\(\int \frac {(A+B x) (d+e x)^6}{(a+b x+c x^2)^{7/2}} \, dx\) [994]

Optimal result

Integrand size = 27, antiderivative size = 1202 \[ \int \frac {(A+B x) (d+e x)^6}{\left (a+b x+c x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Output:

-2/5*(A*b-2*B*a-(-2*A*c+B*b)*x)*(e*x+d)^6/(-4*a*c+b^2)/(c*x^2+b*x+a)^(5/2) 
+8/15*(e*x+d)^5*(4*a*A*c*e-b^2*(3*A*e+2*B*d)+4*b*(A*c*d+B*a*e)-(b^2*B*e+4* 
b*c*(A*e+B*d)-4*c*(2*A*c*d+3*B*a*e))*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(3/2) 
-16/15*(e*x+d)^4*(b^3*e*(15*A*e+8*B*d)+24*a*c*e*(2*A*c*d+3*B*a*e)+4*A*b*c* 
(-11*a*e^2+4*c*d^2)-2*b^2*(14*A*c*d*e+13*B*a*e^2+4*B*c*d^2)-(3*b^3*B*e^2-2 
*b^2*c*e*(9*A*e+2*B*d)-8*c^2*(-5*A*a*e^2+4*A*c*d^2+6*B*a*d*e)+4*b*c*(8*A*c 
*d*e+B*a*e^2+4*B*c*d^2))*x)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)^(1/2)+8/15*e*(7*b 
^4*B*e^3+2*b^3*c*e^2*(-A*e+3*B*d)+12*b^2*c*e*(3*A*c*d*e-5*B*a*e^2+2*B*c*d^ 
2)-8*b*c^2*(3*A*a*e^3+12*A*c*d^2*e+15*B*a*d*e^2+4*B*c*d^3)+16*c^2*(6*a*B*e 
*(2*a*e^2+c*d^2)+A*c*d*(3*a*e^2+4*c*d^2)))*(e*x+d)^2*(c*x^2+b*x+a)^(1/2)/c 
^2/(-4*a*c+b^2)^3-16/15*e*(3*b^3*B*e^2-2*b^2*c*e*(9*A*e+2*B*d)-8*c^2*(-5*A 
*a*e^2+4*A*c*d^2+6*B*a*d*e)+4*b*c*(8*A*c*d*e+B*a*e^2+4*B*c*d^2))*(e*x+d)^3 
*(c*x^2+b*x+a)^(1/2)/c/(-4*a*c+b^2)^3+1/15*e*(105*b^6*B*e^5-30*b^5*c*e^4*( 
A*e+6*B*d)-56*b^4*B*c*e^3*(20*a*e^2+c*d^2)+16*b^3*c^2*e^3*(20*A*a*e^2+A*c* 
d^2+120*B*a*d*e)+256*c^3*(6*a*B*e*(-2*a^2*e^4+4*a*c*d^2*e^2+c^2*d^4)+A*c*d 
*(12*a^2*e^4+11*a*c*d^2*e^2+4*c^2*d^4))+48*b^2*c^2*e*(28*A*c^2*d^3*e+B*(77 
*a^2*e^4+10*a*c*d^2*e^2+16*c^2*d^4))-32*b*c^3*(3*A*e*(11*a^2*e^4+38*a*c*d^ 
2*e^2+24*c^2*d^4)+2*B*(99*a^2*d*e^4+64*a*c*d^3*e^2+8*c^2*d^5))-2*c*e*(35*b 
^5*B*e^4-2*b^4*c*e^3*(5*A*e+2*B*d)-8*b^3*c*e^2*(2*A*c*d*e+42*B*a*e^2+3*B*c 
*d^2)-16*b^2*c^2*e*(-6*A*a*e^3+15*A*c*d^2*e-6*B*a*d*e^2+10*B*c*d^3)-32*...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(3199\) vs. \(2(1202)=2404\).

Time = 22.90 (sec) , antiderivative size = 3199, normalized size of antiderivative = 2.66 \[ \int \frac {(A+B x) (d+e x)^6}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[((A + B*x)*(d + e*x)^6)/(a + b*x + c*x^2)^(7/2),x]
 

Output:

