\(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^6} \, dx\) [420]

Optimal result

Integrand size = 33, antiderivative size = 422 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {b^5 B x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}-\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}+\frac {5 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}-\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {b^4 (6 b B d-A b e-5 a B e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \] Output:

b^5*B*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-1/5*(-a*e+b*d)^5*(-A*e+B*d)*((b*x+a) 
^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^5+1/4*(-a*e+b*d)^4*(-5*A*b*e-B*a*e+6*B*b*d)* 
((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^4-5/3*b*(-a*e+b*d)^3*(-2*A*b*e-B*a*e 
+3*B*b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^3+5*b^2*(-a*e+b*d)^2*(-A*b 
*e-B*a*e+2*B*b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^2-5*b^3*(-a*e+b*d) 
*(-A*b*e-2*B*a*e+3*B*b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)-b^4*(-A*b* 
e-5*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 = 1.32 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.16 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=-\frac {\sqrt {(a+b x)^2} \left (3 a^5 e^5 (4 A e+B (d+5 e x))+5 a^4 b e^4 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+30 a^2 b^3 e^2 \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+5 a b^4 e \left (12 A e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-B d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+b^5 \left (-A d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+6 B \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )+60 b^4 (6 b B d-A b e-5 a B e) (d+e x)^5 \log (d+e x)\right )}{60 e^7 (a+b x) (d+e x)^5} \] Input:

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^6,x]
 

Output:

-1/60*(Sqrt[(a + b*x)^2]*(3*a^5*e^5*(4*A*e + B*(d + 5*e*x)) + 5*a^4*b*e^4* 
(3*A*e*(d + 5*e*x) + 2*B*(d^2 + 5*d*e*x + 10*e^2*x^2)) + 10*a^3*b^2*e^3*(2 
*A*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*B*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 
10*e^3*x^3)) + 30*a^2*b^3*e^2*(A*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^ 
3*x^3) + 4*B*(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) 
) + 5*a*b^4*e*(12*A*e*(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) - B*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 
+ 300*e^4*x^4)) + b^5*(-(A*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 
 900*d*e^3*x^3 + 300*e^4*x^4)) + 6*B*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x 
^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6)) + 60*b 
^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^5*Log[d + e*x]))/(e^7*(a + b*x)*( 
d + e*x)^5)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 261, 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.

\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{(d+e x)^6} \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{(d+e x)^6}dx}{b^5 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{(d+e x)^6}dx}{a+b x}\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^6,x]
 

Output:

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

Time = 2.44 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.50

method	result	size
risch	\(\frac {b^{5} B x \sqrt {\left (b x +a \right )^{2}}}{e^{6} \left (b x +a \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-5 A a \,b^{4} e^{5}+5 A \,b^{5} d \,e^{4}-10 B \,e^{5} a^{2} b^{3}+25 B a \,b^{4} d \,e^{4}-15 B \,b^{5} d^{2} e^{3}\right ) x^{4}-5 b^{2} e^{2} \left (A \,a^{2} b \,e^{3}+2 A a \,b^{2} d \,e^{2}-3 A \,b^{3} d^{2} e +B \,e^{3} a^{3}+4 B \,a^{2} b d \,e^{2}-15 B a \,b^{2} d^{2} e +10 B \,b^{3} d^{3}\right ) x^{3}-\frac {5 b e \left (2 A \,a^{3} b \,e^{4}+3 A \,a^{2} b^{2} d \,e^{3}+6 A a \,b^{3} d^{2} e^{2}-11 A \,b^{4} d^{3} e +B \,e^{4} a^{4}+3 B \,a^{3} b d \,e^{3}+12 B \,a^{2} b^{2} d^{2} e^{2}-55 B a \,b^{3} d^{3} e +39 B \,b^{4} d^{4}\right ) x^{2}}{3}+\left (-\frac {5}{4} A \,a^{4} b \,e^{5}-\frac {5}{3} A \,a^{3} b^{2} d \,e^{4}-\frac {5}{2} A \,a^{2} b^{3} d^{2} e^{3}-5 A a \,b^{4} d^{3} e^{2}+\frac {125}{12} A \,b^{5} d^{4} e -\frac {1}{4} B \,a^{5} e^{5}-\frac {5}{6} B \,a^{4} b d \,e^{4}-\frac {5}{2} B \,a^{3} b^{2} d^{2} e^{3}-10 B \,a^{2} b^{3} d^{3} e^{2}+\frac {625}{12} B a \,b^{4} d^{4} e -\frac {77}{2} B \,b^{5} d^{5}\right ) x -\frac {12 A \,a^{5} e^{6}+15 A \,a^{4} b d \,e^{5}+20 A \,a^{3} b^{2} d^{2} e^{4}+30 A \,a^{2} b^{3} d^{3} e^{3}+60 A a \,b^{4} d^{4} e^{2}-137 A \,b^{5} d^{5} e +3 B \,a^{5} d \,e^{5}+10 B \,a^{4} b \,d^{2} e^{4}+30 B \,a^{3} b^{2} d^{3} e^{3}+120 B \,a^{2} b^{3} d^{4} e^{2}-685 B a \,b^{4} d^{5} e +522 b^{5} B \,d^{6}}{60 e}\right )}{\left (b x +a \right ) e^{6} \left (e x +d \right )^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} \left (A b e +5 B a e -6 B b d \right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\)	\(634\)
default	\(\text {Expression too large to display}\)	\(1012\)

