\(\int \frac {(A+B x) (b x+c x^2)^{5/2}}{(d+e x)^5} \, dx\) [118]

Optimal result

Integrand size = 26, antiderivative size = 780 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=-\frac {5 \left (B d \left (192 c^3 d^3-304 b c^2 d^2 e+120 b^2 c d e^2-7 b^3 e^3\right )-A e \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right )\right ) \sqrt {b x+c x^2}}{64 d^2 e^6 (c d-b e)}+\frac {5 \left (B d \left (288 c^3 d^3-432 b c^2 d^2 e+154 b^2 c d e^2-7 b^3 e^3\right )-A e \left (96 c^3 d^3-112 b c^2 d^2 e+18 b^2 c d e^2+b^3 e^3\right )\right ) x \sqrt {b x+c x^2}}{192 d^3 e^5 (c d-b e)}+\frac {\left (A e \left (40 c^2 d^2-24 b c d e-b^2 e^2\right )-B d \left (120 c^2 d^2-112 b c d e+7 b^2 e^2\right )\right ) x^3 \sqrt {b x+c x^2}}{96 d^3 e^3 (d+e x)^2}-\frac {\left (B d \left (960 c^3 d^3-1400 b c^2 d^2 e+476 b^2 c d e^2-21 b^3 e^3\right )-A e \left (320 c^3 d^3-360 b c^2 d^2 e+52 b^2 c d e^2+3 b^3 e^3\right )\right ) x^2 \sqrt {b x+c x^2}}{192 d^3 e^4 (c d-b e) (d+e x)}-\frac {(B d (12 c d-7 b e)-A e (4 c d+b e)) x^2 \left (b x+c x^2\right )^{3/2}}{24 d^2 e^2 (d+e x)^3}-\frac {(B d-A e) x \left (b x+c x^2\right )^{5/2}}{4 d e (d+e x)^4}-\frac {5 \sqrt {c} \left (4 A c e (2 c d-b e)-B \left (24 c^2 d^2-20 b c d e+3 b^2 e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 e^7}+\frac {5 \left (A e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B d \left (384 c^4 d^4-896 b c^3 d^3 e+672 b^2 c^2 d^2 e^2-168 b^3 c d e^3+7 b^4 e^4\right )\right ) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{64 d^{3/2} e^7 (c d-b e)^{3/2}} \] Output:

-5/64*(B*d*(-7*b^3*e^3+120*b^2*c*d*e^2-304*b*c^2*d^2*e+192*c^3*d^3)-A*e*(b 
^3*e^3+16*b^2*c*d*e^2-80*b*c^2*d^2*e+64*c^3*d^3))*(c*x^2+b*x)^(1/2)/d^2/e^ 
6/(-b*e+c*d)+5/192*(B*d*(-7*b^3*e^3+154*b^2*c*d*e^2-432*b*c^2*d^2*e+288*c^ 
3*d^3)-A*e*(b^3*e^3+18*b^2*c*d*e^2-112*b*c^2*d^2*e+96*c^3*d^3))*x*(c*x^2+b 
*x)^(1/2)/d^3/e^5/(-b*e+c*d)+1/96*(A*e*(-b^2*e^2-24*b*c*d*e+40*c^2*d^2)-B* 
d*(7*b^2*e^2-112*b*c*d*e+120*c^2*d^2))*x^3*(c*x^2+b*x)^(1/2)/d^3/e^3/(e*x+ 
d)^2-1/192*(B*d*(-21*b^3*e^3+476*b^2*c*d*e^2-1400*b*c^2*d^2*e+960*c^3*d^3) 
-A*e*(3*b^3*e^3+52*b^2*c*d*e^2-360*b*c^2*d^2*e+320*c^3*d^3))*x^2*(c*x^2+b* 
x)^(1/2)/d^3/e^4/(-b*e+c*d)/(e*x+d)-1/24*(B*d*(-7*b*e+12*c*d)-A*e*(b*e+4*c 
*d))*x^2*(c*x^2+b*x)^(3/2)/d^2/e^2/(e*x+d)^3-1/4*(-A*e+B*d)*x*(c*x^2+b*x)^ 
(5/2)/d/e/(e*x+d)^4-5/4*c^(1/2)*(4*A*c*e*(-b*e+2*c*d)-B*(3*b^2*e^2-20*b*c* 
d*e+24*c^2*d^2))*arctanh(c^(1/2)*x/(c*x^2+b*x)^(1/2))/e^7+5/64*(A*e*(-b^4* 
e^4-16*b^3*c*d*e^3+144*b^2*c^2*d^2*e^2-256*b*c^3*d^3*e+128*c^4*d^4)-B*d*(7 
*b^4*e^4-168*b^3*c*d*e^3+672*b^2*c^2*d^2*e^2-896*b*c^3*d^3*e+384*c^4*d^4)) 
*arctanh((-b*e+c*d)^(1/2)*x/d^(1/2)/(c*x^2+b*x)^(1/2))/d^(3/2)/e^7/(-b*e+c 
*d)^(3/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2578\) vs. \(2(780)=1560\).