((a + b*x + c*x^2)^4*((B*e^6)/c^4 + ((-2*(-(A*b*c^6*d^6) + 2*a*B*c^6*d^6 - 
 6*a*b*B*c^5*d^5*e + 12*a*A*c^6*d^5*e + 15*a*b^2*B*c^4*d^4*e^2 - 15*a*A*b* 
c^5*d^4*e^2 - 30*a^2*B*c^5*d^4*e^2 - 20*a*b^3*B*c^3*d^3*e^3 + 20*a*A*b^2*c 
^4*d^3*e^3 + 60*a^2*b*B*c^4*d^3*e^3 - 40*a^2*A*c^5*d^3*e^3 + 15*a*b^4*B*c^ 
2*d^2*e^4 - 15*a*A*b^3*c^3*d^2*e^4 - 60*a^2*b^2*B*c^3*d^2*e^4 + 45*a^2*A*b 
*c^4*d^2*e^4 + 30*a^3*B*c^4*d^2*e^4 - 6*a*b^5*B*c*d*e^5 + 6*a*A*b^4*c^2*d* 
e^5 + 30*a^2*b^3*B*c^2*d*e^5 - 24*a^2*A*b^2*c^3*d*e^5 - 30*a^3*b*B*c^3*d*e 
^5 + 12*a^3*A*c^4*d*e^5 + a*b^6*B*e^6 - a*A*b^5*c*e^6 - 6*a^2*b^4*B*c*e^6 
+ 5*a^2*A*b^3*c^2*e^6 + 9*a^3*b^2*B*c^2*e^6 - 5*a^3*A*b*c^3*e^6 - 2*a^4*B* 
c^3*e^6))/(5*c^6*(-b^2 + 4*a*c)) - (2*(b*B*c^6*d^6 - 2*A*c^7*d^6 - 6*b^2*B 
*c^5*d^5*e + 6*A*b*c^6*d^5*e + 12*a*B*c^6*d^5*e + 15*b^3*B*c^4*d^4*e^2 - 1 
5*A*b^2*c^5*d^4*e^2 - 45*a*b*B*c^5*d^4*e^2 + 30*a*A*c^6*d^4*e^2 - 20*b^4*B 
*c^3*d^3*e^3 + 20*A*b^3*c^4*d^3*e^3 + 80*a*b^2*B*c^4*d^3*e^3 - 60*a*A*b*c^ 
5*d^3*e^3 - 40*a^2*B*c^5*d^3*e^3 + 15*b^5*B*c^2*d^2*e^4 - 15*A*b^4*c^3*d^2 
*e^4 - 75*a*b^3*B*c^3*d^2*e^4 + 60*a*A*b^2*c^4*d^2*e^4 + 75*a^2*b*B*c^4*d^ 
2*e^4 - 30*a^2*A*c^5*d^2*e^4 - 6*b^6*B*c*d*e^5 + 6*A*b^5*c^2*d*e^5 + 36*a* 
b^4*B*c^2*d*e^5 - 30*a*A*b^3*c^3*d*e^5 - 54*a^2*b^2*B*c^3*d*e^5 + 30*a^2*A 
*b*c^4*d*e^5 + 12*a^3*B*c^4*d*e^5 + b^7*B*e^6 - A*b^6*c*e^6 - 7*a*b^5*B*c* 
e^6 + 6*a*A*b^4*c^2*e^6 + 14*a^2*b^3*B*c^2*e^6 - 9*a^2*A*b^2*c^3*e^6 - 7*a 
^3*b*B*c^3*e^6 + 2*a^3*A*c^4*e^6)*x)/(5*c^6*(-b^2 + 4*a*c)))/(a + b*x +...
 

Rubi [A] (verified)

Time = 4.71 (sec) , antiderivative size = 1448, normalized size of antiderivative = 1.20, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1233, 27, 1233, 27, 1233, 27, 27, 1160, 1092, 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.

