3.18.36 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^{11}} \, dx\) [1736]

3.18.36.1 Optimal result

Integrand size = 33, antiderivative size = 298 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=-\frac {(b d-a e)^3 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^5 (a+b x) (d+e x)^{10}}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x) (d+e x)^9}-\frac {3 b (b d-a e) (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x) (d+e x)^8}+\frac {b^2 (4 b B d-A b e-3 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x) (d+e x)^6} \]

output

-1/10*(-a*e+b*d)^3*(-A*e+B*d)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^10+1/9 
*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b*d)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+ 
d)^9-3/8*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*((b*x+a)^2)^(1/2)/e^5/(b*x+a) 
/(e*x+d)^8+1/7*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/ 
(e*x+d)^7-1/6*b^3*B*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^6
 

3.18.36.2 Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.78 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (28 a^3 e^3 (9 A e+B (d+10 e x))+21 a^2 b e^2 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+3 a b^2 e \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )\right )+b^3 \left (3 A e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )\right )}{2520 e^5 (a+b x) (d+e x)^{10}} \]

input

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^11,x]
 

output

-1/2520*(Sqrt[(a + b*x)^2]*(28*a^3*e^3*(9*A*e + B*(d + 10*e*x)) + 21*a^2*b 
*e^2*(4*A*e*(d + 10*e*x) + B*(d^2 + 10*d*e*x + 45*e^2*x^2)) + 3*a*b^2*e*(7 
*A*e*(d^2 + 10*d*e*x + 45*e^2*x^2) + 3*B*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 
+ 120*e^3*x^3)) + b^3*(3*A*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^ 
3) + 2*B*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) 
)))/(e^5*(a + b*x)*(d + e*x)^10)
 

3.18.36.3 Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.64, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 86, 2009}

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.

\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A+B x)}{(d+e x)^{11}} \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^3 (a+b x)^3 (A+B x)}{(d+e x)^{11}}dx}{b^3 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{11}}dx}{a+b x}\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B b^3}{e^4 (d+e x)^7}+\frac {(-4 b B d+A b e+3 a B e) b^2}{e^4 (d+e x)^8}-\frac {3 (b d-a e) (-2 b B d+A b e+a B e) b}{e^4 (d+e x)^9}+\frac {(a e-b d)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^{10}}+\frac {(a e-b d)^3 (A e-B d)}{e^4 (d+e x)^{11}}\right )dx}{a+b x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {b^2 (-3 a B e-A b e+4 b B d)}{7 e^5 (d+e x)^7}-\frac {3 b (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5 (d+e x)^8}+\frac {(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5 (d+e x)^9}-\frac {(b d-a e)^3 (B d-A e)}{10 e^5 (d+e x)^{10}}-\frac {b^3 B}{6 e^5 (d+e x)^6}\right )}{a+b x}\)

input

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^11,x]
 

output

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/10*((b*d - a*e)^3*(B*d - A*e))/(e^5*(d 
+ e*x)^10) + ((b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e))/(9*e^5*(d + e*x)^ 
9) - (3*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e))/(8*e^5*(d + e*x)^8) + (b^ 
2*(4*b*B*d - A*b*e - 3*a*B*e))/(7*e^5*(d + e*x)^7) - (b^3*B)/(6*e^5*(d + e 
*x)^6)))/(a + b*x)
 

3.18.36.3.1 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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 1187

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ 
IntPart[p]*(b/2 + c*x)^(2*FracPart[p]))   Int[(d + e*x)^m*(f + g*x)^n*(b/2 
+ c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 
 - 4*a*c, 0] &&  !IntegerQ[p]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.18.36.4 Maple [A] (verified)

