\(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^2} \, dx\) [416]

Optimal result

Integrand size = 33, antiderivative size = 423 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {5 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {b^4 (6 b B d-A b e-5 a B e) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}+\frac {b^5 B (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}-\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \] Output:

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

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.20 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {\sqrt {(a+b x)^2} \left (60 a^5 e^5 (B d-A e)+300 a^4 b e^4 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+300 a^3 b^2 e^3 \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )+100 a^2 b^3 e^2 \left (3 A e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+2 B \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )+25 a b^4 e \left (4 A e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+B \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )+b^5 \left (5 A e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )-6 B \left (10 d^6-50 d^5 e x-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4+3 d e^5 x^5-2 e^6 x^6\right )\right )-60 (b d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x) \log (d+e x)\right )}{60 e^7 (a+b x) (d+e x)} \] Input:

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

Output:

(Sqrt[(a + b*x)^2]*(60*a^5*e^5*(B*d - A*e) + 300*a^4*b*e^4*(A*d*e + B*(-d^ 
2 + d*e*x + e^2*x^2)) + 300*a^3*b^2*e^3*(2*A*e*(-d^2 + d*e*x + e^2*x^2) + 
B*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + 100*a^2*b^3*e^2*(3*A*e*(2 
*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 2*B*(-3*d^4 + 9*d^3*e*x + 6*d^ 
2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)) + 25*a*b^4*e*(4*A*e*(-3*d^4 + 9*d^3*e* 
x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*e*x - 30*d 
^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + b^5*(5*A*e*(12*d 
^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^ 
5) - 6*B*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*d^3*e^3*x^3 - 5*d^2*e^ 
4*x^4 + 3*d*e^5*x^5 - 2*e^6*x^6)) - 60*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - 
a*B*e)*(d + e*x)*Log[d + e*x]))/(60*e^7*(a + b*x)*(d + e*x))
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.62, 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.

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

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{(d+e x)^2}dx}{b^5 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{(d+e x)^2}dx}{a+b x}\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(684\) vs. \(2(340)=680\).

Time = 1.52 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.62

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b \left (-20 B a \,b^{3} d^{3} e x +\frac {5}{3} A a \,b^{3} e^{4} x^{3}-20 A \,a^{2} b^{2} d \,e^{3} x +30 B \,a^{2} b^{2} d^{2} e^{2} x +10 A \,a^{3} b \,e^{4} x +\frac {1}{5} b^{4} B \,x^{5} e^{4}+\frac {1}{4} A \,b^{4} e^{4} x^{4}+\frac {5}{4} B a \,b^{3} e^{4} x^{4}-4 A \,b^{4} d^{3} e x -20 B \,a^{3} b d \,e^{3} x +15 A a \,b^{3} d^{2} e^{2} x +\frac {10}{3} B \,a^{2} b^{2} e^{4} x^{3}+B \,b^{4} d^{2} e^{2} x^{3}+5 A \,a^{2} b^{2} e^{4} x^{2}-\frac {2}{3} A \,b^{4} d \,e^{3} x^{3}-\frac {1}{2} B \,b^{4} d \,e^{3} x^{4}+\frac {3}{2} A \,b^{4} d^{2} e^{2} x^{2}+5 B \,a^{3} b \,e^{4} x^{2}-2 B \,b^{4} d^{3} e \,x^{2}-\frac {10}{3} B a \,b^{3} d \,e^{3} x^{3}+5 B \,a^{4} e^{4} x +5 B \,b^{4} d^{4} x -10 B \,a^{2} b^{2} d \,e^{3} x^{2}+\frac {15}{2} B a \,b^{3} d^{2} e^{2} x^{2}-5 A a \,b^{3} d \,e^{3} x^{2}\right )}{\left (b x +a \right ) e^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +B \,a^{5} e^{5}-10 B \,a^{4} b d \,e^{4}+30 B \,a^{3} b^{2} d^{2} e^{3}-40 B \,a^{2} b^{3} d^{3} e^{2}+25 B a \,b^{4} d^{4} e -6 B \,b^{5} d^{5}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A \,a^{5} e^{6}-5 A \,a^{4} b d \,e^{5}+10 A \,a^{3} b^{2} d^{2} e^{4}-10 A \,a^{2} b^{3} d^{3} e^{3}+5 A a \,b^{4} d^{4} e^{2}-A \,b^{5} d^{5} e -B \,a^{5} d \,e^{5}+5 B \,a^{4} b \,d^{2} e^{4}-10 B \,a^{3} b^{2} d^{3} e^{3}+10 B \,a^{2} b^{3} d^{4} e^{2}-5 B a \,b^{4} d^{5} e +b^{5} B \,d^{6}\right )}{\left (b x +a \right ) e^{7} \left (e x +d \right )}\)	\(685\)
default	\(\text {Expression too large to display}\)	\(1084\)

