3.25.88 \(\int \frac {(A+B x) (d+e x)^5}{(a+b x+c x^2)^{7/2}} \, dx\) [2488]

3.25.88.1 Optimal result

Integrand size = 27, antiderivative size = 942 \[ \int \frac {(A+B x) (d+e x)^5}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 (d+e x)^2 \left (b^3 B e \left (3 c d^2-5 a e^2\right )-4 b^2 c d \left (2 B c d^2+4 A c d e+a B e^2\right )-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+4 b c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right )+\left (2 b^3 B c d e^2-5 b^4 B e^3+2 b^2 c e \left (7 B c d^2+8 A c d e+19 a B e^2\right )-8 b c^2 \left (2 B c d^3+6 A c d^2 e+7 a B d e^2+2 a A e^3\right )+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 b^4 B c^2 d^3 e^2+5 b^5 B e^3 \left (c d^2-3 a e^2\right )+32 b^2 c^3 d^2 \left (2 B c d^3+8 A c d^2 e+17 a B d e^2+16 a A e^3\right )+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )-8 b^3 c e \left (16 A c^2 d^3 e+B \left (11 c^2 d^4+7 a c d^2 e^2-20 a^2 e^4\right )\right )-16 b c^2 \left (8 A c d \left (c^2 d^4+6 a c d^2 e^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c d^2 e^2+33 a^2 e^4\right )\right )+\left (10 b^5 B c d e^4-15 b^6 B e^5+2 b^4 B c e^3 \left (3 c d^2+85 a e^2\right )+16 b^3 c^2 d e^2 \left (6 B c d^2+8 A c d e-7 a B e^2\right )-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c d^2 e^2-3 a^2 e^4\right )\right )-16 b^2 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c d^2 e^2+39 a^2 e^4\right )\right )+32 b c^3 \left (4 A e \left (5 c^2 d^4+6 a c d^2 e^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c d^3 e^2+29 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {B e^5 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{7/2}} \]

output

2/5*(e*x+d)^4*(2*a*c*(A*e+B*d)-b*(A*c*d+B*a*e)-(b^2*B*e-b*c*(A*e+B*d)+2*c* 
(A*c*d-B*a*e))*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+a)^(5/2)+2/15*(e*x+d)^2*(b^3*B 
*e*(-5*a*e^2+3*c*d^2)-4*b^2*c*d*(4*A*c*d*e+B*a*e^2+2*B*c*d^2)-16*a*c^2*e*( 
5*B*a*d*e+2*A*(a*e^2+c*d^2))+4*b*c*(9*a*B*e*(a*e^2+c*d^2)+4*A*c*d*(3*a*e^2 
+c*d^2))+(2*b^3*B*c*d*e^2-5*b^4*B*e^3+2*b^2*c*e*(8*A*c*d*e+19*B*a*e^2+7*B* 
c*d^2)-8*b*c^2*(2*A*a*e^3+6*A*c*d^2*e+7*B*a*d*e^2+2*B*c*d^3)+8*c^2*(5*a*B* 
e*(-a*e^2+c*d^2)+4*A*c*d*(a*e^2+c*d^2)))*x)/c^2/(-4*a*c+b^2)^2/(c*x^2+b*x+ 
a)^(3/2)+B*e^5*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)+ 
2/15*(4*b^4*B*c^2*d^3*e^2+5*b^5*B*e^3*(-3*a*e^2+c*d^2)+32*b^2*c^3*d^2*(16* 
A*a*e^3+8*A*c*d^2*e+17*B*a*d*e^2+2*B*c*d^3)+64*a*c^3*e*(4*A*(a*e^2+c*d^2)^ 
2+5*a*B*d*e*(4*a*e^2+c*d^2))-8*b^3*c*e*(16*A*c^2*d^3*e+B*(-20*a^2*e^4+7*a* 
c*d^2*e^2+11*c^2*d^4))-16*b*c^2*(8*A*c*d*(5*a^2*e^4+6*a*c*d^2*e^2+c^2*d^4) 
+a*B*e*(33*a^2*e^4+71*a*c*d^2*e^2+18*c^2*d^4))+(10*b^5*B*c*d*e^4-15*b^6*B* 
e^5+2*b^4*B*c*e^3*(85*a*e^2+3*c*d^2)+16*b^3*c^2*d*e^2*(8*A*c*d*e-7*B*a*e^2 
+6*B*c*d^2)-32*c^3*(8*A*c*d*(a*e^2+c*d^2)^2+5*a*B*e*(-3*a^2*e^4+5*a*c*d^2* 
e^2+2*c^2*d^4))-16*b^2*c^2*e*(16*A*c*d*e*(a*e^2+2*c*d^2)+B*(39*a^2*e^4+29* 
a*c*d^2*e^2+15*c^2*d^4))+32*b*c^3*(4*A*e*(a^2*e^4+6*a*c*d^2*e^2+5*c^2*d^4) 
+B*(29*a^2*d*e^4+28*a*c*d^3*e^2+4*c^2*d^5)))*x)/c^3/(-4*a*c+b^2)^3/(c*x^2+ 
b*x+a)^(1/2)
 