\(\displaystyle \int \frac {(A+B x) (d+e x)^6}{\left (a+b x+c x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {2 \int -\frac {(d+e x)^4 \left (16 A c^2 d^2-2 b c (4 B d+9 A e) d+2 b B e \left (\frac {3 b d}{2}-5 a e\right )+4 a c e (6 B d+5 A e)-e \left (7 B e b^2-2 c (B d+A e) b+4 c (A c d-6 a B e)\right ) x\right )}{2 \left (c x^2+b x+a\right )^{5/2}}dx}{5 c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^5 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^5 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^4 \left (16 A c^2 d^2-2 b c (4 B d+9 A e) d+b B e (3 b d-10 a e)+4 a c e (6 B d+5 A e)-e \left (7 B e b^2-2 c (B d+A e) b+4 c (A c d-6 a B e)\right ) x\right )}{\left (c x^2+b x+a\right )^{5/2}}dx}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {2 (d+e x)^5 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\frac {2 \int \frac {(d+e x)^2 \left (7 B d e^3 b^4-2 e^3 (A c d+21 a B e) b^3-12 c e \left (8 B c d^3+14 A c e d^2+3 a B e^2 d-a A e^3\right ) b^2+8 c \left (9 A c d e \left (4 c d^2+5 a e^2\right )+B \left (8 c^2 d^4+60 a c e^2 d^2+33 a^2 e^4\right )\right ) b-16 c^2 \left (6 a B d e \left (2 c d^2+7 a e^2\right )+A \left (8 c^2 d^4+18 a c e^2 d^2+15 a^2 e^4\right )\right )-e \left (35 B e^3 b^4-2 c e^2 (9 B d+5 A e) b^3-12 c e \left (2 B c d^2+A c e d+21 a B e^2\right ) b^2+8 c^2 \left (4 B c d^3+12 A c e d^2+21 a B e^2 d+9 a A e^3\right ) b-16 c^2 \left (6 a B e \left (c d^2-4 a e^2\right )+A c d \left (4 c d^2+9 a e^2\right )\right )\right ) x\right )}{2 \left (c x^2+b x+a\right )^{3/2}}dx}{3 c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^3 \left (2 b^2 c \left (-a A e^3+9 A c d^2 e+4 B c d^3\right )+x \left (-12 b^2 c e \left (4 a B e^2+A c d e+B c d^2\right )+8 b c^2 \left (3 a A e^3+9 a B d e^2+6 A c d^2 e+2 B c d^3\right )-16 c^2 \left (A c d \left (3 a e^2+2 c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )-2 b^3 c e^2 (A e+3 B d)+7 b^4 B e^3\right )-4 b c \left (4 A c d \left (3 a e^2+c d^2\right )+a B e \left (11 a e^2+9 c d^2\right )\right )+8 a c^2 e \left (5 a A e^2+12 a B d e+3 A c d^2\right )+b^3 (-B) \left (3 c d^2 e-7 a e^3\right )\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^5 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\frac {\int \frac {(d+e x)^2 \left (7 B d e^3 b^4-2 e^3 (A c d+21 a B e) b^3-12 c e \left (8 B c d^3+14 A c e d^2+3 a B e^2 d-a A e^3\right ) b^2+8 c \left (9 A c d e \left (4 c d^2+5 a e^2\right )+B \left (8 c^2 d^4+60 a c e^2 d^2+33 a^2 e^4\right )\right ) b-16 c^2 \left (6 a B d e \left (2 c d^2+7 a e^2\right )+A \left (8 c^2 d^4+18 a c e^2 d^2+15 a^2 e^4\right )\right )-e \left (35 B e^3 b^4-2 c e^2 (9 B d+5 A e) b^3-12 c e \left (2 B c d^2+A c e d+21 a B e^2\right ) b^2+8 c^2 \left (4 B c d^3+12 A c e d^2+21 a B e^2 d+9 a A e^3\right ) b-16 c^2 \left (6 a B e \left (c d^2-4 a e^2\right )+A c d \left (4 c d^2+9 a e^2\right )\right )\right ) x\right )}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^3 \left (2 b^2 c \left (-a A e^3+9 A c d^2 e+4 B c d^3\right )+x \left (-12 b^2 c e \left (4 a B e^2+A c d e+B c d^2\right )+8 b c^2 \left (3 a A e^3+9 a B d e^2+6 A c d^2 e+2 B c d^3\right )-16 c^2 \left (A c d \left (3 a e^2+2 c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )-2 b^3 c e^2 (A e+3 B d)+7 b^4 B e^3\right )-4 b c \left (4 A c d \left (3 a e^2+c d^2\right )+a B e \left (11 a e^2+9 c d^2\right )\right )+8 a c^2 e \left (5 a A e^2+12 a B d e+3 A c d^2\right )+b^3 (-B) \left (3 c d^2 e-7 a e^3\right )\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}}-\frac {\frac {2 \left (-B \left (3 c d^2 e-7 a e^3\right ) b^3+2 c \left (4 B c d^3+9 A c e d^2-a A e^3\right ) b^2-4 c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right ) b+8 a c^2 e \left (3 A c d^2+12 a B e d+5 a A e^2\right )+\left (7 B e^3 b^4-2 c e^2 (3 B d+A e) b^3-12 c e \left (B c d^2+A c e d+4 a B e^2\right ) b^2+8 c^2 \left (2 B c d^3+6 A c e d^2+9 a B e^2 d+3 a A e^3\right ) b-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right ) (d+e x)^3}{3 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}+\frac {\frac {2 \int -\frac {35 B d e^5 b^6-2 e^4 \left (16 B c d^2+5 A c e d-35 a B e^2\right ) b^5-4 c e^3 \left (6 B c d^3+2 A c e d^2+114 a B e^2 d+5 a A e^3\right ) b^4+16 c e^2 \left (3 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (2 c^2 d^4+3 a c e^2 d^2-6 a^2 e^4\right )\right ) b^3-16 c^2 e \left (2 A e \left (20 c^2 d^4+39 a c e^2 d^2-6 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3-129 a^2 e^4 d\right )\right ) b^2+32 c^2 e \left (3 a B e \left (4 c^2 d^4+18 a c e^2 d^2+19 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right ) b-960 a^3 c^3 e^5 (6 B d+A e)+e \left (105 B e^5 b^6-10 c e^4 (11 B d+3 A e) b^5-4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right ) b^4-16 c^2 e^2 \left (3 B c d^3+A c e d^2-78 a B e^2 d-20 a A e^3\right ) b^3+16 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c e^2 d^2+33 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (40 c^2 d^4+78 a c e^2 