Input:

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

Output:

((b*x+a)^2)^(1/2)/(b*x+a)*b/e^6*(-20*B*a*b^3*d^3*e*x+5/3*A*a*b^3*e^4*x^3-2 
0*A*a^2*b^2*d*e^3*x+30*B*a^2*b^2*d^2*e^2*x+10*A*a^3*b*e^4*x+1/5*b^4*B*x^5* 
e^4+1/4*A*b^4*e^4*x^4+5/4*B*a*b^3*e^4*x^4-4*A*b^4*d^3*e*x-20*B*a^3*b*d*e^3 
*x+15*A*a*b^3*d^2*e^2*x+10/3*B*a^2*b^2*e^4*x^3+B*b^4*d^2*e^2*x^3+5*A*a^2*b 
^2*e^4*x^2-2/3*A*b^4*d*e^3*x^3-1/2*B*b^4*d*e^3*x^4+3/2*A*b^4*d^2*e^2*x^2+5 
*B*a^3*b*e^4*x^2-2*B*b^4*d^3*e*x^2-10/3*B*a*b^3*d*e^3*x^3+5*B*a^4*e^4*x+5* 
B*b^4*d^4*x-10*B*a^2*b^2*d*e^3*x^2+15/2*B*a*b^3*d^2*e^2*x^2-5*A*a*b^3*d*e^ 
3*x^2)+((b*x+a)^2)^(1/2)/(b*x+a)/e^7*(5*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+30* 
A*a^2*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+B*a^5*e^5-10*B*a^4*b*d* 
e^4+30*B*a^3*b^2*d^2*e^3-40*B*a^2*b^3*d^3*e^2+25*B*a*b^4*d^4*e-6*B*b^5*d^5 
)*ln(e*x+d)-((b*x+a)^2)^(1/2)/(b*x+a)/e^7*(A*a^5*e^6-5*A*a^4*b*d*e^5+10*A* 
a^3*b^2*d^2*e^4-10*A*a^2*b^3*d^3*e^3+5*A*a*b^4*d^4*e^2-A*b^5*d^5*e-B*a^5*d 
*e^5+5*B*a^4*b*d^2*e^4-10*B*a^3*b^2*d^3*e^3+10*B*a^2*b^3*d^4*e^2-5*B*a*b^4 
*d^5*e+B*b^5*d^6)/(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 797 vs. \(2 (340) = 680\).

Time = 0.09 (sec) , antiderivative size = 797, normalized size of antiderivative = 1.88 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/60*(12*B*b^5*e^6*x^6 - 60*B*b^5*d^6 - 60*A*a^5*e^6 + 60*(5*B*a*b^4 + A*b 
^5)*d^5*e - 300*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 600*(B*a^3*b^2 + A*a^2*b 
^3)*d^3*e^3 - 300*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 60*(B*a^5 + 5*A*a^4*b) 
*d*e^5 - 3*(6*B*b^5*d*e^5 - 5*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(6*B*b^5*d^ 
2*e^4 - 5*(5*B*a*b^4 + A*b^5)*d*e^5 + 20*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 
- 10*(6*B*b^5*d^3*e^3 - 5*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 20*(2*B*a^2*b^3 + 
A*a*b^4)*d*e^5 - 30*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 30*(6*B*b^5*d^4*e^2 
 - 5*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 20*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 30 
*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 10*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 60* 
(5*B*b^5*d^5*e - 4*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 15*(2*B*a^2*b^3 + A*a*b^4 
)*d^3*e^3 - 20*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 5*(B*a^4*b + 2*A*a^3*b^2) 
*d*e^5)*x - 60*(6*B*b^5*d^6 - 5*(5*B*a*b^4 + A*b^5)*d^5*e + 20*(2*B*a^2*b^ 
3 + A*a*b^4)*d^4*e^2 - 30*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 10*(B*a^4*b + 
2*A*a^3*b^2)*d^2*e^4 - (B*a^5 + 5*A*a^4*b)*d*e^5 + (6*B*b^5*d^5*e - 5*(5*B 
*a*b^4 + A*b^5)*d^4*e^2 + 20*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 30*(B*a^3*b 
^2 + A*a^2*b^3)*d^2*e^4 + 10*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - (B*a^5 + 5*A* 
a^4*b)*e^6)*x)*log(e*x + d))/(e^8*x + d*e^7)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 944 vs. \(2 (340) = 680\).