3.25.88.2 Mathematica [A] (verified)

Time = 17.57 (sec) , antiderivative size = 1610, normalized size of antiderivative = 1.71 \[ \int \frac {(A+B x) (d+e x)^5}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {\frac {2 \sqrt {c} \left (A c^3 \left (10 b^4 (a e-c d x) \left (d^4+20 d^3 e x-90 d^2 e^2 x^2+20 d e^3 x^3+e^4 x^4\right )+b^5 \left (3 d^5+25 d^4 e x+150 d^3 e^2 x^2-150 d^2 e^3 x^3-25 d e^4 x^4-3 e^5 x^5\right )+40 b^3 (d-e x) \left (2 a^2 e^2 \left (d^2-14 d e x+e^2 x^2\right )+2 c^2 d^2 x^2 \left (d^2-14 d e x+e^2 x^2\right )-a c (d-e x)^2 \left (d^2+18 d e x+e^2 x^2\right )\right )+80 b (d-e x) \left (8 a^4 e^4+8 c^4 d^4 x^4+3 a^2 c^2 (d-e x)^4+4 a^3 c e^2 \left (3 d^2-2 d e x+3 e^2 x^2\right )+4 a c^3 d^2 x^2 \left (3 d^2-2 d e x+3 e^2 x^2\right )\right )+80 b^2 \left (-2 a^3 e^3 \left (3 d^2-10 d e x+3 e^2 x^2\right )+2 c^3 d^3 x^3 \left (3 d^2-10 d e x+3 e^2 x^2\right )-3 a^2 c e \left (d^4-10 d^3 e x+10 d^2 e^2 x^2-10 d e^3 x^3+e^4 x^4\right )+3 a c^2 d x \left (d^4-10 d^3 e x+10 d^2 e^2 x^2-10 d e^3 x^3+e^4 x^4\right )\right )+32 \left (-8 a^5 e^5+8 c^5 d^5 x^5-20 a^4 c e^3 \left (d^2+e^2 x^2\right )+20 a c^4 d^3 x^3 \left (d^2+e^2 x^2\right )-5 a^3 c^2 e \left (3 d^4+10 d^2 e^2 x^2+3 e^4 x^4\right )+5 a^2 c^3 d x \left (3 d^4+10 d^2 e^2 x^2+3 e^4 x^4\right )\right )\right )+B \left (-16 a^5 c^2 e^4 (80 c d-33 b e+30 c e x)-80 a^4 c e^2 \left (2 b^3 e^3-21 b^2 c e^3 x+b c^2 e \left (-16 d^2+40 d e x-3 e^2 x^2\right )+2 c^3 \left (4 d^3+20 d e^2 x^2+7 e^3 x^3\right )\right )+b x \left (15 b^7 e^5 