\(\int \frac {(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^8} \, dx\) [111]

Optimal result

Integrand size = 26, antiderivative size = 968 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx=-\frac {(B d (8 c d-5 b e)+3 A e (2 c d-3 b e)) x^2 \sqrt {b x+c x^2}}{84 d^2 e^2 (d+e x)^6}-\frac {\left (B d \left (48 c^2 d^2-10 b c d e-35 b^2 e^2\right )+3 A e \left (12 c^2 d^2+8 b c d e-21 b^2 e^2\right )\right ) x \sqrt {b x+c x^2}}{840 d^2 e^3 (c d-b e) (d+e x)^5}-\frac {\left (3 A e \left (16 c^3 d^3-8 b c^2 d^2 e-26 b^2 c d e^2+21 b^3 e^3\right )+B d \left (64 c^3 d^3-88 b c^2 d^2 e-20 b^2 c d e^2+35 b^3 e^3\right )\right ) \sqrt {b x+c x^2}}{2240 d^2 e^4 (c d-b e)^2 (d+e x)^4}+\frac {\left (B d \left (128 c^4 d^4-368 b c^3 d^3 e+288 b^2 c^2 d^2 e^2+50 b^3 c d e^3-35 b^4 e^4\right )+3 A e \left (32 c^4 d^4-64 b c^3 d^3 e-12 b^2 c^2 d^2 e^2+44 b^3 c d e^3-21 b^4 e^4\right )\right ) \sqrt {b x+c x^2}}{13440 d^3 e^4 (c d-b e)^3 (d+e x)^3}+\frac {\left (3 A e \left (128 c^5 d^5-320 b c^4 d^4 e+32 b^2 c^3 d^3 e^2+272 b^3 c^2 d^2 e^3-322 b^4 c d e^4+105 b^5 e^5\right )+B d \left (512 c^5 d^5-1728 b c^4 d^4 e+1696 b^2 c^3 d^3 e^2+80 b^3 c^2 d^2 e^3-420 b^4 c d e^4+175 b^5 e^5\right )\right ) \sqrt {b x+c x^2}}{53760 d^4 e^4 (c d-b e)^4 (d+e x)^2}+\frac {\left (B d \left (1024 c^6 d^6-3968 b c^5 d^5 e+4864 b^2 c^4 d^4 e^2-800 b^3 c^3 d^3 e^3-1400 b^4 c^2 d^2 e^4+1750 b^5 c d e^5-525 b^6 e^6\right )+3 A e \left (256 c^6 d^6-768 b c^5 d^5 e+320 b^2 c^4 d^4 e^2+640 b^3 c^3 d^3 e^3-1708 b^4 c^2 d^2 e^4+1260 b^5 c d e^5-315 b^6 e^6\right )\right ) \sqrt {b x+c x^2}}{107520 d^5 e^4 (c d-b e)^5 (d+e x)}-\frac {(B d-A e) x \left (b x+c x^2\right )^{3/2}}{7 d e (d+e x)^7}+\frac {b^4 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{1024 d^{11/2} (c d-b e)^{11/2}} \] Output:

-1/84*(B*d*(-5*b*e+8*c*d)+3*A*e*(-3*b*e+2*c*d))*x^2*(c*x^2+b*x)^(1/2)/d^2/ 
e^2/(e*x+d)^6-1/840*(B*d*(-35*b^2*e^2-10*b*c*d*e+48*c^2*d^2)+3*A*e*(-21*b^ 
2*e^2+8*b*c*d*e+12*c^2*d^2))*x*(c*x^2+b*x)^(1/2)/d^2/e^3/(-b*e+c*d)/(e*x+d 
)^5-1/2240*(3*A*e*(21*b^3*e^3-26*b^2*c*d*e^2-8*b*c^2*d^2*e+16*c^3*d^3)+B*d 
*(35*b^3*e^3-20*b^2*c*d*e^2-88*b*c^2*d^2*e+64*c^3*d^3))*(c*x^2+b*x)^(1/2)/ 
d^2/e^4/(-b*e+c*d)^2/(e*x+d)^4+1/13440*(B*d*(-35*b^4*e^4+50*b^3*c*d*e^3+28 
8*b^2*c^2*d^2*e^2-368*b*c^3*d^3*e+128*c^4*d^4)+3*A*e*(-21*b^4*e^4+44*b^3*c 
*d*e^3-12*b^2*c^2*d^2*e^2-64*b*c^3*d^3*e+32*c^4*d^4))*(c*x^2+b*x)^(1/2)/d^ 
3/e^4/(-b*e+c*d)^3/(e*x+d)^3+1/53760*(3*A*e*(105*b^5*e^5-322*b^4*c*d*e^4+2 
72*b^3*c^2*d^2*e^3+32*b^2*c^3*d^3*e^2-320*b*c^4*d^4*e+128*c^5*d^5)+B*d*(17 
5*b^5*e^5-420*b^4*c*d*e^4+80*b^3*c^2*d^2*e^3+1696*b^2*c^3*d^3*e^2-1728*b*c 
^4*d^4*e+512*c^5*d^5))*(c*x^2+b*x)^(1/2)/d^4/e^4/(-b*e+c*d)^4/(e*x+d)^2+1/ 
107520*(B*d*(-525*b^6*e^6+1750*b^5*c*d*e^5-1400*b^4*c^2*d^2*e^4-800*b^3*c^ 
3*d^3*e^3+4864*b^2*c^4*d^4*e^2-3968*b*c^5*d^5*e+1024*c^6*d^6)+3*A*e*(-315* 
b^6*e^6+1260*b^5*c*d*e^5-1708*b^4*c^2*d^2*e^4+640*b^3*c^3*d^3*e^3+320*b^2* 
c^4*d^4*e^2-768*b*c^5*d^5*e+256*c^6*d^6))*(c*x^2+b*x)^(1/2)/d^5/e^4/(-b*e+ 
c*d)^5/(e*x+d)-1/7*(-A*e+B*d)*x*(c*x^2+b*x)^(3/2)/d/e/(e*x+d)^7+1/1024*b^4 
*(48*A*c^3*d^3-24*b*c^2*d^2*(3*A*e+B*d)-b^3*e^2*(9*A*e+5*B*d)+2*b^2*c*d*e* 
(21*A*e+10*B*d))*arctanh((-b*e+c*d)^(1/2)*x/d^(1/2)/(c*x^2+b*x)^(1/2))/d^( 
11/2)/(-b*e+c*d)^(11/2)
 

Mathematica [A] (verified)

Time = 13.45 (sec) , antiderivative size = 505, normalized size of antiderivative = 0.52 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx=\frac {(x (b+c x))^{3/2} \left (-\frac {(B d-A e) x^{5/2} (b+c x)}{(d+e x)^7}-\frac {(9 A e (-2 c d+b e)+B d (4 c d+5 b e)) x^{5/2} (b+c x)}{12 d (c d-b e) (d+e x)^6}-\frac {\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) x^{5/2} (b+c x)}{120 d^2 (c d-b e)^2 (d+e x)^5}-\frac {7 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) \left (16 d^{5/2} (c d-b e)^{3/2} x^{3/2} (b+c x)^{5/2}-b (d+e x) \left (8 d^{5/2} \sqrt {c d-b e} \sqrt {x} (b+c x)^{5/2}-b (d+e x) \left (2 d^{3/2} \sqrt {c d-b e} \sqrt {x} (b+c x)^{3/2}+3 b (d+e x) \left (\sqrt {d} \sqrt {c d-b e} \sqrt {x} \sqrt {b+c x}+b (d+e x) \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )\right )\right )\right )}{3072 d^{9/2} (c d-b e)^{9/2} (b+c x)^{3/2} (d+e x)^4}\right )}{7 d (-c d+b e) x^{3/2}} \] Input:

Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^8,x]
 

Output:

((x*(b + c*x))^(3/2)*(-(((B*d - A*e)*x^(5/2)*(b + c*x))/(d + e*x)^7) - ((9 
*A*e*(-2*c*d + b*e) + B*d*(4*c*d + 5*b*e))*x^(5/2)*(b + c*x))/(12*d*(c*d - 
 b*e)*(d + e*x)^6) - ((B*d*(8*c^2*d^2 + 90*b*c*d*e - 35*b^2*e^2) - 3*A*e*( 
68*c^2*d^2 - 68*b*c*d*e + 21*b^2*e^2))*x^(5/2)*(b + c*x))/(120*d^2*(c*d - 
b*e)^2*(d + e*x)^5) - (7*(48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3* 
e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e))*(16*d^(5/2)*(c*d - b* 
e)^(3/2)*x^(3/2)*(b + c*x)^(5/2) - b*(d + e*x)*(8*d^(5/2)*Sqrt[c*d - b*e]* 
Sqrt[x]*(b + c*x)^(5/2) - b*(d + e*x)*(2*d^(3/2)*Sqrt[c*d - b*e]*Sqrt[x]*( 
b + c*x)^(3/2) + 3*b*(d + e*x)*(Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[x]*Sqrt[b + c 
*x] + b*(d + e*x)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x] 
)])))))/(3072*d^(9/2)*(c*d - b*e)^(9/2)*(b + c*x)^(3/2)*(d + e*x)^4)))/(7* 
d*(-(c*d) + b*e)*x^(3/2))
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 515, normalized size of antiderivative = 0.53, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1237, 27, 25, 1237, 27, 1228, 1152, 1152, 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.

