\(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^5} \, dx\) [419]

Optimal result

Integrand size = 33, antiderivative size = 421 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=-\frac {b^4 (5 b B d-A b e-5 a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}-\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {10 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \] Output:

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

Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.18 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a^5 e^5 (3 A e+B (d+4 e x))+5 a^4 b e^4 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+10 a^2 b^3 e^2 \left (3 A e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-B d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )-5 a b^4 e \left (A d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-B \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )+b^5 \left (A e \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )-3 B \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )\right )-60 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^4 \log (d+e x)\right )}{12 e^7 (a+b x) (d+e x)^4} \] Input:

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

Output:

-1/12*(Sqrt[(a + b*x)^2]*(a^5*e^5*(3*A*e + B*(d + 4*e*x)) + 5*a^4*b*e^4*(A 
*e*(d + 4*e*x) + B*(d^2 + 4*d*e*x + 6*e^2*x^2)) + 10*a^3*b^2*e^3*(A*e*(d^2 
 + 4*d*e*x + 6*e^2*x^2) + 3*B*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) 
 + 10*a^2*b^3*e^2*(3*A*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - B*d 
*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) - 5*a*b^4*e*(A*d*e*(2 
5*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) - B*(77*d^5 + 248*d^4*e*x 
 + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5)) + b^5*(A 
*e*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 
 - 12*e^5*x^5) - 3*B*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3* 
x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*x^6)) - 60*b^3*(b*d - a*e)*(3* 
b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4*Log[d + e*x]))/(e^7*(a + b*x)*(d + e* 
x)^4)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 260, 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.

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

\(\Big \downarrow \) 1187

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-((b^4*(5*b*B*d - A*b*e - 5*a*B*e)*x)/e^6) 
 + (b^5*B*x^2)/(2*e^5) - ((b*d - a*e)^5*(B*d - A*e))/(4*e^7*(d + e*x)^4) + 
 ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e))/(3*e^7*(d + e*x)^3) - (5*b*(b 
*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e))/(2*e^7*(d + e*x)^2) + (10*b^2*(b* 
d - a*e)^2*(2*b*B*d - A*b*e - a*B*e))/(e^7*(d + e*x)) + (5*b^3*(b*d - a*e) 
*(3*b*B*d - A*b*e - 2*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 = 1.92 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.50

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} \left (\frac {1}{2} B b e \,x^{2}+A b e x +5 B a e x -5 B b d x \right )}{\left (b x +a \right ) e^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-10 A \,a^{2} b^{3} e^{5}+20 A a \,b^{4} d \,e^{4}-10 A \,b^{5} d^{2} e^{3}-10 B \,e^{5} a^{3} b^{2}+40 B \,a^{2} b^{3} d \,e^{4}-50 B a \,b^{4} d^{2} e^{3}+20 B \,b^{5} d^{3} e^{2}\right ) x^{3}-\frac {5 e b \left (2 A \,a^{3} b \,e^{4}+6 A \,a^{2} b^{2} d \,e^{3}-18 A a \,b^{3} d^{2} e^{2}+10 A \,b^{4} d^{3} e +B \,e^{4} a^{4}+6 B \,a^{3} b d \,e^{3}-36 B \,a^{2} b^{2} d^{2} e^{2}+50 B a \,b^{3} d^{3} e -21 B \,b^{4} d^{4}\right ) x^{2}}{2}+\left (-\frac {5}{3} A \,a^{4} b \,e^{5}-\frac {10}{3} A \,a^{3} b^{2} d \,e^{4}-10 A \,a^{2} b^{3} d^{2} e^{3}+\frac {110}{3} A a \,b^{4} d^{3} e^{2}-\frac {65}{3} A \,b^{5} d^{4} e -\frac {1}{3} B \,a^{5} e^{5}-\frac {5}{3} B \,a^{4} b d \,e^{4}-10 B \,a^{3} b^{2} d^{2} e^{3}+\frac {220}{3} B \,a^{2} b^{3} d^{3} e^{2}-\frac {325}{3} B a \,b^{4} d^{4} e +47 B \,b^{5} d^{5}\right ) x -\frac {3 A \,a^{5} e^{6}+5 A \,a^{4} b d \,e^{5}+10 A \,a^{3} b^{2} d^{2} e^{4}+30 A \,a^{2} b^{3} d^{3} e^{3}-125 A a \,b^{4} d^{4} e^{2}+77 A \,b^{5} d^{5} e +B \,a^{5} d \,e^{5}+5 B \,a^{4} b \,d^{2} e^{4}+30 B \,a^{3} b^{2} d^{3} e^{3}-250 B \,a^{2} b^{3} d^{4} e^{2}+385 B a \,b^{4} d^{5} e -171 b^{5} B \,d^{6}}{12 e}\right )}{\left (b x +a \right ) e^{6} \left (e x +d \right )^{4}}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, b^{3} \left (A a b \,e^{2}-A \,b^{2} d e +2 B \,e^{2} a^{2}-5 B a b d e +3 B \,b^{2} d^{2}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\)	\(633\)
default	\(\text {Expression too large to display}\)	\(1163\)