x^2+35 b^6 c e^5 x^3-128 c^7 d^5 x^4+23 b^5 c^2 e^5 x^4+80 b c^6 d^4 x^3 (-4 d+3 e x)-40 b^2 c^5 d^3 x^2 \left (6 d^2-15 d e x+2 e^2 x^2\right )-10 b^3 c^4 d^2 x \left (4 d^3-45 d^2 e x+20 d e^2 x^2+2 e^3 x^3\right )+5 b^4 c^3 d \left (d^4+15 d^3 e x-30 d^2 e^2 x^2-10 d e^3 x^3-3 e^4 x^4\right )\right )+a \left (45 b^7 e^5 x^2-100 b^6 c e^5 x^3-375 b^5 c^2 e^5 x^4+320 c^7 d^4 e x^5-160 b c^6 d^3 x^3 \left (2 d^2-5 d e x+6 e^2 x^2\right )+240 b^2 c^5 d^2 x^2 \left (-2 d^3+5 d^2 e x-10 d e^2 x^2+2 e^3 x^3\right )+40 b^3 c^4 d x \left (-3 d^4+25 d^3 e x-50 d^2 e^2 x^2+30 d e^3 x^3+5 e^4 x^4\right )+2 b^4 c^3 \left (d^5+50 d^4 e x-450 d^3 e^2 x^2+200 d^2 e^3 x^3+25 d e^4 x^4-129 e^5 x^5\right )\right )+a^2 \left (45 b^6 e^5 x-465 b^5 c e^5 x^2-150 b^4 c^2 e^5 x^3+160 c^6 d^2 e x^3 \left (5 d^2+6 e^2 x^2\right )-240 b c^5 d x \left (d^4-5 d^3 e x+10 d^2 e^2 x^2-10 d e^3 x^3+5 e^4 x^4\right )+40 b^3 c^3 e \left (d^4-30 d^3 e x+60 d^2 e^2 x^2-10 d e^3 x^3+30 e^4 x^4\right )-48 b^2 c^4 \left (d^5-25 d^4 e x+50 d^3 e^2 x^2-100 d^2 e^3 x^3+25 d e^4 x^4-19 e^5 x^5\right )\right )-a^3 \left (-15 b^5 e^5+490 b^4 c e^5 x-1400 b^3 c^2 e^5 x^2+80 b^2 c^3 e^2 \left (6 d^3-40 d^2 e x+30 d e^2 x^2-27 e^3 x^3\right )+80 b c^4 e \left (-6 d^4+20 d^3 e x-40 d^2 e^2 x^2+60 d e^3 x^3+5 e^4 x^4\right )+32 c^5 \left (3 d^5+50 d^3 e^2 x^2+75 d e^4 x^4+23 e^5 x^5\right )\right )\right )\right )}{(a+x (b+c x))^{5/2}}-15 B \left (b^2-4 a c\right )^3 e^5 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{15 c^{7/2} \left (-b^2+4 a c\right )^3} \]