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

\(\Big \downarrow \) 1237

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)}-\frac {\int -\frac {(14 A c d-b (5 B d+9 A e)+4 c (B d-A e) x) \left (c x^2+b x\right )^{3/2}}{2 (d+e x)^7}dx}{7 d (c d-b e)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int -\frac {(5 b B d-14 A c d+9 A b e-4 c (B d-A e) x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^7}dx}{14 d (c d-b e)}+\frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)}-\frac {\int \frac {(5 b B d-14 A c d+9 A b e-4 c (B d-A e) x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^7}dx}{14 d (c d-b e)}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{6 d (d+e x)^6 (c d-b e)}-\frac {\int \frac {\left (7 e (5 B d+9 A e) b^2-2 c d (40 B d+93 A e) b+168 A c^2 d^2-2 c (9 A e (2 c d-b e)-B d (4 c d+5 b e)) x\right ) \left (c x^2+b x\right )^{3/2}}{2 (d+e x)^6}dx}{6 d (c d-b e)}}{14 d (c d-b e)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{6 d (d+e x)^6 (c d-b e)}-\frac {\int \frac {\left (7 e (5 B d+9 A e) b^2-2 c d (40 B d+93 A e) b+168 A c^2 d^2-2 c (9 A e (2 c d-b e)-B d (4 c d+5 b e)) x\right ) \left (c x^2+b x\right )^{3/2}}{(d+e x)^6}dx}{12 d (c d-b e)}}{14 d (c d-b e)}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{6 d (d+e x)^6 (c d-b e)}-\frac {\frac {7 \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right ) \int \frac {\left (c x^2+b x\right )^{3/2}}{(d+e x)^5}dx}{2 d (c d-b e)}+\frac {\left (b x+c x^2\right )^{5/2} \left (B d \left (-35 b^2 e^2+90 b c d e+8 c^2 d^2\right )-3 A e \left (21 b^2 e^2-68 b c d e+68 c^2 d^2\right )\right )}{5 d (d+e x)^5 (c d-b e)}}{12 d (c d-b e)}}{14 d (c d-b e)}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{6 d (d+e x)^6 (c d-b e)}-\frac {\frac {7 \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right ) \left (\frac {\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d)}{8 d (d+e x)^4 (c d-b e)}-\frac {3 b^2 \int \frac {\sqrt {c x^2+b x}}{(d+e x)^3}dx}{16 d (c d-b e)}\right )}{2 d (c d-b e)}+\frac {\left (b x+c x^2\right )^{5/2} \left (B d \left (-35 b^2 e^2+90 b c d e+8 c^2 d^2\right )-3 A e \left (21 b^2 e^2-68 b c d e+68 c^2 d^2\right )\right )}{5 d (d+e x)^5 (c d-b e)}}{12 d (c d-b e)}}{14 d (c d-b e)}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{6 d (d+e x)^6 (c d-b e)}-\frac {\frac {7 \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right ) \left (\frac {\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d)}{8 d (d+e x)^4 (c d-b e)}-\frac {3 b^2 \left (\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{8 d (c d-b e)}\right )}{16 d (c d-b e)}\right )}{2 d (c d-b e)}+\frac {\left (b x+c x^2\right )^{5/2} \left (B d \left (-35 b^2 e^2+90 b c d e+8 c^2 d^2\right )-3 A e \left (21 b^2 e^2-68 b c d e+68 c^2 d^2\right )\right )}{5 d (d+e x)^5 (c d-b e)}}{12 d (c d-b e)}}{14 d (c d-b e)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{6 d (d+e x)^6 (c d-b e)}-\frac {\frac {7 \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right ) \left (\frac {\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d)}{8 d (d+e x)^4 (c d-b e)}-\frac {3 b^2 \left (\frac {b^2 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{4 d (c d-b e)}+\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}\right )}{16 d (c d-b e)}\right )}{2 d (c d-b e)}+\frac {\left (b x+c x^2\right )^{5/2} \left (B d \left (-35 b^2 e^2+90 b c d e+8 c^2 d^2\right )-3 A e \left (21 b^2 e^2-68 b c d e+68 c^2 d^2\right )\right )}{5 d (d+e x)^5 (c d-b e)}}{12 d (c d-b e)}}{14 d (c d-b e)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{6 d (d+e x)^6 (c d-b e)}-\frac {\frac {7 \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right ) \left (\frac {\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d)}{8 d (d+e x)^4 (c d-b e)}-\frac {3 b^2 \left (\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{3/2} (c d-b e)^{3/2}}\right )}{16 d (c d-b e)}\right )}{2 d (c d-b e)}+\frac {\left (b x+c x^2\right )^{5/2} \left (B d \left (-35 b^2 e^2+90 b c d e+8 c^2 d^2\right )-3 A e \left (21 b^2 e^2-68 b c d e+68 c^2 d^2\right )\right )}{5 d (d+e x)^5 (c d-b e)}}{12 d (c d-b e)}}{14 d (c d-b e)}\)

Input:

Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^8,x]
 

Output:

((B*d - A*e)*(b*x + c*x^2)^(5/2))/(7*d*(c*d - b*e)*(d + e*x)^7) - (((9*A*e 
*(2*c*d - b*e) - B*d*(4*c*d + 5*b*e))*(b*x + c*x^2)^(5/2))/(6*d*(c*d - b*e 
)*(d + e*x)^6) - (((B*d*(8*c^2*d^2 + 90*b*c*d*e - 35*b^2*e^2) - 3*A*e*(68* 
c^2*d^2 - 68*b*c*d*e + 21*b^2*e^2))*(b*x + c*x^2)^(5/2))/(5*d*(c*d - b*e)* 
(d + e*x)^5) + (7*(48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5* 
B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e))*(((b*d + (2*c*d - b*e)*x)*(b 
*x + c*x^2)^(3/2))/(8*d*(c*d - b*e)*(d + e*x)^4) - (3*b^2*(((b*d + (2*c*d 
- b*e)*x)*Sqrt[b*x + c*x^2])/(4*d*(c*d - b*e)*(d + e*x)^2) - (b^2*ArcTanh[ 
(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8 
*d^(3/2)*(c*d - b*e)^(3/2))))/(16*d*(c*d - b*e))))/(2*d*(c*d - b*e)))/(12* 
d*(c*d - b*e)))/(14*d*(c*d - b*e))
 