Time = 0.18 (sec) , antiderivative size = 944, normalized size of antiderivative = 2.23 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-(6*B*b^5*d^5*sgn(b*x + a) - 25*B*a*b^4*d^4*e*sgn(b*x + a) - 5*A*b^5*d^4*e 
*sgn(b*x + a) + 40*B*a^2*b^3*d^3*e^2*sgn(b*x + a) + 20*A*a*b^4*d^3*e^2*sgn 
(b*x + a) - 30*B*a^3*b^2*d^2*e^3*sgn(b*x + a) - 30*A*a^2*b^3*d^2*e^3*sgn(b 
*x + a) + 10*B*a^4*b*d*e^4*sgn(b*x + a) + 20*A*a^3*b^2*d*e^4*sgn(b*x + a) 
- B*a^5*e^5*sgn(b*x + a) - 5*A*a^4*b*e^5*sgn(b*x + a))*log(abs(e*x + d))/e 
^7 - (B*b^5*d^6*sgn(b*x + a) - 5*B*a*b^4*d^5*e*sgn(b*x + a) - A*b^5*d^5*e* 
sgn(b*x + a) + 10*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 5*A*a*b^4*d^4*e^2*sgn(b 
*x + a) - 10*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 10*A*a^2*b^3*d^3*e^3*sgn(b*x 
 + a) + 5*B*a^4*b*d^2*e^4*sgn(b*x + a) + 10*A*a^3*b^2*d^2*e^4*sgn(b*x + a) 
 - B*a^5*d*e^5*sgn(b*x + a) - 5*A*a^4*b*d*e^5*sgn(b*x + a) + A*a^5*e^6*sgn 
(b*x + a))/((e*x + d)*e^7) + 1/60*(12*B*b^5*e^8*x^5*sgn(b*x + a) - 30*B*b^ 
5*d*e^7*x^4*sgn(b*x + a) + 75*B*a*b^4*e^8*x^4*sgn(b*x + a) + 15*A*b^5*e^8* 
x^4*sgn(b*x + a) + 60*B*b^5*d^2*e^6*x^3*sgn(b*x + a) - 200*B*a*b^4*d*e^7*x 
^3*sgn(b*x + a) - 40*A*b^5*d*e^7*x^3*sgn(b*x + a) + 200*B*a^2*b^3*e^8*x^3* 
sgn(b*x + a) + 100*A*a*b^4*e^8*x^3*sgn(b*x + a) - 120*B*b^5*d^3*e^5*x^2*sg 
n(b*x + a) + 450*B*a*b^4*d^2*e^6*x^2*sgn(b*x + a) + 90*A*b^5*d^2*e^6*x^2*s 
gn(b*x + a) - 600*B*a^2*b^3*d*e^7*x^2*sgn(b*x + a) - 300*A*a*b^4*d*e^7*x^2 
*sgn(b*x + a) + 300*B*a^3*b^2*e^8*x^2*sgn(b*x + a) + 300*A*a^2*b^3*e^8*x^2 
*sgn(b*x + a) + 300*B*b^5*d^4*e^4*x*sgn(b*x + a) - 1200*B*a*b^4*d^3*e^5*x* 
sgn(b*x + a) - 240*A*b^5*d^3*e^5*x*sgn(b*x + a) + 1800*B*a^2*b^3*d^2*e^...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.30 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {60 b^{6} d^{6} e x +30 b^{6} d^{5} e^{2} x^{2}-10 b^{6} d^{4} e^{3} x^{3}+5 b^{6} d^{3} e^{4} x^{4}-3 b^{6} d^{2} e^{5} x^{5}+2 b^{6} d \,e^{6} x^{6}+60 \,\mathrm {log}\left (e x +d \right ) a^{5} b d \,e^{6} x -300 \,\mathrm {log}\left (e x +d \right ) a^{4} b^{2} d^{2} e^{5} x +600 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{3} d^{3} e^{4} x -600 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{4} e^{3} x +300 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{5} e^{2} x +50 a \,b^{5} d^{3} e^{4} x^{3}-25 a \,b^{5} d^{2} e^{5} x^{4}+15 a \,b^{5} d \,e^{6} x^{5}-60 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{7}+10 a^{6} e^{7} x +60 \,\mathrm {log}\left (e x +d \right ) a^{5} b \,d^{2} e^{5}-300 \,\mathrm {log}\left (e x +d \right ) a^{4} b^{2} d^{3} e^{4}+600 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{3} d^{4} e^{3}-600 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{5} e^{2}+300 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{6} e -60 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{6} e x -60 a^{5} b d \,e^{6} x +300 a^{4} b^{2} d^{2} e^{5} x +150 a^{4} b^{2} d \,e^{6} x^{2}-600 a^{3} b^{3} d^{3} e^{4} x -300 a^{3} b^{3} d^{2} e^{5} x^{2}+100 a^{3} b^{3} d \,e^{6} x^{3}+600 a^{2} b^{4} d^{4} e^{3} x +300 a^{2} b^{4} d^{3} e^{4} x^{2}-100 a^{2} b^{4} d^{2} e^{5} x^{3}+50 a^{2} b^{4} d \,e^{6} x^{4}-300 a \,b^{5} d^{5} e^{2} x -150 a \,b^{5} d^{4} e^{3} x^{2}}{10 d \,e^{7} \left (e x +d \right )} \] Input:

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

Output:

(60*log(d + e*x)*a**5*b*d**2*e**5 + 60*log(d + e*x)*a**5*b*d*e**6*x - 300* 
log(d + e*x)*a**4*b**2*d**3*e**4 - 300*log(d + e*x)*a**4*b**2*d**2*e**5*x 
+ 600*log(d + e*x)*a**3*b**3*d**4*e**3 + 600*log(d + e*x)*a**3*b**3*d**3*e 
**4*x - 600*log(d + e*x)*a**2*b**4*d**5*e**2 - 600*log(d + e*x)*a**2*b**4* 
d**4*e**3*x + 300*log(d + e*x)*a*b**5*d**6*e + 300*log(d + e*x)*a*b**5*d** 
5*e**2*x - 60*log(d + e*x)*b**6*d**7 - 60*log(d + e*x)*b**6*d**6*e*x + 10* 
a**6*e**7*x - 60*a**5*b*d*e**6*x + 300*a**4*b**2*d**2*e**5*x + 150*a**4*b* 
*2*d*e**6*x**2 - 600*a**3*b**3*d**3*e**4*x - 300*a**3*b**3*d**2*e**5*x**2 
+ 100*a**3*b**3*d*e**6*x**3 + 600*a**2*b**4*d**4*e**3*x + 300*a**2*b**4*d* 
*3*e**4*x**2 - 100*a**2*b**4*d**2*e**5*x**3 + 50*a**2*b**4*d*e**6*x**4 - 3 
00*a*b**5*d**5*e**2*x - 150*a*b**5*d**4*e**3*x**2 + 50*a*b**5*d**3*e**4*x* 
*3 - 25*a*b**5*d**2*e**5*x**4 + 15*a*b**5*d*e**6*x**5 + 60*b**6*d**6*e*x + 
 30*b**6*d**5*e**2*x**2 - 10*b**6*d**4*e**3*x**3 + 5*b**6*d**3*e**4*x**4 - 
 3*b**6*d**2*e**5*x**5 + 2*b**6*d*e**6*x**6)/(10*d*e**7*(d + e*x))