input

Integrate[((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2),x]
 

output

((2*Sqrt[c]*(A*c^3*(10*b^4*(a*e - c*d*x)*(d^4 + 20*d^3*e*x - 90*d^2*e^2*x^ 
2 + 20*d*e^3*x^3 + e^4*x^4) + b^5*(3*d^5 + 25*d^4*e*x + 150*d^3*e^2*x^2 - 
150*d^2*e^3*x^3 - 25*d*e^4*x^4 - 3*e^5*x^5) + 40*b^3*(d - e*x)*(2*a^2*e^2* 
(d^2 - 14*d*e*x + e^2*x^2) + 2*c^2*d^2*x^2*(d^2 - 14*d*e*x + e^2*x^2) - a* 
c*(d - e*x)^2*(d^2 + 18*d*e*x + e^2*x^2)) + 80*b*(d - e*x)*(8*a^4*e^4 + 8* 
c^4*d^4*x^4 + 3*a^2*c^2*(d - e*x)^4 + 4*a^3*c*e^2*(3*d^2 - 2*d*e*x + 3*e^2 
*x^2) + 4*a*c^3*d^2*x^2*(3*d^2 - 2*d*e*x + 3*e^2*x^2)) + 80*b^2*(-2*a^3*e^ 
3*(3*d^2 - 10*d*e*x + 3*e^2*x^2) + 2*c^3*d^3*x^3*(3*d^2 - 10*d*e*x + 3*e^2 
*x^2) - 3*a^2*c*e*(d^4 - 10*d^3*e*x + 10*d^2*e^2*x^2 - 10*d*e^3*x^3 + e^4* 
x^4) + 3*a*c^2*d*x*(d^4 - 10*d^3*e*x + 10*d^2*e^2*x^2 - 10*d*e^3*x^3 + e^4 
*x^4)) + 32*(-8*a^5*e^5 + 8*c^5*d^5*x^5 - 20*a^4*c*e^3*(d^2 + e^2*x^2) + 2 
0*a*c^4*d^3*x^3*(d^2 + e^2*x^2) - 5*a^3*c^2*e*(3*d^4 + 10*d^2*e^2*x^2 + 3* 
e^4*x^4) + 5*a^2*c^3*d*x*(3*d^4 + 10*d^2*e^2*x^2 + 3*e^4*x^4))) + B*(-16*a 
^5*c^2*e^4*(80*c*d - 33*b*e + 30*c*e*x) - 80*a^4*c*e^2*(2*b^3*e^3 - 21*b^2 
*c*e^3*x + b*c^2*e*(-16*d^2 + 40*d*e*x - 3*e^2*x^2) + 2*c^3*(4*d^3 + 20*d* 
e^2*x^2 + 7*e^3*x^3)) + b*x*(15*b^7*e^5*x^2 + 35*b^6*c*e^5*x^3 - 128*c^7*d 
^5*x^4 + 23*b^5*c^2*e^5*x^4 + 80*b*c^6*d^4*x^3*(-4*d + 3*e*x) - 40*b^2*c^5 
*d^3*x^2*(6*d^2 - 15*d*e*x + 2*e^2*x^2) - 10*b^3*c^4*d^2*x*(4*d^3 - 45*d^2 
*e*x + 20*d*e^2*x^2 + 2*e^3*x^3) + 5*b^4*c^3*d*(d^4 + 15*d^3*e*x - 30*d^2* 
e^2*x^2 - 10*d*e^3*x^3 - 3*e^4*x^4)) + a*(45*b^7*e^5*x^2 - 100*b^6*c*e^...
 

3.25.88.3 Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 989, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1233, 27, 1233, 27, 1224, 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.

\(\displaystyle \int \frac {(A+B x) (d+e x)^5}{\left (a+b x+c x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {2 \int -\frac {(d+e x)^3 \left (3 B d e b^2-8 \left (B c d^2+2 A c e d+a B e^2\right ) b+4 c \left (5 a B d e+4 A \left (c d^2+a e^2\right )\right )-5 B \left (b^2-4 a c\right ) e^2 x\right )}{2 \left (c x^2+b x+a\right )^{5/2}}dx}{5 c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^4 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^4 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^3 \left (3 B d e b^2-8 \left (B c d^2+2 A c e d+a B e^2\right ) b+4 c \left (5 a B d e+4 A \left (c d^2+a e^2\right )\right )-5 B \left (b^2-4 a c\right ) e^2 x\right )}{\left (c x^2+b x+a\right )^{5/2}}dx}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {2 (d+e x)^4 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\frac {2 \int \frac {(d+e x) \left (5 B d e^3 b^4+4 B e^2 \left (c d^2-5 a e^2\right ) b^3-8 c d e \left (11 B c d^2+16 A c e d+6 a B e^2\right ) b^2+16 c \left (16 A c d e \left (c d^2+a e^2\right )+B \left (4 c^2 d^4+23 a c e^2 d^2+9 a^2 e^4\right )\right ) b-16 c^2 \left (8 A \left (c d^2+a e^2\right )^2+5 a B d e \left (2 c d^2+5 a e^2\right )\right )-15 B \left (b^2-4 a c\right )^2 e^4 x\right )}{2 \left (c x^2+b x+a\right )^{3/2}}dx}{3 c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^2 \left (4 b^2 c d \left (a B e^2+4 A c d e+2 B c d^2\right )-x \left (2 b^2 c e \left (19 a B e^2+8 A c d e+7 B c d^2\right )-8 b c^2 \left (2 a A e^3+7 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c^2 \left (4 A c d \left (a e^2+c d^2\right )+5 a B e \left (c d^2-a e^2\right )\right )-5 b^4 B e^3+2 b^3 B c d e^2\right )-4 b c \left (4 A c d \left (3 a e^2+c d^2\right )+9 a B e \left (a e^2+c d^2\right )\right )+16 a c^2 e \left (2 A \left (a e^2+c d^2\right )+5 a B d e\right )+b^3 (-B) \left (3 c d^2 e-5 a e^3\right )\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^4 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\frac {\int \frac {(d+e x) \left (5 B d e^3 b^4+4 B e^2 \left (c d^2-5 a e^2\right ) b^3-8 c d e \left (11 B c d^2+16 A c e d+6 a B e^2\right ) b^2+16 c \left (16 A c d e \left (c d^2+a e^2\right )+B \left (4 c^2 d^4+23 a c e^2 d^2+9 a^2 e^4\right )\right ) b-16 c^2 \left (8 A \left (c d^2+a e^2\right )^2+5 a B d e \left (2 c d^2+5 a e^2\right )\right )-15 B \left (b^2-4 a c\right )^2 e^4 x\right )}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^2 \left (4 b^2 c d \left (a B e^2+4 A c d e+2 B c d^2\right )-x \left (2 b^2 c e \left (19 a B e^2+8 A c d e+7 B c d^2\right )-8 b c^2 \left (2 a A e^3+7 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c^2 \left (4 A c d \left (a e^2+c d^2\right )+5 a B e \left (c d^2-a e^2\right )\right )-5 b^4 B e^3+2 b^3 B c d e^2\right )-4 b c \left (4 A c d \left (3 a e^2+c d^2\right )+9 a B e \left (a e^2+c d^2\right )\right )+16 a c^2 e \left (2 A \left (a e^2+c d^2\right )+5 a B d e\right )+b^3 (-B) \left (3 c d^2 e-5 a e^3\right )\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}}{5 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1224