Time = 16.64 (sec) , antiderivative size = 2578, normalized size of antiderivative = 3.31 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=\text {Result too large to show} \] Input:

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

Output:

((-(B*d) + A*e)*x*(b + c*x)*(x*(b + c*x))^(5/2))/(4*d*(-(c*d) + b*e)*(d + 
e*x)^4) + ((x*(b + c*x))^(5/2)*(((-3*c*d*(B*d - A*e) + (e*(7*b*B*d - 8*A*c 
*d + A*b*e))/2)*x^(7/2)*(b + c*x)^(7/2))/(3*d*(-(c*d) + b*e)*(d + e*x)^3) 
+ (((-2*c*d*(B*d*(6*c*d - 7*b*e) + A*e*(2*c*d - b*e)) + (e*(-7*b^2*B*d*e + 
 A*(48*c^2*d^2 - 40*b*c*d*e - b^2*e^2)))/4)*x^(7/2)*(b + c*x)^(7/2))/(2*d* 
(-(c*d) + b*e)*(d + e*x)^2) + ((((e*(-192*A*c^3*d^3 - 4*b^2*c*d*e*(91*B*d 
- 17*A*e) + 3*b^3*e^2*(7*B*d + A*e) + 16*b*c^2*d^2*(21*B*d + 8*A*e)))/8 + 
(5*c*d*(A*e*(32*c^2*d^2 - 32*b*c*d*e - b^2*e^2) - B*d*(48*c^2*d^2 - 56*b*c 
*d*e + 7*b^2*e^2)))/4)*x^(7/2)*(b + c*x)^(7/2))/(d*(-(c*d) + b*e)*(d + e*x 
)) + ((-1/8*(c*d*(-192*A*c^3*d^3 - 4*b^2*c*d*e*(91*B*d - 17*A*e) + 3*b^3*e 
^2*(7*B*d + A*e) + 16*b*c^2*d^2*(21*B*d + 8*A*e))) + (b*e*(-192*A*c^3*d^3 
- 4*b^2*c*d*e*(91*B*d - 17*A*e) + 3*b^3*e^2*(7*B*d + A*e) + 16*b*c^2*d^2*( 
21*B*d + 8*A*e)))/8 - (7*b*((e*(-192*A*c^3*d^3 - 4*b^2*c*d*e*(91*B*d - 17* 
A*e) + 3*b^3*e^2*(7*B*d + A*e) + 16*b*c^2*d^2*(21*B*d + 8*A*e)))/8 + (5*c* 
d*(A*e*(32*c^2*d^2 - 32*b*c*d*e - b^2*e^2) - B*d*(48*c^2*d^2 - 56*b*c*d*e 
+ 7*b^2*e^2)))/4))/2)*((2*b^2*x^(5/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((5/(1 
6*(1 + (c*x)/b)^3) + 5/(8*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 - (15*b 
^3*((2*c*x)/b - (4*c^2*x^2)/(3*b^2) - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]* 
Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(512*c^3*x^3*(1 + (c*x)/b 
)^3)))/(5*e) - (d*(((5*b^3*Sqrt[x]*Sqrt[b + c*x])/(64*c) + (59*b^2*x^(3...
 