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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

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 1228

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

rule 1237

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

Maple [A] (verified)

Time = 1.88 (sec) , antiderivative size = 839, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(839\)
default	\(\text {Expression too large to display}\)	\(22885\)

Input:

int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^8,x,method=_RETURNVERBOSE)
 

Output:

-9/1024*((e*x+d)^7*(8/3*(-2*A*c^3+B*b*c^2)*d^3+8*c*(A*c-5/18*B*b)*e*b*d^2- 
14/3*e^2*(A*c-5/42*B*b)*b^2*d+A*b^3*e^3)*b^4*arctan((x*(c*x+b))^(1/2)/x*d/ 
(d*(b*e-c*d))^(1/2))+(-16/3*c^2*(-1/2*B*b^4+c*(1/3*B*x+A)*b^3-2/3*c^2*x*(2 
/5*B*x+A)*b^2-8*c^3*x^2*(11/15*B*x+A)*b-16/3*c^4*(4/5*B*x+A)*x^3)*d^9+8*c* 
e*(-5/18*b^5*B+c*(65/27*B*x+A)*b^4-46/9*c^2*x*(109/345*B*x+A)*b^3-1124/45* 
c^3*x^2*(214/281*B*x+A)*b^2-176/15*c^4*(97/99*B*x+A)*x^3*b+32/15*c^5*(4/9* 
B*x+A)*x^4)*d^8-14/3*e^2*(-5/42*B*b^6+c*(205/63*B*x+A)*b^5-254/21*c^2*x*(2 
038/1905*B*x+A)*b^4-2376/35*c^3*x^2*(16787/18711*B*x+A)*b^3-1424/105*c^4*x 
^3*(3638/1869*B*x+A)*b^2+1216/105*c^5*(221/399*B*x+A)*x^4*b-128/105*c^6*(4 
/21*B*x+A)*x^5)*d^7+e^3*((A+100/27*B*x)*b^6-286/9*c*(599/429*B*x+A)*x*b^5- 
2548/9*c^2*(51448/66885*B*x+A)*x^2*b^4+5648/105*c^3*(-2207/3177*B*x+A)*x^3 
*b^3+1056/35*c^4*x^4*(1126/891*B*x+A)*b^2-5504/315*c^5*x^5*(31/129*B*x+A)* 
b+256/315*x^6*c^6*A)*d^6+20/3*e^4*x*((283/180*B*x+A)*b^5+304/15*c*x*(4285/ 
6384*B*x+A)*b^4-2522/105*c^2*(3097/18915*B*x+A)*x^2*b^3+3008/525*c^3*x^3*( 
-251/1128*B*x+A)*b^2+656/525*(76/123*B*x+A)*c^4*x^4*b-64/175*x^5*c^5*A)*b* 
d^5-1199/45*e^5*x^2*b^2*((5120/8393*B*x+A)*b^4-37084/8393*c*x*(8290/27813* 
B*x+A)*b^3+32528/8393*c^2*(1175/12198*B*x+A)*x^2*b^2-4320/8393*c^3*x^3*(-5 
/81*B*x+A)*b-320/8393*A*c^4*x^4)*d^4-1024/35*e^6*x^3*b^3*((9905/27648*B*x+ 
A)*b^3-11929/4608*c*(5845/35787*B*x+A)*x*b^2+953/768*c^2*(350/8577*B*x+A)* 
x^2*b-5/72*A*c^3*x^3)*d^3-283/15*e^7*((500/2547*B*x+A)*b^2-1202/849*c*x...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2690 vs. \(2 (928) = 1856\).