\(\displaystyle \frac {2 (d+e x)^4 \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-5 a e^3\right ) b^3+4 c d \left (2 B c d^2+4 A c e d+a B e^2\right ) b^2-4 c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right ) b+16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )-\left (-5 B e^3 b^4+2 B c d e^2 b^3+2 c e \left (7 B c d^2+8 A c e d+19 a B e^2\right ) b^2-8 c^2 \left (2 B c d^3+6 A c e d^2+7 a B e^2 d+2 a A e^3\right ) b+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right ) (d+e x)^2}{3 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}+\frac {-\frac {15 B \left (b^2-4 a c\right )^2 \int \frac {1}{\sqrt {c x^2+b x+a}}dx e^5}{c}-\frac {2 \left (5 B e^3 \left (c d^2-3 a e^2\right ) b^5+4 B c^2 d^3 e^2 b^4-8 c e \left (16 A c^2 e d^3+B \left (11 c^2 d^4+7 a c e^2 d^2-20 a^2 e^4\right )\right ) b^3+32 c^3 d^2 \left (2 B c d^3+8 A c e d^2+17 a B e^2 d+16 a A e^3\right ) b^2-16 c^2 \left (8 A c d \left (c^2 d^4+6 a c e^2 d^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c e^2 d^2+33 a^2 e^4\right )\right ) b+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )+\left (-15 B e^5 b^6+10 B c d e^4 b^5+2 B c e^3 \left (3 c d^2+85 a e^2\right ) b^4+16 c^2 d e^2 \left (6 B c d^2+8 A c e d-7 a B e^2\right ) b^3-16 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c e^2 d^2+39 a^2 e^4\right )\right ) b^2+32 c^3 \left (4 A e \left (5 c^2 d^4+6 a c e^2 d^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c e^2 d^3+29 a^2 e^4 d\right )\right ) b-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c e^2 d^2-3 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 )}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {2 (d+e x)^4 \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-5 a e^3\right ) b^3+4 c d \left (2 B c d^2+4 A c e d+a B e^2\right ) b^2-4 c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right ) b+16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )-\left (-5 B e^3 b^4+2 B c d e^2 b^3+2 c e \left (7 B c d^2+8 A c e d+19 a B e^2\right ) b^2-8 c^2 \left (2 B c d^3+6 A c e d^2+7 a B e^2 d+2 a A e^3\right ) b+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right ) (d+e x)^2}{3 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}+\frac {-\frac {30 B \left (b^2-4 a c\right )^2 \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^5}{c}-\frac {2 \left (5 B e^3 \left (c d^2-3 a e^2\right ) b^5+4 B c^2 d^3 e^2 b^4-8 c e \left (16 A c^2 e d^3+B \left (11 c^2 d^4+7 a c e^2 d^2-20 a^2 e^4\right )\right ) b^3+32 c^3 d^2 \left (2 B c d^3+8 A c e d^2+17 a B e^2 d+16 a A e^3\right ) b^2-16 c^2 \left (8 A c d \left (c^2 d^4+6 a c e^2 d^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c e^2 d^2+33 a^2 e^4\right )\right ) b+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )+\left (-15 B e^5 b^6+10 B c d e^4 b^5+2 B c e^3 \left (3 c d^2+85 a e^2\right ) b^4+16 c^2 d e^2 \left (6 B c d^2+8 A c e d-7 a B e^2\right ) b^3-16 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c e^2 d^2+39 a^2 e^4\right )\right ) b^2+32 c^3 \left (4 A e \left (5 c^2 d^4+6 a c e^2 d^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c e^2 d^3+29 a^2 e^4 d\right )\right ) b-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c e^2 d^2-3 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 )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 (d+e x)^4 \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-5 a e^3\right ) b^3+4 c d \left (2 B c d^2+4 A c e d+a B e^2\right ) b^2-4 c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right ) b+16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )-\left (-5 B e^3 b^4+2 B c d e^2 b^3+2 c e \left (7 B c d^2+8 A c e d+19 a B e^2\right ) b^2-8 c^2 \left (2 B c d^3+6 A c e d^2+7 a B e^2 d+2 a A e^3\right ) b+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right ) (d+e x)^2}{3 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}+\frac {-\frac {15 B \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^5}{c^{3/2}}-\frac {2 \left (5 B e^3 \left (c d^2-3 a e^2\right ) b^5+4 B c^2 d^3 e^2 b^4-8 c e \left (16 A c^2 e d^3+B \left (11 c^2 d^4+7 a c e^2 d^2-20 a^2 e^4\right )\right ) b^3+32 c^3 d^2 \left (2 B c d^3+8 A c e d^2+17 a B e^2 d+16 a A e^3\right ) b^2-16 c^2 \left (8 A c d \left (c^2 d^4+6 a c e^2 d^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c e^2 d^2+33 a^2 e^4\right )\right ) b+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )+\left (-15 B e^5 b^6+10 B c d e^4 b^5+2 B c e^3 \left (3 c d^2+85 a e^2\right ) b^4+16 c^2 d e^2 \left (6 B c d^2+8 A c e d-7 a B e^2\right ) b^3-16 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c e^2 d^2+39 a^2 e^4\right )\right ) b^2+32 c^3 \left (4 A e \left (5 c^2 d^4+6 a c e^2 d^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c e^2 d^3+29 a^2 e^4 d\right )\right ) b-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c e^2 d^2-3 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 )}\)