Time = 3.57 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {B \,b^{3} x^{4}}{6 e}-\frac {b^{2} \left (3 A b e +9 B a e +2 B b d \right ) x^{3}}{21 e^{2}}-\frac {b \left (21 A a b \,e^{2}+3 A \,b^{2} d e +21 a^{2} B \,e^{2}+9 B a b d e +2 B \,b^{2} d^{2}\right ) x^{2}}{56 e^{3}}-\frac {\left (84 A \,a^{2} b \,e^{3}+21 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +28 B \,e^{3} a^{3}+21 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e +2 B \,b^{3} d^{3}\right ) x}{252 e^{4}}-\frac {252 A \,a^{3} e^{4}+84 A \,a^{2} b d \,e^{3}+21 A a \,b^{2} d^{2} e^{2}+3 A \,b^{3} d^{3} e +28 B \,a^{3} d \,e^{3}+21 B \,a^{2} b \,d^{2} e^{2}+9 B a \,b^{2} d^{3} e +2 B \,b^{3} d^{4}}{2520 e^{5}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{10}}\)	\(286\)
gosper	\(-\frac {\left (420 B \,b^{3} e^{4} x^{4}+360 A \,b^{3} e^{4} x^{3}+1080 B a \,b^{2} e^{4} x^{3}+240 B \,b^{3} d \,e^{3} x^{3}+945 A a \,b^{2} e^{4} x^{2}+135 A \,b^{3} d \,e^{3} x^{2}+945 B \,a^{2} b \,e^{4} x^{2}+405 B a \,b^{2} d \,e^{3} x^{2}+90 B \,b^{3} d^{2} e^{2} x^{2}+840 A \,a^{2} b \,e^{4} x +210 A a \,b^{2} d \,e^{3} x +30 A \,b^{3} d^{2} e^{2} x +280 B \,a^{3} e^{4} x +210 B \,a^{2} b d \,e^{3} x +90 B a \,b^{2} d^{2} e^{2} x +20 B \,b^{3} d^{3} e x +252 A \,a^{3} e^{4}+84 A \,a^{2} b d \,e^{3}+21 A a \,b^{2} d^{2} e^{2}+3 A \,b^{3} d^{3} e +28 B \,a^{3} d \,e^{3}+21 B \,a^{2} b \,d^{2} e^{2}+9 B a \,b^{2} d^{3} e +2 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{2520 e^{5} \left (e x +d \right )^{10} \left (b x +a \right )^{3}}\)	\(317\)
default	\(-\frac {\left (420 B \,b^{3} e^{4} x^{4}+360 A \,b^{3} e^{4} x^{3}+1080 B a \,b^{2} e^{4} x^{3}+240 B \,b^{3} d \,e^{3} x^{3}+945 A a \,b^{2} e^{4} x^{2}+135 A \,b^{3} d \,e^{3} x^{2}+945 B \,a^{2} b \,e^{4} x^{2}+405 B a \,b^{2} d \,e^{3} x^{2}+90 B \,b^{3} d^{2} e^{2} x^{2}+840 A \,a^{2} b \,e^{4} x +210 A a \,b^{2} d \,e^{3} x +30 A \,b^{3} d^{2} e^{2} x +280 B \,a^{3} e^{4} x +210 B \,a^{2} b d \,e^{3} x +90 B a \,b^{2} d^{2} e^{2} x +20 B \,b^{3} d^{3} e x +252 A \,a^{3} e^{4}+84 A \,a^{2} b d \,e^{3}+21 A a \,b^{2} d^{2} e^{2}+3 A \,b^{3} d^{3} e +28 B \,a^{3} d \,e^{3}+21 B \,a^{2} b \,d^{2} e^{2}+9 B a \,b^{2} d^{3} e +2 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{2520 e^{5} \left (e x +d \right )^{10} \left (b x +a \right )^{3}}\)	\(317\)

input

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^11,x,method=_RETURNVERBOSE 
)
 

output

((b*x+a)^2)^(1/2)/(b*x+a)*(-1/6/e*B*b^3*x^4-1/21*b^2/e^2*(3*A*b*e+9*B*a*e+ 
2*B*b*d)*x^3-1/56*b/e^3*(21*A*a*b*e^2+3*A*b^2*d*e+21*B*a^2*e^2+9*B*a*b*d*e 
+2*B*b^2*d^2)*x^2-1/252/e^4*(84*A*a^2*b*e^3+21*A*a*b^2*d*e^2+3*A*b^3*d^2*e 
+28*B*a^3*e^3+21*B*a^2*b*d*e^2+9*B*a*b^2*d^2*e+2*B*b^3*d^3)*x-1/2520/e^5*( 
252*A*a^3*e^4+84*A*a^2*b*d*e^3+21*A*a*b^2*d^2*e^2+3*A*b^3*d^3*e+28*B*a^3*d 
*e^3+21*B*a^2*b*d^2*e^2+9*B*a*b^2*d^3*e+2*B*b^3*d^4))/(e*x+d)^10
 