d^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3+141 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c e^2 d^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right )\right ) x}{2 \sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}-\frac {2 (d+e x) \left (7 B e^3 \left (c d^2-5 a e^2\right ) b^5-2 c e^3 \left (A c d^2-16 a B e d-5 a A e^2\right ) b^4-8 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c e^2 d^2-7 a^2 e^4\right )\right ) b^3+16 c^2 \left (6 A e \left (3 c^2 d^4+6 a c e^2 d^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c e^2 d^3-21 a^2 e^4 d\right )\right ) b^2-16 c^2 \left (2 A c d \left (4 c^2 d^4+19 a c e^2 d^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c e^2 d^2+57 a^2 e^4\right )\right ) b+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c e^2 d^2+15 a^2 e^4\right )\right )-\left (35 B e^5 b^6-2 c e^4 (23 B d+5 A e) b^5-4 c e^3 \left (5 B c d^2+A c e d+91 a B e^2\right ) b^4-8 c^2 e^2 \left (5 B c d^3+7 A c e d^2-63 a B e^2 d-13 a A e^3\right ) b^3+16 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c e^2 d^2+72 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (20 c^2 d^4+33 a c e^2 d^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c e^2 d^3+60 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c e^2 d^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c e^2 d^2+12 a^2 e^4\right )\right )\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}}{3 c \left (b^2-4 a c\right )}}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}}-\frac {\frac {2 \left (-B \left (3 c d^2 e-7 a e^3\right ) b^3+2 c \left (4 B c d^3+9 A c e d^2-a A e^3\right ) b^2-4 c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right ) b+8 a c^2 e \left (3 A c d^2+12 a B e d+5 a A e^2\right )+\left (7 B e^3 b^4-2 c e^2 (3 B d+A e) b^3-12 c e \left (B c d^2+A c e d+4 a B e^2\right ) b^2+8 c^2 \left (2 B c d^3+6 A c e d^2+9 a B e^2 d+3 a A e^3\right ) b-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right ) (d+e x)^3}{3 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}+\frac {-\frac {2 (d+e x) \left (7 B e^3 \left (c d^2-5 a e^2\right ) b^5-2 c e^3 \left (A c d^2-16 a B e d-5 a A e^2\right ) b^4-8 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c e^2 d^2-7 a^2 e^4\right )\right ) b^3+16 c^2 \left (6 A e \left (3 c^2 d^4+6 a c e^2 d^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c e^2 d^3-21 a^2 e^4 d\right )\right ) b^2-16 c^2 \left (2 A c d \left (4 c^2 d^4+19 a c e^2 d^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c e^2 d^2+57 a^2 e^4\right )\right ) b+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c e^2 d^2+15 a^2 e^4\right )\right )-\left (35 B e^5 b^6-2 c e^4 (23 B d+5 A e) b^5-4 c e^3 \left (5 B c d^2+A c e d+91 a B e^2\right ) b^4-8 c^2 e^2 \left (5 B c d^3+7 A c e d^2-63 a B e^2 d-13 a A e^3\right ) b^3+16 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c e^2 d^2+72 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (20 c^2 d^4+33 a c e^2 d^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c e^2 d^3+60 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c e^2 d^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c e^2 d^2+12 a^2 e^4\right )\right )\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {\int \frac {e \left (35 B d e^4 b^6-2 e^3 \left (16 B c d^2+5 A c e d-35 a B e^2\right ) b^5-4 c e^2 \left (6 B c d^3+2 A c e d^2+114 a B e^2 d+5 a A e^3\right ) b^4+16 c e \left (3 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (2 c^2 d^4+3 a c e^2 d^2-6 a^2 e^4\right )\right ) b^3-16 c^2 \left (2 A e \left (20 c^2 d^4+39 a c e^2 d^2-6 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3-129 a^2 e^4 d\right )\right ) b^2+32 c^2 \left (3 a B e \left (4 c^2 d^4+18 a c e^2 d^2+19 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right ) b-960 a^3 c^3 e^4 (6 B d+A e)+\left (105 B e^5 b^6-10 c e^4 (11 B d+3 A e) b^5-4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right ) b^4-16 c^2 e^2 \left (3 B c d^3+A c e d^2-78 a B e^2 d-20 a A e^3\right ) b^3+16 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c e^2 d^2+33 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (40 c^2 d^4+78 a c e^2 d^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3+141 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c e^2 d^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right )\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}}{3 c \left (b^2-4 a c\right )}}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}}-\frac {\frac {2 \left (-B \left (3 c d^2 e-7 a e^3\right ) b^3+2 c \left (4 B c d^3+9 A c e d^2-a A e^3\right ) b^2-4 c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right ) b+8 a c^2 e \left (3 A c d^2+12 a B e d+5 a A e^2\right )+\left (7 B e^3 b^4-2 c e^2 (3 B d+A e) b^3-12 c e \left (B c d^2+A c e d+4 a B e^2\right ) b^2+8 c^2 \left (2 B c d^3+6 A c e d^2+9 a B e^2 d+3 a A e^3\right ) b-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right ) (d+e x)^3}{3 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}+\frac {-\frac {2 (d+e x) \left (7 B e^3 \left (c d^2-5 a e^2\right ) b^5-2 c e^3 \left (A c d^2-16 a B e d-5 a A e^2\right ) b^4-8 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c e^2 d^2-7 a^2 e^4\right )\right ) b^3+16 c^2 \left (6 A e \left (3 c^2 d^4+6 a c e^2 d^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c e^2 d^3-21 a^2 e^4 d\right )\right ) b^2-16 c^2 \left (2 A c d \left (4 c^2 d^4+19 a c e^2 d^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c e^2 d^2+57 a^2 e^4\right )\right ) b+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c e^2 d^2+15 a^2 e^4\right )\right )-\left (35 B e^5 b^6-2 c e^4 (23 B d+5 A e) b^5-4 c e^3 \left (5 B c d^2+A c e d+91 a B e^2\right ) b^4-8 c^2 e^2 \left (5 B c d^3+7 A c e d^2-63 a B e^2 d-13 a A e^3\right ) b^3+16 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c e^2 d^2+72 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (20 c^2 d^4+33 a c e^2 d^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c e^2 d^3+60 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c e^2 d^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c e^2 d^2+12 a^2 e^4\right )\right )\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {e \int \frac {35 B d e^4 b^6-2 e^3 \left (16 B c d^2+5 A c e d-35 a B e^2\right ) b^5-4 c e^2 \left (6 B c d^3+2 A c e d^2+114 a B e^2 d+5 a A e^3\right ) b^4+16 c e \left (3 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (2 c^2 d^4+3 a c e^2 d^2-6 a^2 e^4\right )\right ) b^3-16 c^2 \left (2 A e \left (20 c^2 d^4+39 a c e^2 d^2-6 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3-129 a^2 e^4 d\right )\right ) b^2+32 c^2 \left (3 a B e \left (4 c^2 d^4+18 a c e^2 d^2+19 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right ) b-960 a^3 c^3 e^4 (6 B d+A e)+\left (105 B e^5 b^6-10 c e^4 (11 B d+3 A e) b^5-4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right ) b^4-16 c^2 e^2 \left (3 B c d^3+A c e d^2-78 a B e^2 d-20 a A e^3\right ) b^3+16 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c e^2 d^2+33 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (40 c^2 d^4+78 a c e^2 d^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3+141 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c e^2 d^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right )\right ) x}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}}{3 c \left (b^2-4 a c\right )}}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}}-\frac {\frac {2 \left (-B \left (3 c d^2 e-7 a e^3\right ) b^3+2 c \left (4 B c d^3+9 A c e d^2-a A e^3\right ) b^2-4 c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right ) b+8 a c^2 e \left (3 A c d^2+12 a B e d+5 a A e^2\right )+\left (7 B e^3 b^4-2 c e^2 (3 B d+A e) b^3-12 c e \left (B c d^2+A c e d+4 a B e^2\right ) b^2+8 c^2 \left (2 B c d^3+6 A c e d^2+9 a B e^2 d+3 a A e^3\right ) b-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right ) (d+e x)^3}{3 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}+\frac {-\frac {2 (d+e x) \left (7 B e^3 \left (c d^2-5 a e^2\right ) b^5-2 c e^3 \left (A c d^2-16 a B e d-5 a A e^2\right ) b^4-8 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c e^2 d^2-7 a^2 e^4\right )\right ) b^3+16 c^2 \left (6 A e \left (3 c^2 d^4+6 a c e^2 d^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c e^2 d^3-21 a^2 e^4 d\right )\right ) b^2-16 c^2 \left (2 A c d \left (4 c^2 d^4+19 a c e^2 d^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c e^2 d^2+57 a^2 e^4\right )\right ) b+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c e^2 d^2+15 a^2 e^4\right )\right )-\left (35 B e^5 b^6-2 c e^4 (23 B d+5 A e) b^5-4 c e^3 \left (5 B c d^2+A c e d+91 a B e^2\right ) b^4-8 c^2 e^2 \left (5 B c d^3+7 A c e d^2-63 a B e^2 d-13 a A e^3\right ) b^3+16 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c e^2 d^2+72 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (20 c^2 d^4+33 a c e^2 d^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c e^2 d^3+60 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c e^2 d^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c e^2 d^2+12 a^2 e^4\right )\right )\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {e \left (\frac {15 \left (b^2-4 a c\right )^3 (12 B c d-7 b B e+2 A c e) \int \frac {1}{\sqrt {c x^2+b x+a}}dx e^4}{2 c}+\frac {\left (105 B e^5 b^6-10 c e^4 (11 B d+3 A e) b^5-4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right ) b^4-16 c^2 e^2 \left (3 B c d^3+A c e d^2-78 a B e^2 d-20 a A e^3\right ) b^3+16 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c e^2 d^2+33 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (40 c^2 d^4+78 a c e^2 d^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3+141 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c e^2 d^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right )\right ) \sqrt {c x^2+b x+a}}{c}\right )}{c \left (b^2-4 a c\right )}}{3 c \left (b^2-4 a c\right )}}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}}-\frac {\frac {2 \left (-B \left (3 c d^2 e-7 a e^3\right ) b^3+2 c \left (4 B c d^3+9 A c e d^2-a A e^3\right ) b^2-4 c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right ) b+8 a c^2 e \left (3 A c d^2+12 a B e d+5 a A e^2\right )+\left (7 B e^3 b^4-2 c e^2 (3 B d+A e) b^3-12 c e \left (B c d^2+A c e d+4 a B e^2\right ) b^2+8 c^2 \left (2 B c d^3+6 A c e d^2+9 a B e^2 d+3 a A e^3\right ) b-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right ) (d+e x)^3}{3 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}+\frac {-\frac {2 (d+e x) \left (7 B e^3 \left (c d^2-5 a e^2\right ) b^5-2 c e^3 \left (A c d^2-16 a B e d-5 a A e^2\right ) b^4-8 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c e^2 d^2-7 a^2 e^4\right )\right ) b^3+16 c^2 \left (6 A e \left (3 c^2 d^4+6 a c e^2 d^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c e^2 d^3-21 a^2 e^4 d\right )\right ) b^2-16 c^2 \left (2 A c d \left (4 c^2 d^4+19 a c e^2 d^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c e^2 d^2+57 a^2 e^4\right )\right ) b+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c e^2 d^2+15 a^2 e^4\right )\right )-\left (35 B e^5 b^6-2 c e^4 (23 B d+5 A e) b^5-4 c e^3 \left (5 B c d^2+A c e d+91 a B e^2\right ) b^4-8 c^2 e^2 \left (5 B c d^3+7 A c e d^2-63 a B e^2 d-13 a A e^3\right ) b^3+16 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c e^2 d^2+72 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (20 c^2 d^4+33 a c e^2 d^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c e^2 d^3+60 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c e^2 d^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c e^2 d^2+12 a^2 e^4\right )\right )\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {e \left (\frac {15 \left (b^2-4 a c\right )^3 (12 B c d-7 b B e+2 A c e) \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^4}{c}+\frac {\left (105 B e^5 b^6-10 c e^4 (11 B d+3 A e) b^5-4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right ) b^4-16 c^2 e^2 \left (3 B c d^3+A c e d^2-78 a B e^2 d-20 a A e^3\right ) b^3+16 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c e^2 d^2+33 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (40 c^2 d^4+78 a c e^2 d^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3+141 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c e^2 d^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right )\right ) \sqrt {c x^2+b x+a}}{c}\right )}{c \left (b^2-4 a c\right )}}{3 c \left (b^2-4 a c\right )}}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}}-\frac {\frac {2 \left (-B \left (3 c d^2 e-7 a e^3\right ) b^3+2 c \left (4 B c d^3+9 A c e d^2-a A e^3\right ) b^2-4 c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right ) b+8 a c^2 e \left (3 A c d^2+12 a B e d+5 a A e^2\right )+\left (7 B e^3 b^4-2 c e^2 (3 B d+A e) b^3-12 c e \left (B c d^2+A c e d+4 a B e^2\right ) b^2+8 c^2 \left (2 B c d^3+6 A c e d^2+9 a B e^2 d+3 a A e^3\right ) b-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right ) (d+e x)^3}{3 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}+\frac {-\frac {2 (d+e x) \left (7 B e^3 \left (c d^2-5 a e^2\right ) b^5-2 c e^3 \left (A c d^2-16 a B e d-5 a A e^2\right ) b^4-8 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c e^2 d^2-7 a^2 e^4\right )\right ) b^3+16 c^2 \left (6 A e \left (3 c^2 d^4+6 a c e^2 d^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c e^2 d^3-21 a^2 e^4 d\right )\right ) b^2-16 c^2 \left (2 A c d \left (4 c^2 d^4+19 a c e^2 d^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c e^2 d^2+57 a^2 e^4\right )\right ) b+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c e^2 d^2+15 a^2 e^4\right )\right )-\left (35 B e^5 b^6-2 c e^4 (23 B d+5 A e) b^5-4 c e^3 \left (5 B c d^2+A c e d+91 a B e^2\right ) b^4-8 c^2 e^2 \left (5 B c d^3+7 A c e d^2-63 a B e^2 d-13 a A e^3\right ) b^3+16 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c e^2 d^2+72 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (20 c^2 d^4+33 a c e^2 d^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c e^2 d^3+60 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c e^2 d^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c e^2 d^2+12 a^2 e^4\right )\right )\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}-\frac {e \left (\frac {15 \left (b^2-4 a c\right )^3 (12 B c d-7 b B e+2 A c e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^4}{2 c^{3/2}}+\frac {\left (105 B e^5 b^6-10 c e^4 (11 B d+3 A e) b^5-4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right ) b^4-16 c^2 e^2 \left (3 B c d^3+A c e d^2-78 a B e^2 d-20 a A e^3\right ) b^3+16 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c e^2 d^2+33 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (40 c^2 d^4+78 a c e^2 d^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3+141 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c e^2 d^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right )\right ) \sqrt {c x^2+b x+a}}{c}\right )}{c \left (b^2-4 a c\right )}}{3 c \left (b^2-4 a c\right )}}{5 c \left (b^2-4 a c\right )}\)