input

Int[((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2),x]
 

output

(2*(d + e*x)^4*(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)^2*(4*b^2*c*d*(2*B*c*d^2 + 4*A*c*d*e + a*B*e^2) - b^3 
*B*(3*c*d^2*e - 5*a*e^3) + 16*a*c^2*e*(5*a*B*d*e + 2*A*(c*d^2 + a*e^2)) - 
4*b*c*(9*a*B*e*(c*d^2 + a*e^2) + 4*A*c*d*(c*d^2 + 3*a*e^2)) - (2*b^3*B*c*d 
*e^2 - 5*b^4*B*e^3 + 2*b^2*c*e*(7*B*c*d^2 + 8*A*c*d*e + 19*a*B*e^2) - 8*b* 
c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 7*a*B*d*e^2 + 2*a*A*e^3) + 8*c^2*(5*a*B*e*( 
c*d^2 - a*e^2) + 4*A*c*d*(c*d^2 + a*e^2)))*x))/(3*c*(b^2 - 4*a*c)*(a + b*x 
 + c*x^2)^(3/2)) + ((-2*(4*b^4*B*c^2*d^3*e^2 + 5*b^5*B*e^3*(c*d^2 - 3*a*e^ 
2) + 32*b^2*c^3*d^2*(2*B*c*d^3 + 8*A*c*d^2*e + 17*a*B*d*e^2 + 16*a*A*e^3) 
+ 64*a*c^3*e*(4*A*(c*d^2 + a*e^2)^2 + 5*a*B*d*e*(c*d^2 + 4*a*e^2)) - 8*b^3 
*c*e*(16*A*c^2*d^3*e + B*(11*c^2*d^4 + 7*a*c*d^2*e^2 - 20*a^2*e^4)) - 16*b 
*c^2*(8*A*c*d*(c^2*d^4 + 6*a*c*d^2*e^2 + 5*a^2*e^4) + a*B*e*(18*c^2*d^4 + 
71*a*c*d^2*e^2 + 33*a^2*e^4)) + (10*b^5*B*c*d*e^4 - 15*b^6*B*e^5 + 2*b^4*B 
*c*e^3*(3*c*d^2 + 85*a*e^2) + 16*b^3*c^2*d*e^2*(6*B*c*d^2 + 8*A*c*d*e - 7* 
a*B*e^2) - 32*c^3*(8*A*c*d*(c*d^2 + a*e^2)^2 + 5*a*B*e*(2*c^2*d^4 + 5*a*c* 
d^2*e^2 - 3*a^2*e^4)) - 16*b^2*c^2*e*(16*A*c*d*e*(2*c*d^2 + a*e^2) + B*(15 
*c^2*d^4 + 29*a*c*d^2*e^2 + 39*a^2*e^4)) + 32*b*c^3*(4*A*e*(5*c^2*d^4 + 6* 
a*c*d^2*e^2 + a^2*e^4) + B*(4*c^2*d^5 + 28*a*c*d^3*e^2 + 29*a^2*d*e^4)))*x 
))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - (15*B*(b^2 - 4*a*c)^2*e^5*...
 

3.25.88.3.1 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 1224

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - ( 
b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2)^(p 
+ 1)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c 
*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(c*(p + 1)*(b^2 - 4*a*c))   Int[(a + 
b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, - 
1] &&  !(IntegerQ[p] && NeQ[a, 0] && NiceSqrtQ[b^2 - 4*a*c])
 

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])
 

3.25.88.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4848\) vs. \(2(922)=1844\).

Time = 1.23 (sec) , antiderivative size = 4849, normalized size of antiderivative = 5.15

method	result	size
default	\(\text {Expression too large to display}\)	\(4849\)

input

int((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),x,method=_RETURNVERBOSE)
 

output

A*d^5*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2 
/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b 
)/(c*x^2+b*x+a)^(1/2)))+B*e^5*(-1/5*x^5/c/(c*x^2+b*x+a)^(5/2)-1/2*b/c*(-x^ 
4/c/(c*x^2+b*x+a)^(5/2)+3/2*b/c*(-1/2*x^3/c/(c*x^2+b*x+a)^(5/2)+1/4*b/c*(- 
1/3*x^2/c/(c*x^2+b*x+a)^(5/2)-1/6*b/c*(-1/4*x/c/(c*x^2+b*x+a)^(5/2)-3/8*b/ 
c*(-1/5/c/(c*x^2+b*x+a)^(5/2)-1/2*b/c*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b* 
x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/ 
2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))+1/4*a/c*(2/5*(2*c 
*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2*c*x+b)/(4 
*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a) 
^(1/2))))+2/3*a/c*(-1/5/c/(c*x^2+b*x+a)^(5/2)-1/2*b/c*(2/5*(2*c*x+b)/(4*a* 
c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2*c*x+b)/(4*a*c-b^2)/( 
c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))))+ 
3/2*a/c*(-1/4*x/c/(c*x^2+b*x+a)^(5/2)-3/8*b/c*(-1/5/c/(c*x^2+b*x+a)^(5/2)- 
1/2*b/c*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)* 
(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x 
+b)/(c*x^2+b*x+a)^(1/2))))+1/4*a/c*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a 
)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+ 
16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))))+4*a/c*(-1/3*x^2/c/( 
c*x^2+b*x+a)^(5/2)-1/6*b/c*(-1/4*x/c/(c*x^2+b*x+a)^(5/2)-3/8*b/c*(-1/5/...
 