Rubi [A] (verified)

Time = 2.12 (sec) , antiderivative size = 660, normalized size of antiderivative = 0.85, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1230, 27, 1229, 27, 1230, 1269, 1091, 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.

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1229

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}-\frac {\int \frac {\left (b \left (A e \left (16 c^2 d^2-12 b c e d-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c e d+7 b^2 e^2\right )\right )+2 c \left (A e \left (16 c^2 d^2-16 b c e d+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c e d+17 b^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x}}{2 (d+e x)^2}dx}{4 d e^2 (c d-b e)}\right )}{8 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}-\frac {\int \frac {\left (b \left (A e \left (16 c^2 d^2-12 b c e d-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c e d+7 b^2 e^2\right )\right )+2 c \left (A e \left (16 c^2 d^2-16 b c e d+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c e d+17 b^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x}}{(d+e x)^2}dx}{8 d e^2 (c d-b e)}\right )}{8 e^2}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}-\frac {-\frac {\int \frac {b \left (-64 c^3 (3 B d-A e) d^3+16 b c^2 e (19 B d-5 A e) d^2-8 b^2 c e^2 (15 B d-2 A e) d+b^3 e^3 (7 B d+A e)\right )+16 c d (c d-b e) \left (4 A c e (2 c d-b e)-B \left (24 c^2 d^2-20 b c e d+3 b^2 e^2\right )\right ) x}{(d+e x) \sqrt {c x^2+b x}}dx}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (b^3 e^3+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (-7 b^3 e^3+120 b^2 c d e^2-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{e^2 (d+e x)}}{8 d e^2 (c d-b e)}\right )}{8 e^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}-\frac {-\frac {\frac {16 c d (c d-b e) \left (4 A c e (2 c d-b e)-B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}-\frac {\left (A e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B d \left (7 b^4 e^4-168 b^3 c d e^3+672 b^2 c^2 d^2 e^2-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (b^3 e^3+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (-7 b^3 e^3+120 b^2 c d e^2-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{e^2 (d+e x)}}{8 d e^2 (c d-b e)}\right )}{8 e^2}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}-\frac {-\frac {\frac {32 c d (c d-b e) \left (4 A c e (2 c d-b e)-B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}-\frac {\left (A e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B d \left (7 b^4 e^4-168 b^3 c d e^3+672 b^2 c^2 d^2 e^2-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (b^3 e^3+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (-7 b^3 e^3+120 b^2 c d e^2-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{e^2 (d+e x)}}{8 d e^2 (c d-b e)}\right )}{8 e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}-\frac {-\frac {\frac {32 \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (c d-b e) \left (4 A c e (2 c d-b e)-B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right )}{e}-\frac {\left (A e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B d \left (7 b^4 e^4-168 b^3 c d e^3+672 b^2 c^2 d^2 e^2-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (b^3 e^3+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (-7 b^3 e^3+120 b^2 c d e^2-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{e^2 (d+e x)}}{8 d e^2 (c d-b e)}\right )}{8 e^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}-\frac {-\frac {\frac {2 \left (A e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B d \left (7 b^4 e^4-168 b^3 c d e^3+672 b^2 c^2 d^2 e^2-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}+\frac {32 \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (c d-b e) \left (4 A c e (2 c d-b e)-B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right )}{e}}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (b^3 e^3+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (-7 b^3 e^3+120 b^2 c d e^2-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{e^2 (d+e x)}}{8 d e^2 (c d-b e)}\right )}{8 e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{12 d e^2 (d+e x)^3 (c d-b e)}-\frac {-\frac {\frac {32 \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (c d-b e) \left (4 A c e (2 c d-b e)-B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right )}{e}-\frac {\left (A e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B d \left (7 b^4 e^4-168 b^3 c d e^3+672 b^2 c^2 d^2 e^2-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{\sqrt {d} e \sqrt {c d-b e}}}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (b^3 e^3+16 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (-7 b^3 e^3+120 b^2 c d e^2-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{e^2 (d+e x)}}{8 d e^2 (c d-b e)}\right )}{8 e^2}\)

Input:

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

Output:

((3*B*d - A*e + 2*B*e*x)*(b*x + c*x^2)^(5/2))/(4*e^2*(d + e*x)^4) - (5*((( 
d*(A*e*(16*c^2*d^2 - 12*b*c*d*e - b^2*e^2) - B*d*(48*c^2*d^2 - 52*b*c*d*e 
+ 7*b^2*e^2)) + 3*e*(A*e*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - B*d*(24*c^2*d 
^2 - 32*b*c*d*e + 9*b^2*e^2))*x)*(b*x + c*x^2)^(3/2))/(12*d*e^2*(c*d - b*e 
)*(d + e*x)^3) - (-(((B*d*(192*c^3*d^3 - 304*b*c^2*d^2*e + 120*b^2*c*d*e^2 
 - 7*b^3*e^3) - A*e*(64*c^3*d^3 - 80*b*c^2*d^2*e + 16*b^2*c*d*e^2 + b^3*e^ 
3) - 2*c*e*(A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B*d*(48*c^2*d^2 - 64 
*b*c*d*e + 17*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(e^2*(d + e*x))) - ((32*Sqrt 
[c]*d*(c*d - b*e)*(4*A*c*e*(2*c*d - b*e) - B*(24*c^2*d^2 - 20*b*c*d*e + 3* 
b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/e - ((A*e*(128*c^4*d^4 - 
 256*b*c^3*d^3*e + 144*b^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4*e^4) - B*d*( 
384*c^4*d^4 - 896*b*c^3*d^3*e + 672*b^2*c^2*d^2*e^2 - 168*b^3*c*d*e^3 + 7* 
b^4*e^4))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[ 
b*x + c*x^2])])/(Sqrt[d]*e*Sqrt[c*d - b*e]))/(2*e^2))/(8*d*e^2*(c*d - b*e) 
)))/(8*e^2)
 

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 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{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 1229

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/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

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 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 [A] (verified)

Time = 1.70 (sec) , antiderivative size = 593, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(-\frac {5 \left (\left (e x +d \right )^{4} \left (3 b^{2} \left (-3 A \,d^{2} e^{3}+14 B \,d^{3} e^{2}\right ) c^{\frac {5}{2}}+16 e \,d^{3} \left (A e -\frac {7 B d}{2}\right ) b \,c^{\frac {7}{2}}+b^{3} d \,e^{3} \left (A e -\frac {21 B d}{2}\right ) c^{\frac {3}{2}}+8 \left (-A \,d^{4} e +3 B \,d^{5}\right ) c^{\frac {9}{2}}+\frac {b^{4} e^{4} \sqrt {c}\, \left (A e +7 B d \right )}{16}\right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )+\sqrt {d \left (b e -c d \right )}\, \left (-4 c d \left (e x +d \right )^{4} \left (b e -c d \right ) \left (6 B \,c^{2} d^{2}+\left (-2 A \,c^{2}-5 B b c \right ) e d +b \,e^{2} \left (A c +\frac {3 B b}{4}\right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )+e \sqrt {x \left (c x +b \right )}\, \left (-5 \left (-\frac {19 B \,d^{5}}{5}+e \left (-\frac {67 B x}{5}+A \right ) d^{4}+\frac {53 \left (-\frac {503 B x}{106}+A \right ) e^{2} x \,d^{3}}{15}+\frac {133 e^{3} \left (-\frac {1231 B x}{665}+A \right ) x^{2} d^{2}}{30}+\frac {109 e^{4} x^{3} \left (-\frac {42 B x}{109}+A \right ) d}{50}+\frac {4 e^{5} x^{4} \left (\frac {B x}{2}+A \right )}{25}\right ) e d b \,c^{\frac {5}{2}}+4 d^{2} \left (-3 B \,d^{5}+e \left (-\frac {21 B x}{2}+A \right ) d^{4}+\frac {7 e^{2} \left (-\frac {26 B x}{7}+A \right ) x \,d^{3}}{2}+\frac {13 e^{3} \left (-\frac {75 B x}{52}+A \right ) x^{2} d^{2}}{3}+\frac {25 \left (-\frac {36 B x}{125}+A \right ) e^{4} x^{3} d}{12}+\frac {e^{5} x^{4} \left (\frac {B x}{2}+A \right )}{5}\right ) c^{\frac {7}{2}}+\left (d \left (-\frac {15 B \,d^{4}}{2}+e \left (-\frac {643 B x}{24}+A \right ) d^{3}+\frac {29 e^{2} x \left (-\frac {1366 B x}{145}+A \right ) d^{2}}{8}+\frac {283 e^{3} x^{2} \left (-\frac {2071 B x}{566}+A \right ) d}{60}+\frac {323 e^{4} \left (-\frac {216 B x}{323}+A \right ) x^{3}}{120}\right ) c^{\frac {3}{2}}+\frac {e \left (7 B \,d^{4}+e \left (\frac {77 B x}{3}+A \right ) d^{3}+\frac {11 e^{2} \left (\frac {511 B x}{55}+A \right ) x \,d^{2}}{3}+\frac {73 e^{3} x^{2} \left (\frac {279 B x}{73}+A \right ) d}{15}-A \,e^{4} x^{3}\right ) b \sqrt {c}}{16}\right ) e^{2} b^{2}\right )\right )\right )}{4 \sqrt {c}\, \sqrt {d \left (b e -c d \right )}\, \left (e x +d \right )^{4} e^{7} d \left (b e -c d \right )}\)	\(593\)
risch	\(\text {Expression too large to display}\)	\(3596\)
default	\(\text {Expression too large to display}\)	\(10740\)