Input:

Int[((A + B*x)*(d + e*x)^6)/(a + b*x + c*x^2)^(7/2),x]
 

Output:

(2*(d + e*x)^5*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B* 
d + A*e) + 2*c*(A*c*d - a*B*e))*x))/(5*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^( 
5/2)) - ((2*(d + e*x)^3*(8*a*c^2*e*(3*A*c*d^2 + 12*a*B*d*e + 5*a*A*e^2) - 
b^3*B*(3*c*d^2*e - 7*a*e^3) + 2*b^2*c*(4*B*c*d^3 + 9*A*c*d^2*e - a*A*e^3) 
- 4*b*c*(4*A*c*d*(c*d^2 + 3*a*e^2) + a*B*e*(9*c*d^2 + 11*a*e^2)) + (7*b^4* 
B*e^3 - 2*b^3*c*e^2*(3*B*d + A*e) - 12*b^2*c*e*(B*c*d^2 + A*c*d*e + 4*a*B* 
e^2) + 8*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 9*a*B*d*e^2 + 3*a*A*e^3) - 16*c^ 
2*(3*a*B*e*(c*d^2 - a*e^2) + A*c*d*(2*c*d^2 + 3*a*e^2)))*x))/(3*c*(b^2 - 4 
*a*c)*(a + b*x + c*x^2)^(3/2)) + ((-2*(d + e*x)*(7*b^5*B*e^3*(c*d^2 - 5*a* 
e^2) - 2*b^4*c*e^3*(A*c*d^2 - 16*a*B*d*e - 5*a*A*e^2) - 8*b^3*c*e*(A*c*d*e 
*(21*c*d^2 - a*e^2) + 6*B*(2*c^2*d^4 + a*c*d^2*e^2 - 7*a^2*e^4)) + 32*a*c^ 
3*e*(6*a*B*d*e*(c*d^2 + 11*a*e^2) + A*(4*c^2*d^4 + 9*a*c*d^2*e^2 + 15*a^2* 
e^4)) - 16*b*c^2*(2*A*c*d*(4*c^2*d^4 + 19*a*c*d^2*e^2 + 21*a^2*e^4) + a*B* 
e*(16*c^2*d^4 + 75*a*c*d^2*e^2 + 57*a^2*e^4)) + 16*b^2*c^2*(6*A*e*(3*c^2*d 
^4 + 6*a*c*d^2*e^2 - a^2*e^4) + B*(4*c^2*d^5 + 37*a*c*d^3*e^2 - 21*a^2*d*e 
^4)) - (35*b^6*B*e^5 - 2*b^5*c*e^4*(23*B*d + 5*A*e) - 4*b^4*c*e^3*(5*B*c*d 
^2 + A*c*d*e + 91*a*B*e^2) - 8*b^3*c^2*e^2*(5*B*c*d^3 + 7*A*c*d^2*e - 63*a 
*B*d*e^2 - 13*a*A*e^3) + 64*c^3*(6*a*B*e*(c^2*d^4 + 4*a*c*d^2*e^2 - 2*a^2* 
e^4) + A*c*d*(4*c^2*d^4 + 11*a*c*d^2*e^2 + 12*a^2*e^4)) + 16*b^2*c^2*e*(A* 
c*d*e*(29*c*d^2 + 9*a*e^2) + B*(14*c^2*d^4 + 21*a*c*d^2*e^2 + 72*a^2*e^...
 

Defintions of rubi rules used

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 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1233

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) 
^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(7878\) vs. \(2(1168)=2336\).