3.25.88.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2625 vs. \(2 (922) = 1844\).

Time = 46.80 (sec) , antiderivative size = 5253, normalized size of antiderivative = 5.58 \[ \int \frac {(A+B x) (d+e x)^5}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \]

input

integrate((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")
 

output

Too large to include
 

3.25.88.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^5}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Timed out} \]

input

integrate((B*x+A)*(e*x+d)**5/(c*x**2+b*x+a)**(7/2),x)
 

output

Timed out
 

3.25.88.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) (d+e x)^5}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \]

input

integrate((B*x+A)*(e*x+d)^5/(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
 

3.25.88.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2612 vs. \(2 (922) = 1844\).

Time = 0.32 (sec) , antiderivative size = 2612, normalized size of antiderivative = 2.77 \[ \int \frac {(A+B x) (d+e x)^5}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \]

input

integrate((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")
 

output

-B*e^5*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(7/2) 
 + 2/15*((((((128*B*b*c^7*d^5 - 256*A*c^8*d^5 - 240*B*b^2*c^6*d^4*e - 320* 
B*a*c^7*d^4*e + 640*A*b*c^7*d^4*e + 80*B*b^3*c^5*d^3*e^2 + 960*B*a*b*c^6*d 
^3*e^2 - 480*A*b^2*c^6*d^3*e^2 - 640*A*a*c^7*d^3*e^2 + 20*B*b^4*c^4*d^2*e^ 
3 - 480*B*a*b^2*c^5*d^2*e^3 + 80*A*b^3*c^5*d^2*e^3 - 960*B*a^2*c^6*d^2*e^3 
 + 960*A*a*b*c^6*d^2*e^3 + 15*B*b^5*c^3*d*e^4 - 200*B*a*b^3*c^4*d*e^4 + 10 
*A*b^4*c^4*d*e^4 + 1200*B*a^2*b*c^5*d*e^4 - 240*A*a*b^2*c^5*d*e^4 - 480*A* 
a^2*c^6*d*e^4 - 23*B*b^6*c^2*e^5 + 258*B*a*b^4*c^3*e^5 + 3*A*b^5*c^3*e^5 - 
 912*B*a^2*b^2*c^4*e^5 - 40*A*a*b^3*c^4*e^5 + 736*B*a^3*c^5*e^5 + 240*A*a^ 
2*b*c^5*e^5)*x/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6) + 5* 
(64*B*b^2*c^6*d^5 - 128*A*b*c^7*d^5 - 120*B*b^3*c^5*d^4*e - 160*B*a*b*c^6* 
d^4*e + 320*A*b^2*c^6*d^4*e + 40*B*b^4*c^4*d^3*e^2 + 480*B*a*b^2*c^5*d^3*e 
^2 - 240*A*b^3*c^5*d^3*e^2 - 320*A*a*b*c^6*d^3*e^2 + 10*B*b^5*c^3*d^2*e^3 
- 240*B*a*b^3*c^4*d^2*e^3 + 40*A*b^4*c^4*d^2*e^3 - 480*B*a^2*b*c^5*d^2*e^3 
 + 480*A*a*b^2*c^5*d^2*e^3 - 10*B*a*b^4*c^3*d*e^4 + 5*A*b^5*c^3*d*e^4 + 24 
0*B*a^2*b^2*c^4*d*e^4 - 120*A*a*b^3*c^4*d*e^4 + 480*B*a^3*c^5*d*e^4 - 240* 
A*a^2*b*c^5*d*e^4 - 7*B*b^7*c*e^5 + 75*B*a*b^5*c^2*e^5 - 240*B*a^2*b^3*c^3 
*e^5 - 2*A*a*b^4*c^3*e^5 + 80*B*a^3*b*c^4*e^5 + 48*A*a^2*b^2*c^4*e^5 + 96* 
A*a^3*c^5*e^5)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x + 
 5*(48*B*b^3*c^5*d^5 + 64*B*a*b*c^6*d^5 - 96*A*b^2*c^6*d^5 - 128*A*a*c^...
 

3.25.88.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^5}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^5}{{\left (c\,x^2+b\,x+a\right )}^{7/2}} \,d x \]

input

int(((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2),x)
 

output

int(((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2), x)