Input:

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

Output:

((b*x+a)^2)^(1/2)/(b*x+a)*b^4/e^6*(1/2*B*b*e*x^2+A*b*e*x+5*B*a*e*x-5*B*b*d 
*x)+((b*x+a)^2)^(1/2)/(b*x+a)*((-10*A*a^2*b^3*e^5+20*A*a*b^4*d*e^4-10*A*b^ 
5*d^2*e^3-10*B*a^3*b^2*e^5+40*B*a^2*b^3*d*e^4-50*B*a*b^4*d^2*e^3+20*B*b^5* 
d^3*e^2)*x^3-5/2*e*b*(2*A*a^3*b*e^4+6*A*a^2*b^2*d*e^3-18*A*a*b^3*d^2*e^2+1 
0*A*b^4*d^3*e+B*a^4*e^4+6*B*a^3*b*d*e^3-36*B*a^2*b^2*d^2*e^2+50*B*a*b^3*d^ 
3*e-21*B*b^4*d^4)*x^2+(-5/3*A*a^4*b*e^5-10/3*A*a^3*b^2*d*e^4-10*A*a^2*b^3* 
d^2*e^3+110/3*A*a*b^4*d^3*e^2-65/3*A*b^5*d^4*e-1/3*B*a^5*e^5-5/3*B*a^4*b*d 
*e^4-10*B*a^3*b^2*d^2*e^3+220/3*B*a^2*b^3*d^3*e^2-325/3*B*a*b^4*d^4*e+47*B 
*b^5*d^5)*x-1/12*(3*A*a^5*e^6+5*A*a^4*b*d*e^5+10*A*a^3*b^2*d^2*e^4+30*A*a^ 
2*b^3*d^3*e^3-125*A*a*b^4*d^4*e^2+77*A*b^5*d^5*e+B*a^5*d*e^5+5*B*a^4*b*d^2 
*e^4+30*B*a^3*b^2*d^3*e^3-250*B*a^2*b^3*d^4*e^2+385*B*a*b^4*d^5*e-171*B*b^ 
5*d^6)/e)/e^6/(e*x+d)^4+5*((b*x+a)^2)^(1/2)/(b*x+a)/e^7*b^3*(A*a*b*e^2-A*b 
^2*d*e+2*B*a^2*e^2-5*B*a*b*d*e+3*B*b^2*d^2)*ln(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 871 vs. \(2 (336) = 672\).

Time = 0.09 (sec) , antiderivative size = 871, normalized size of antiderivative = 2.07 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

1/12*(6*B*b^5*e^6*x^6 + 171*B*b^5*d^6 - 3*A*a^5*e^6 - 77*(5*B*a*b^4 + A*b^ 
5)*d^5*e + 125*(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 - 5*(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 
- 12*(3*B*b^5*d*e^5 - (5*B*a*b^4 + A*b^5)*e^6)*x^5 - 12*(17*B*b^5*d^2*e^4 
- 4*(5*B*a*b^4 + A*b^5)*d*e^5)*x^4 - 24*(4*B*b^5*d^3*e^3 + 2*(5*B*a*b^4 + 
A*b^5)*d^2*e^4 - 10*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + 5*(B*a^3*b^2 + A*a^2*b 
^3)*e^6)*x^3 + 6*(66*B*b^5*d^4*e^2 - 42*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 90*( 
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 - 5*(B*a 
^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 4*(126*B*b^5*d^5*e - 62*(5*B*a*b^4 + A*b^5) 
*d^4*e^2 + 110*(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 - 5*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - (B*a^5 + 5*A*a^4*b)*e^6)*x + 
 60*(3*B*b^5*d^6 - (5*B*a*b^4 + A*b^5)*d^5*e + (2*B*a^2*b^3 + A*a*b^4)*d^4 
*e^2 + (3*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 + 4*(3*B*b^5*d^3*e^3 - (5*B*a*b^4 + A*b^5)*d^2*e^4 + (2*B*a^2 
*b^3 + A*a*b^4)*d*e^5)*x^3 + 6*(3*B*b^5*d^4*e^2 - (5*B*a*b^4 + A*b^5)*d^3* 
e^3 + (2*B*a^2*b^3 + A*a*b^4)*d^2*e^4)*x^2 + 4*(3*B*b^5*d^5*e - (5*B*a*b^4 
 + A*b^5)*d^4*e^2 + (2*B*a^2*b^3 + A*a*b^4)*d^3*e^3)*x)*log(e*x + d))/(e^1 
1*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^5,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. 907 vs. \(2 (336) = 672\).