Input:

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)
 

Output:

b^5*B*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+((b*x+a)^2)^(1/2)/(b*x+a)*((-5*A*a*b 
^4*e^5+5*A*b^5*d*e^4-10*B*a^2*b^3*e^5+25*B*a*b^4*d*e^4-15*B*b^5*d^2*e^3)*x 
^4-5*b^2*e^2*(A*a^2*b*e^3+2*A*a*b^2*d*e^2-3*A*b^3*d^2*e+B*a^3*e^3+4*B*a^2* 
b*d*e^2-15*B*a*b^2*d^2*e+10*B*b^3*d^3)*x^3-5/3*b*e*(2*A*a^3*b*e^4+3*A*a^2* 
b^2*d*e^3+6*A*a*b^3*d^2*e^2-11*A*b^4*d^3*e+B*a^4*e^4+3*B*a^3*b*d*e^3+12*B* 
a^2*b^2*d^2*e^2-55*B*a*b^3*d^3*e+39*B*b^4*d^4)*x^2+(-5/4*A*a^4*b*e^5-5/3*A 
*a^3*b^2*d*e^4-5/2*A*a^2*b^3*d^2*e^3-5*A*a*b^4*d^3*e^2+125/12*A*b^5*d^4*e- 
1/4*B*a^5*e^5-5/6*B*a^4*b*d*e^4-5/2*B*a^3*b^2*d^2*e^3-10*B*a^2*b^3*d^3*e^2 
+625/12*B*a*b^4*d^4*e-77/2*B*b^5*d^5)*x-1/60/e*(12*A*a^5*e^6+15*A*a^4*b*d* 
e^5+20*A*a^3*b^2*d^2*e^4+30*A*a^2*b^3*d^3*e^3+60*A*a*b^4*d^4*e^2-137*A*b^5 
*d^5*e+3*B*a^5*d*e^5+10*B*a^4*b*d^2*e^4+30*B*a^3*b^2*d^3*e^3+120*B*a^2*b^3 
*d^4*e^2-685*B*a*b^4*d^5*e+522*B*b^5*d^6))/e^6/(e*x+d)^5+((b*x+a)^2)^(1/2) 
/(b*x+a)*b^4/e^7*(A*b*e+5*B*a*e-6*B*b*d)*ln(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (339) = 678\).

Time = 0.10 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.90 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^6,x, algorithm="fric 
as")
 

Output:

1/60*(60*B*b^5*e^6*x^6 + 300*B*b^5*d*e^5*x^5 - 522*B*b^5*d^6 - 12*A*a^5*e^ 
6 + 137*(5*B*a*b^4 + A*b^5)*d^5*e - 60*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 - 3 
0*(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 - 3 
*(B*a^5 + 5*A*a^4*b)*d*e^5 - 300*(B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^ 
5 + (2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 300*(8*B*b^5*d^3*e^3 - 3*(5*B*a*b^4 
 + A*b^5)*d^2*e^4 + 2*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + (B*a^3*b^2 + A*a^2*b 
^3)*e^6)*x^3 - 100*(36*B*b^5*d^4*e^2 - 11*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 6* 
(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 + 3*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + (B*a^4 
*b + 2*A*a^3*b^2)*e^6)*x^2 - 5*(450*B*b^5*d^5*e - 125*(5*B*a*b^4 + A*b^5)* 
d^4*e^2 + 60*(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 + 3*(B*a^5 + 5*A*a^4*b)*e^6)*x 
- 60*(6*B*b^5*d^6 - (5*B*a*b^4 + A*b^5)*d^5*e + (6*B*b^5*d*e^5 - (5*B*a*b^ 
4 + A*b^5)*e^6)*x^5 + 5*(6*B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^5)*x^4 
+ 10*(6*B*b^5*d^3*e^3 - (5*B*a*b^4 + A*b^5)*d^2*e^4)*x^3 + 10*(6*B*b^5*d^4 
*e^2 - (5*B*a*b^4 + A*b^5)*d^3*e^3)*x^2 + 5*(6*B*b^5*d^5*e - (5*B*a*b^4 + 
A*b^5)*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^ 
3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{6}}\, dx \] Input:

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**6,x)
 

Output:

Integral((A + B*x)*((a + b*x)**2)**(5/2)/(d + e*x)**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^6,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. 899 vs. \(2 (339) = 678\).

Time = 0.16 (sec) , antiderivative size = 899, normalized size of antiderivative = 2.13 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^6,x, algorithm="giac 
")
 

Output:

B*b^5*x*sgn(b*x + a)/e^6 - (6*B*b^5*d*sgn(b*x + a) - 5*B*a*b^4*e*sgn(b*x + 
 a) - A*b^5*e*sgn(b*x + a))*log(abs(e*x + d))/e^7 - 1/60*(522*B*b^5*d^6*sg 
n(b*x + a) - 685*B*a*b^4*d^5*e*sgn(b*x + a) - 137*A*b^5*d^5*e*sgn(b*x + a) 
 + 120*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 60*A*a*b^4*d^4*e^2*sgn(b*x + a) + 
30*B*a^3*b^2*d^3*e^3*sgn(b*x + a) + 30*A*a^2*b^3*d^3*e^3*sgn(b*x + a) + 10 
*B*a^4*b*d^2*e^4*sgn(b*x + a) + 20*A*a^3*b^2*d^2*e^4*sgn(b*x + a) + 3*B*a^ 
5*d*e^5*sgn(b*x + a) + 15*A*a^4*b*d*e^5*sgn(b*x + a) + 12*A*a^5*e^6*sgn(b* 
x + a) + 300*(3*B*b^5*d^2*e^4*sgn(b*x + a) - 5*B*a*b^4*d*e^5*sgn(b*x + a) 
- A*b^5*d*e^5*sgn(b*x + a) + 2*B*a^2*b^3*e^6*sgn(b*x + a) + A*a*b^4*e^6*sg 
n(b*x + a))*x^4 + 300*(10*B*b^5*d^3*e^3*sgn(b*x + a) - 15*B*a*b^4*d^2*e^4* 
sgn(b*x + a) - 3*A*b^5*d^2*e^4*sgn(b*x + a) + 4*B*a^2*b^3*d*e^5*sgn(b*x + 
a) + 2*A*a*b^4*d*e^5*sgn(b*x + a) + B*a^3*b^2*e^6*sgn(b*x + a) + A*a^2*b^3 
*e^6*sgn(b*x + a))*x^3 + 100*(39*B*b^5*d^4*e^2*sgn(b*x + a) - 55*B*a*b^4*d 
^3*e^3*sgn(b*x + a) - 11*A*b^5*d^3*e^3*sgn(b*x + a) + 12*B*a^2*b^3*d^2*e^4 
*sgn(b*x + a) + 6*A*a*b^4*d^2*e^4*sgn(b*x + a) + 3*B*a^3*b^2*d*e^5*sgn(b*x 
 + a) + 3*A*a^2*b^3*d*e^5*sgn(b*x + a) + B*a^4*b*e^6*sgn(b*x + a) + 2*A*a^ 
3*b^2*e^6*sgn(b*x + a))*x^2 + 5*(462*B*b^5*d^5*e*sgn(b*x + a) - 625*B*a*b^ 
4*d^4*e^2*sgn(b*x + a) - 125*A*b^5*d^4*e^2*sgn(b*x + a) + 120*B*a^2*b^3*d^ 
3*e^3*sgn(b*x + a) + 60*A*a*b^4*d^3*e^3*sgn(b*x + a) + 30*B*a^3*b^2*d^2*e^ 
4*sgn(b*x + a) + 30*A*a^2*b^3*d^2*e^4*sgn(b*x + a) + 10*B*a^4*b*d*e^5*s...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^6} \,d x \] Input:

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^6,x)
 

Output:

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.36 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {-325 b^{6} d^{6} e x -500 b^{6} d^{5} e^{2} x^{2}-300 b^{6} d^{4} e^{3} x^{3}+60 b^{6} d^{2} e^{5} x^{5}+10 b^{6} d \,e^{6} x^{6}+600 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{4} e^{3} x^{2}-60 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{2} e^{5} x^{5}+300 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{5} e^{2} x +300 a \,b^{5} d^{3} e^{4} x^{3}-60 a \,b^{5} d \,e^{6} x^{5}-2 a^{6} d \,e^{6}-3 a^{5} b \,d^{2} e^{5}-300 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{3} e^{4} x^{4}-60 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{7}+60 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{6} e -300 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{6} e x -15 a^{5} b d \,e^{6} x -25 a^{4} b^{2} d^{2} e^{5} x -50 a^{4} b^{2} d \,e^{6} x^{2}-50 a^{3} b^{3} d^{3} e^{4} x -100 a^{3} b^{3} d^{2} e^{5} x^{2}-100 a^{3} b^{3} d \,e^{6} x^{3}+325 a \,b^{5} d^{5} e^{2} x +500 a \,b^{5} d^{4} e^{3} x^{2}+30 a^{2} b^{4} e^{7} x^{5}-5 a^{4} b^{2} d^{3} e^{4}-10 a^{3} b^{3} d^{4} e^{3}+77 a \,b^{5} d^{6} e +300 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{2} e^{5} x^{4}+60 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d \,e^{6} x^{5}-77 b^{6} d^{7}+600 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{3} e^{4} x^{3}-600 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{5} e^{2} x^{2}-600 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{4} e^{3} x^{3}}{10 d \,e^{7} \left (e^{5} x^{5}+5 d \,e^{4} x^{4}+10 d^{2} e^{3} x^{3}+10 d^{3} e^{2} x^{2}+5 d^{4} e x +d^{5}\right )} \] Input:

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^6,x)
 

Output:

(60*log(d + e*x)*a*b**5*d**6*e + 300*log(d + e*x)*a*b**5*d**5*e**2*x + 600 
*log(d + e*x)*a*b**5*d**4*e**3*x**2 + 600*log(d + e*x)*a*b**5*d**3*e**4*x* 
*3 + 300*log(d + e*x)*a*b**5*d**2*e**5*x**4 + 60*log(d + e*x)*a*b**5*d*e** 
6*x**5 - 60*log(d + e*x)*b**6*d**7 - 300*log(d + e*x)*b**6*d**6*e*x - 600* 
log(d + e*x)*b**6*d**5*e**2*x**2 - 600*log(d + e*x)*b**6*d**4*e**3*x**3 - 
300*log(d + e*x)*b**6*d**3*e**4*x**4 - 60*log(d + e*x)*b**6*d**2*e**5*x**5 
 - 2*a**6*d*e**6 - 3*a**5*b*d**2*e**5 - 15*a**5*b*d*e**6*x - 5*a**4*b**2*d 
**3*e**4 - 25*a**4*b**2*d**2*e**5*x - 50*a**4*b**2*d*e**6*x**2 - 10*a**3*b 
**3*d**4*e**3 - 50*a**3*b**3*d**3*e**4*x - 100*a**3*b**3*d**2*e**5*x**2 - 
100*a**3*b**3*d*e**6*x**3 + 30*a**2*b**4*e**7*x**5 + 77*a*b**5*d**6*e + 32 
5*a*b**5*d**5*e**2*x + 500*a*b**5*d**4*e**3*x**2 + 300*a*b**5*d**3*e**4*x* 
*3 - 60*a*b**5*d*e**6*x**5 - 77*b**6*d**7 - 325*b**6*d**6*e*x - 500*b**6*d 
**5*e**2*x**2 - 300*b**6*d**4*e**3*x**3 + 60*b**6*d**2*e**5*x**5 + 10*b**6 
*d*e**6*x**6)/(10*d*e**7*(d**5 + 5*d**4*e*x + 10*d**3*e**2*x**2 + 10*d**2* 
e**3*x**3 + 5*d*e**4*x**4 + e**5*x**5))