3.18.36.5 Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.22 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=-\frac {420 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 252 \, A a^{3} e^{4} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 21 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 28 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 120 \, {\left (2 \, B b^{3} d e^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 45 \, {\left (2 \, B b^{3} d^{2} e^{2} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 21 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 10 \, {\left (2 \, B b^{3} d^{3} e + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 21 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 28 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{2520 \, {\left (e^{15} x^{10} + 10 \, d e^{14} x^{9} + 45 \, d^{2} e^{13} x^{8} + 120 \, d^{3} e^{12} x^{7} + 210 \, d^{4} e^{11} x^{6} + 252 \, d^{5} e^{10} x^{5} + 210 \, d^{6} e^{9} x^{4} + 120 \, d^{7} e^{8} x^{3} + 45 \, d^{8} e^{7} x^{2} + 10 \, d^{9} e^{6} x + d^{10} e^{5}\right )}} \]

input

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^11,x, algorithm="fri 
cas")
 

output

-1/2520*(420*B*b^3*e^4*x^4 + 2*B*b^3*d^4 + 252*A*a^3*e^4 + 3*(3*B*a*b^2 + 
A*b^3)*d^3*e + 21*(B*a^2*b + A*a*b^2)*d^2*e^2 + 28*(B*a^3 + 3*A*a^2*b)*d*e 
^3 + 120*(2*B*b^3*d*e^3 + 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 45*(2*B*b^3*d^2 
*e^2 + 3*(3*B*a*b^2 + A*b^3)*d*e^3 + 21*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 10* 
(2*B*b^3*d^3*e + 3*(3*B*a*b^2 + A*b^3)*d^2*e^2 + 21*(B*a^2*b + A*a*b^2)*d* 
e^3 + 28*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^15*x^10 + 10*d*e^14*x^9 + 45*d^2*e 
^13*x^8 + 120*d^3*e^12*x^7 + 210*d^4*e^11*x^6 + 252*d^5*e^10*x^5 + 210*d^6 
*e^9*x^4 + 120*d^7*e^8*x^3 + 45*d^8*e^7*x^2 + 10*d^9*e^6*x + d^10*e^5)
 

3.18.36.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=\text {Timed out} \]

input

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**11,x)
 

output

Timed out
 

3.18.36.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=\text {Exception raised: ValueError} \]

input

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^11,x, algorithm="max 
ima")
 

output

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

3.18.36.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (233) = 466\).

Time = 0.29 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=\frac {{\left (2 \, B b^{10} d - 5 \, B a b^{9} e + 3 \, A b^{10} e\right )} \mathrm {sgn}\left (b x + a\right )}{2520 \, {\left (b^{7} d^{7} e^{5} - 7 \, a b^{6} d^{6} e^{6} + 21 \, a^{2} b^{5} d^{5} e^{7} - 35 \, a^{3} b^{4} d^{4} e^{8} + 35 \, a^{4} b^{3} d^{3} e^{9} - 21 \, a^{5} b^{2} d^{2} e^{10} + 7 \, a^{6} b d e^{11} - a^{7} e^{12}\right )}} - \frac {420 \, B b^{3} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 240 \, B b^{3} d e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 1080 \, B a b^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 360 \, A b^{3} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 90 \, B b^{3} d^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 405 \, B a b^{2} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 135 \, A b^{3} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 945 \, B a^{2} b e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 945 \, A a b^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, B b^{3} d^{3} e x \mathrm {sgn}\left (b x + a\right ) + 90 \, B a b^{2} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 30 \, A b^{3} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 210 \, B a^{2} b d e^{3} x \mathrm {sgn}\left (b x + a\right ) + 210 \, A a b^{2} d e^{3} x \mathrm {sgn}\left (b x + a\right ) + 280 \, B a^{3} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 840 \, A a^{2} b e^{4} x \mathrm {sgn}\left (b x + a\right ) + 2 \, B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) + 9 \, B a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, A b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 21 \, B a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, A a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 28 \, B a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 84 \, A a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 252 \, A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )}{2520 \, {\left (e x + d\right )}^{10} e^{5}} \]

input

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^11,x, algorithm="gia 
c")
 