Time = 0.43 (sec) , antiderivative size = 5395, normalized size of antiderivative = 5.57 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^8,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{\left (d + e x\right )^{8}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**8,x)
 

Output:

Integral((x*(b + c*x))**(3/2)*(A + B*x)/(d + e*x)**8, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^8,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8108 vs. \(2 (928) = 1856\).

Time = 0.40 (sec) , antiderivative size = 8108, normalized size of antiderivative = 8.38 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^8,x, algorithm="giac")
 

Output:

-1/1024*(24*B*b^5*c^2*d^3 - 48*A*b^4*c^3*d^3 - 20*B*b^6*c*d^2*e + 72*A*b^5 
*c^2*d^2*e + 5*B*b^7*d*e^2 - 42*A*b^6*c*d*e^2 + 9*A*b^7*e^3)*arctan(-((sqr 
t(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^5*d^1 
0 - 5*b*c^4*d^9*e + 10*b^2*c^3*d^8*e^2 - 10*b^3*c^2*d^7*e^3 + 5*b^4*c*d^6* 
e^4 - b^5*d^5*e^5)*sqrt(-c*d^2 + b*d*e)) + 1/107520*(2520*(sqrt(c)*x - sqr 
t(c*x^2 + b*x))^13*B*b^5*c^2*d^3*e^11 - 5040*(sqrt(c)*x - sqrt(c*x^2 + b*x 
))^13*A*b^4*c^3*d^3*e^11 - 2100*(sqrt(c)*x - sqrt(c*x^2 + b*x))^13*B*b^6*c 
*d^2*e^12 + 7560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^13*A*b^5*c^2*d^2*e^12 + 5 
25*(sqrt(c)*x - sqrt(c*x^2 + b*x))^13*B*b^7*d*e^13 - 4410*(sqrt(c)*x - sqr 
t(c*x^2 + b*x))^13*A*b^6*c*d*e^13 + 945*(sqrt(c)*x - sqrt(c*x^2 + b*x))^13 
*A*b^7*e^14 + 32760*(sqrt(c)*x - sqrt(c*x^2 + b*x))^12*B*b^5*c^(5/2)*d^4*e 
^10 - 65520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^12*A*b^4*c^(7/2)*d^4*e^10 - 27 
300*(sqrt(c)*x - sqrt(c*x^2 + b*x))^12*B*b^6*c^(3/2)*d^3*e^11 + 98280*(sqr 
t(c)*x - sqrt(c*x^2 + b*x))^12*A*b^5*c^(5/2)*d^3*e^11 + 6825*(sqrt(c)*x - 
sqrt(c*x^2 + b*x))^12*B*b^7*sqrt(c)*d^2*e^12 - 57330*(sqrt(c)*x - sqrt(c*x 
^2 + b*x))^12*A*b^6*c^(3/2)*d^2*e^12 + 12285*(sqrt(c)*x - sqrt(c*x^2 + b*x 
))^12*A*b^7*sqrt(c)*d*e^13 + 286720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*c 
^8*d^10*e^4 - 1433600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b*c^7*d^9*e^5 + 
 2867200*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b^2*c^6*d^8*e^6 - 2867200*(s 
qrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b^3*c^5*d^7*e^7 + 1433600*(sqrt(c)*x...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^8} \,d x \] Input:

int(((b*x + c*x^2)^(3/2)*(A + B*x))/(d + e*x)^8,x)
                                                                                    
                                                                                    
 

Output:

int(((b*x + c*x^2)^(3/2)*(A + B*x))/(d + e*x)^8, x)
 

Reduce [F]

\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx=\int \frac {\left (B x +A \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{\left (e x +d \right )^{8}}d x \] Input:

int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^8,x)
 

Output:

int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^8,x)