Input:

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

Output:

-5/4/c^(1/2)/(d*(b*e-c*d))^(1/2)*((e*x+d)^4*(3*b^2*(-3*A*d^2*e^3+14*B*d^3* 
e^2)*c^(5/2)+16*e*d^3*(A*e-7/2*B*d)*b*c^(7/2)+b^3*d*e^3*(A*e-21/2*B*d)*c^( 
3/2)+8*(-A*d^4*e+3*B*d^5)*c^(9/2)+1/16*b^4*e^4*c^(1/2)*(A*e+7*B*d))*arctan 
((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))+(d*(b*e-c*d))^(1/2)*(-4*c*d*(e 
*x+d)^4*(b*e-c*d)*(6*B*c^2*d^2+(-2*A*c^2-5*B*b*c)*e*d+b*e^2*(A*c+3/4*B*b)) 
*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))+e*(x*(c*x+b))^(1/2)*(-5*(-19/5*B*d^5 
+e*(-67/5*B*x+A)*d^4+53/15*(-503/106*B*x+A)*e^2*x*d^3+133/30*e^3*(-1231/66 
5*B*x+A)*x^2*d^2+109/50*e^4*x^3*(-42/109*B*x+A)*d+4/25*e^5*x^4*(1/2*B*x+A) 
)*e*d*b*c^(5/2)+4*d^2*(-3*B*d^5+e*(-21/2*B*x+A)*d^4+7/2*e^2*(-26/7*B*x+A)* 
x*d^3+13/3*e^3*(-75/52*B*x+A)*x^2*d^2+25/12*(-36/125*B*x+A)*e^4*x^3*d+1/5* 
e^5*x^4*(1/2*B*x+A))*c^(7/2)+(d*(-15/2*B*d^4+e*(-643/24*B*x+A)*d^3+29/8*e^ 
2*x*(-1366/145*B*x+A)*d^2+283/60*e^3*x^2*(-2071/566*B*x+A)*d+323/120*e^4*( 
-216/323*B*x+A)*x^3)*c^(3/2)+1/16*e*(7*B*d^4+e*(77/3*B*x+A)*d^3+11/3*e^2*( 
511/55*B*x+A)*x*d^2+73/15*e^3*x^2*(279/73*B*x+A)*d-A*e^4*x^3)*b*c^(1/2))*e 
^2*b^2)))/(e*x+d)^4/e^7/d/(b*e-c*d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2005 vs. \(2 (736) = 1472\).

Time = 8.47 (sec) , antiderivative size = 8041, normalized size of antiderivative = 10.31 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

Integral((x*(b + c*x))**(5/2)*(A + B*x)/(d + e*x)**5, x)
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m 
ore detail
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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