output

1/2520*(2*B*b^10*d - 5*B*a*b^9*e + 3*A*b^10*e)*sgn(b*x + a)/(b^7*d^7*e^5 - 
 7*a*b^6*d^6*e^6 + 21*a^2*b^5*d^5*e^7 - 35*a^3*b^4*d^4*e^8 + 35*a^4*b^3*d^ 
3*e^9 - 21*a^5*b^2*d^2*e^10 + 7*a^6*b*d*e^11 - a^7*e^12) - 1/2520*(420*B*b 
^3*e^4*x^4*sgn(b*x + a) + 240*B*b^3*d*e^3*x^3*sgn(b*x + a) + 1080*B*a*b^2* 
e^4*x^3*sgn(b*x + a) + 360*A*b^3*e^4*x^3*sgn(b*x + a) + 90*B*b^3*d^2*e^2*x 
^2*sgn(b*x + a) + 405*B*a*b^2*d*e^3*x^2*sgn(b*x + a) + 135*A*b^3*d*e^3*x^2 
*sgn(b*x + a) + 945*B*a^2*b*e^4*x^2*sgn(b*x + a) + 945*A*a*b^2*e^4*x^2*sgn 
(b*x + a) + 20*B*b^3*d^3*e*x*sgn(b*x + a) + 90*B*a*b^2*d^2*e^2*x*sgn(b*x + 
 a) + 30*A*b^3*d^2*e^2*x*sgn(b*x + a) + 210*B*a^2*b*d*e^3*x*sgn(b*x + a) + 
 210*A*a*b^2*d*e^3*x*sgn(b*x + a) + 280*B*a^3*e^4*x*sgn(b*x + a) + 840*A*a 
^2*b*e^4*x*sgn(b*x + a) + 2*B*b^3*d^4*sgn(b*x + a) + 9*B*a*b^2*d^3*e*sgn(b 
*x + a) + 3*A*b^3*d^3*e*sgn(b*x + a) + 21*B*a^2*b*d^2*e^2*sgn(b*x + a) + 2 
1*A*a*b^2*d^2*e^2*sgn(b*x + a) + 28*B*a^3*d*e^3*sgn(b*x + a) + 84*A*a^2*b* 
d*e^3*sgn(b*x + a) + 252*A*a^3*e^4*sgn(b*x + a))/((e*x + d)^10*e^5)
 

3.18.36.9 Mupad [B] (verification not implemented)

Time = 10.94 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.94 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11}} \, dx=-\frac {\left (\frac {A\,b^3\,e-3\,B\,b^3\,d+3\,B\,a\,b^2\,e}{7\,e^5}-\frac {B\,b^3\,d}{7\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {\left (\frac {A\,a^3}{10\,e}-\frac {d\,\left (\frac {B\,a^3+3\,A\,b\,a^2}{10\,e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,b^3+3\,B\,a\,b^2}{10\,e}-\frac {B\,b^3\,d}{10\,e^2}\right )}{e}-\frac {3\,a\,b\,\left (A\,b+B\,a\right )}{10\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}-\frac {\left (\frac {B\,a^3\,e^3-3\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e-3\,A\,a\,b^2\,d\,e^2-B\,b^3\,d^3+A\,b^3\,d^2\,e}{9\,e^5}-\frac {d\,\left (\frac {3\,B\,a^2\,b\,e^3-3\,B\,a\,b^2\,d\,e^2+3\,A\,a\,b^2\,e^3+B\,b^3\,d^2\,e-A\,b^3\,d\,e^2}{9\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-B\,b\,d\right )}{9\,e^3}-\frac {B\,b^3\,d}{9\,e^3}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}-\frac {\left (\frac {3\,B\,a^2\,b\,e^2-6\,B\,a\,b^2\,d\,e+3\,A\,a\,b^2\,e^2+3\,B\,b^3\,d^2-2\,A\,b^3\,d\,e}{8\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-2\,B\,b\,d\right )}{8\,e^4}-\frac {B\,b^3\,d}{8\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6} \]

input

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(d + e*x)^11,x)
 

output

- (((A*b^3*e - 3*B*b^3*d + 3*B*a*b^2*e)/(7*e^5) - (B*b^3*d)/(7*e^5))*(a^2 
+ b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - (((A*a^3)/(10*e) - ( 
d*((B*a^3 + 3*A*a^2*b)/(10*e) + (d*((d*((A*b^3 + 3*B*a*b^2)/(10*e) - (B*b^ 
3*d)/(10*e^2)))/e - (3*a*b*(A*b + B*a))/(10*e)))/e))/e)*(a^2 + b^2*x^2 + 2 
*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((B*a^3*e^3 - B*b^3*d^3 + 3*A*a 
^2*b*e^3 + A*b^3*d^2*e - 3*A*a*b^2*d*e^2 + 3*B*a*b^2*d^2*e - 3*B*a^2*b*d*e 
^2)/(9*e^5) - (d*((3*A*a*b^2*e^3 + 3*B*a^2*b*e^3 - A*b^3*d*e^2 + B*b^3*d^2 
*e - 3*B*a*b^2*d*e^2)/(9*e^5) - (d*((b^2*(A*b*e + 3*B*a*e - B*b*d))/(9*e^3 
) - (B*b^3*d)/(9*e^3)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x) 
*(d + e*x)^9) - (((3*B*b^3*d^2 - 2*A*b^3*d*e + 3*A*a*b^2*e^2 + 3*B*a^2*b*e 
^2 - 6*B*a*b^2*d*e)/(8*e^5) - (d*((b^2*(A*b*e + 3*B*a*e - 2*B*b*d))/(8*e^4 
) - (B*b^3*d)/(8*e^4)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d 
+ e*x)^8) - (B*b^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(6*e^5*(a + b*x)*(d + 
e*x)^6)