Time = 4.38 (sec) , antiderivative size = 7879, normalized size of antiderivative = 6.55

method	result	size
default	\(\text {Expression too large to display}\)	\(7879\)
risch	\(\text {Expression too large to display}\)	\(141428\)

Input:

int((B*x+A)*(e*x+d)^6/(c*x^2+b*x+a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3992 vs. \(2 (1168) = 2336\).

Time = 104.39 (sec) , antiderivative size = 7987, normalized size of antiderivative = 6.64 \[ \int \frac {(A+B x) (d+e x)^6}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^6/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^6}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(e*x+d)**6/(c*x**2+b*x+a)**(7/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) (d+e x)^6}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(e*x+d)^6/(c*x^2+b*x+a)^(7/2),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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3496 vs. \(2 (1168) = 2336\).

Time = 0.43 (sec) , antiderivative size = 3496, normalized size of antiderivative = 2.91 \[ \int \frac {(A+B x) (d+e x)^6}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^6/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")
 

Output:

1/15*((((((15*(B*b^6*c^3*e^6 - 12*B*a*b^4*c^4*e^6 + 48*B*a^2*b^2*c^5*e^6 - 
 64*B*a^3*c^6*e^6)*x/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7 
) + (256*B*b*c^8*d^6 - 512*A*c^9*d^6 - 576*B*b^2*c^7*d^5*e - 768*B*a*c^8*d 
^5*e + 1536*A*b*c^8*d^5*e + 240*B*b^3*c^6*d^4*e^2 + 2880*B*a*b*c^7*d^4*e^2 
 - 1440*A*b^2*c^7*d^4*e^2 - 1920*A*a*c^8*d^4*e^2 + 80*B*b^4*c^5*d^3*e^3 - 
1920*B*a*b^2*c^6*d^3*e^3 + 320*A*b^3*c^6*d^3*e^3 - 3840*B*a^2*c^7*d^3*e^3 
+ 3840*A*a*b*c^7*d^3*e^3 + 90*B*b^5*c^4*d^2*e^4 - 1200*B*a*b^3*c^5*d^2*e^4 
 + 60*A*b^4*c^5*d^2*e^4 + 7200*B*a^2*b*c^6*d^2*e^4 - 1440*A*a*b^2*c^6*d^2* 
e^4 - 2880*A*a^2*c^7*d^2*e^4 - 276*B*b^6*c^3*d*e^5 + 3096*B*a*b^4*c^4*d*e^ 
5 + 36*A*b^5*c^4*d*e^5 - 10944*B*a^2*b^2*c^5*d*e^5 - 480*A*a*b^3*c^5*d*e^5 
 + 8832*B*a^3*c^6*d*e^5 + 2880*A*a^2*b*c^6*d*e^5 + 161*B*b^7*c^2*e^6 - 184 
2*B*a*b^5*c^3*e^6 - 46*A*b^6*c^3*e^6 + 6864*B*a^2*b^3*c^4*e^6 + 516*A*a*b^ 
4*c^4*e^6 - 8032*B*a^3*b*c^5*e^6 - 1824*A*a^2*b^2*c^5*e^6 + 1472*A*a^3*c^6 
*e^6)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7))*x + 5*(128*B 
*b^2*c^7*d^6 - 256*A*b*c^8*d^6 - 288*B*b^3*c^6*d^5*e - 384*B*a*b*c^7*d^5*e 
 + 768*A*b^2*c^7*d^5*e + 120*B*b^4*c^5*d^4*e^2 + 1440*B*a*b^2*c^6*d^4*e^2 
- 720*A*b^3*c^6*d^4*e^2 - 960*A*a*b*c^7*d^4*e^2 + 40*B*b^5*c^4*d^3*e^3 - 9 
60*B*a*b^3*c^5*d^3*e^3 + 160*A*b^4*c^5*d^3*e^3 - 1920*B*a^2*b*c^6*d^3*e^3 
+ 1920*A*a*b^2*c^6*d^3*e^3 - 60*B*a*b^4*c^4*d^2*e^4 + 30*A*b^5*c^4*d^2*e^4 
 + 1440*B*a^2*b^2*c^5*d^2*e^4 - 720*A*a*b^3*c^5*d^2*e^4 + 2880*B*a^3*c^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^6}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^6}{{\left (c\,x^2+b\,x+a\right )}^{7/2}} \,d x \] Input:

int(((A + B*x)*(d + e*x)^6)/(a + b*x + c*x^2)^(7/2),x)
 

Output:

int(((A + B*x)*(d + e*x)^6)/(a + b*x + c*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {(A+B x) (d+e x)^6}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\int \frac {\left (B x +A \right ) \left (e x +d \right )^{6}}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}d x \] Input:

int((B*x+A)*(e*x+d)^6/(c*x^2+b*x+a)^(7/2),x)
 

Output:

int((B*x+A)*(e*x+d)^6/(c*x^2+b*x+a)^(7/2),x)