^2 - 4*c*e*(3*b*d - 5*a*e))*(b*e*g + 2*c*(7*e*f - 8*d*g))))*x)*Sqrt[a +...
 

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 219

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])
 

rule 1092

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]
 

rule 1154

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]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2912\) vs. \(2(1261)=2522\).

Time = 2.85 (sec) , antiderivative size = 2913, normalized size of antiderivative = 2.23

method	result	size
risch	\(\text {Expression too large to display}\)	\(2913\)
default	\(\text {Expression too large to display}\)	\(3915\)

Input:

int((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^2,x,method=_RETURNVERBOSE)
 

Output:

1/107520/c^3*(15360*c^6*e^6*g*x^6+55040*b*c^5*e^6*g*x^5-35840*c^6*d*e^5*g* 
x^5+17920*c^6*e^6*f*x^5+67584*a*c^5*e^6*g*x^4+68480*b^2*c^4*e^6*g*x^4-1326 
08*b*c^5*d*e^5*g*x^4+66304*b*c^5*e^6*f*x^4+64512*c^6*d^2*e^4*g*x^4-43008*c 
^6*d*e^5*f*x^4+177728*a*b*c^4*e^6*g*x^3-170240*a*c^5*d*e^5*g*x^3+85120*a*c 
^5*e^6*f*x^3+30480*b^3*c^3*e^6*g*x^3-173376*b^2*c^4*d*e^5*g*x^3+86688*b^2* 
c^4*e^6*f*x^3+249984*b*c^5*d^2*e^4*g*x^3-166656*b*c^5*d*e^5*f*x^3-107520*c 
^6*d^3*e^3*g*x^3+80640*c^6*d^2*e^4*f*x^3+124928*a^2*c^4*e^6*g*x^2+131424*a 
*b^2*c^3*e^6*g*x^2-484736*a*b*c^4*d*e^5*g*x^2+242368*a*b*c^4*e^6*f*x^2+344 
064*a*c^5*d^2*e^4*g*x^2-229376*a*c^5*d*e^5*f*x^2+280*b^4*c^2*e^6*g*x^2-844 
48*b^3*c^3*d*e^5*g*x^2+42224*b^3*c^3*e^6*f*x^2+353472*b^2*c^4*d^2*e^4*g*x^ 
2-235648*b^2*c^4*d*e^5*f*x^2-448000*b*c^5*d^3*e^3*g*x^2+336000*b*c^5*d^2*e 
^4*f*x^2+179200*c^6*d^4*e^2*g*x^2-143360*c^6*d^3*e^3*f*x^2+222368*a^2*b*c^ 
3*e^6*g*x-389760*a^2*c^4*d*e^5*g*x+194880*a^2*c^4*e^6*f*x+5040*a*b^3*c^2*e 
^6*g*x-424256*a*b^2*c^3*d*e^5*g*x+212128*a*b^2*c^3*e^6*f*x+1130304*a*b*c^4 
*d^2*e^4*g*x-753536*a*b*c^4*d*e^5*f*x-698880*a*c^5*d^3*e^3*g*x+524160*a*c^ 
5*d^2*e^4*f*x-350*b^5*c*e^6*g*x-1960*b^4*c^2*d*e^5*g*x+980*b^4*c^2*e^6*f*x 
+203280*b^3*c^3*d^2*e^4*g*x-135520*b^3*c^3*d*e^5*f*x-730240*b^2*c^4*d^3*e^ 
3*g*x+547680*b^2*c^4*d^2*e^4*f*x+851200*b*c^5*d^4*e^2*g*x-680960*b*c^5*d^3 
*e^3*f*x-322560*c^6*d^5*e*g*x+268800*c^6*d^4*e^2*f*x+180224*a^3*c^3*e^6*g+ 
48720*a^2*b^2*c^2*e^6*g-1026368*a^2*b*c^3*d*e^5*g+513184*a^2*b*c^3*e^6*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^2} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^2} \, dx=\int \frac {\left (f + g x\right ) \left (a + b x + c x^{2}\right )^{\frac {7}{2}}}{\left (d + e x\right )^{2}}\, dx \] Input:

integrate((g*x+f)*(c*x**2+b*x+a)**(7/2)/(e*x+d)**2,x)
 

Output:

Integral((f + g*x)*(a + b*x + c*x**2)**(7/2)/(d + e*x)**2, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^2,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
 

Giac [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^2} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^2} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{{\left (d+e\,x\right )}^2} \,d x \] Input:

int(((f + g*x)*(a + b*x + c*x^2)^(7/2))/(d + e*x)^2,x)
 

Output:

int(((f + g*x)*(a + b*x + c*x^2)^(7/2))/(d + e*x)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 10299, normalized size of antiderivative = 7.89 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^2,x)
 

Output:

(215040*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**3*c**4*d*e**6*g 
 + 215040*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**3*c**4*e**7*g 
*x - 1397760*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sq 
rt(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**4*d 
**2*e**5*g + 752640*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**4*d*e**6*f - 1397760*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**4*d*e**6*g*x + 752640*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**4*e**7*f*x + 2150400*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sq 
rt(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**5*d**3*e**4*g - 1505280*sqrt(a*e**2 - b*d*e + c*d**2)*lo 
g(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**5*d**2*e**5*f + 2150400*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**5*d**2*e**5*g*x - 1505280*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) ...