Time = 0.18 (sec) , antiderivative size = 907, normalized size of antiderivative = 2.15 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

5*(3*B*b^5*d^2*sgn(b*x + a) - 5*B*a*b^4*d*e*sgn(b*x + a) - A*b^5*d*e*sgn(b 
*x + a) + 2*B*a^2*b^3*e^2*sgn(b*x + a) + A*a*b^4*e^2*sgn(b*x + a))*log(abs 
(e*x + d))/e^7 + 1/2*(B*b^5*e^5*x^2*sgn(b*x + a) - 10*B*b^5*d*e^4*x*sgn(b* 
x + a) + 10*B*a*b^4*e^5*x*sgn(b*x + a) + 2*A*b^5*e^5*x*sgn(b*x + a))/e^10 
+ 1/12*(171*B*b^5*d^6*sgn(b*x + a) - 385*B*a*b^4*d^5*e*sgn(b*x + a) - 77*A 
*b^5*d^5*e*sgn(b*x + a) + 250*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 125*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) - 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) - 
 3*A*a^5*e^6*sgn(b*x + a) + 120*(2*B*b^5*d^3*e^3*sgn(b*x + a) - 5*B*a*b^4* 
d^2*e^4*sgn(b*x + a) - 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 + 30*(21*B*b^5*d^4*e^2*sgn(b*x + a) - 50*B*a* 
b^4*d^3*e^3*sgn(b*x + a) - 10*A*b^5*d^3*e^3*sgn(b*x + a) + 36*B*a^2*b^3*d^ 
2*e^4*sgn(b*x + a) + 18*A*a*b^4*d^2*e^4*sgn(b*x + a) - 6*B*a^3*b^2*d*e^5*s 
gn(b*x + a) - 6*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 + 4*(141*B*b^5*d^5*e*sgn(b*x + a) - 325* 
B*a*b^4*d^4*e^2*sgn(b*x + a) - 65*A*b^5*d^4*e^2*sgn(b*x + a) + 220*B*a^2*b 
^3*d^3*e^3*sgn(b*x + a) + 110*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) - 5*B*a^4*b*d*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.51 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {200 b^{6} d^{6} e x +180 b^{6} d^{5} e^{2} x^{2}-60 b^{6} d^{3} e^{4} x^{4}-12 b^{6} d^{2} e^{5} x^{5}+2 b^{6} d \,e^{6} x^{6}+360 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{3} e^{4} x^{2}-720 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{4} e^{3} x^{2}+240 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{4} e^{3} x -480 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{5} e^{2} x +120 a \,b^{5} d^{2} e^{5} x^{4}+24 a \,b^{5} d \,e^{6} x^{5}-a^{6} d \,e^{6}-2 a^{5} b \,d^{2} e^{5}+60 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{3} e^{4} x^{4}+60 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{7}+20 a^{3} b^{3} e^{7} x^{4}+60 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{5} e^{2}-120 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{6} e +240 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{6} e x -8 a^{5} b d \,e^{6} x -20 a^{4} b^{2} d^{2} e^{5} x -30 a^{4} b^{2} d \,e^{6} x^{2}+200 a^{2} b^{4} d^{4} e^{3} x +180 a^{2} b^{4} d^{3} e^{4} x^{2}-60 a^{2} b^{4} d \,e^{6} x^{4}-400 a \,b^{5} d^{5} e^{2} x -360 a \,b^{5} d^{4} e^{3} x^{2}-5 a^{4} b^{2} d^{3} e^{4}+65 a^{2} b^{4} d^{5} e^{2}-130 a \,b^{5} d^{6} e +60 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d \,e^{6} x^{4}-120 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{2} e^{5} x^{4}+65 b^{6} d^{7}+240 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{4} d^{2} e^{5} x^{3}-480 \,\mathrm {log}\left (e x +d \right ) a \,b^{5} d^{3} e^{4} x^{3}+360 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{5} e^{2} x^{2}+240 \,\mathrm {log}\left (e x +d \right ) b^{6} d^{4} e^{3} x^{3}}{4 d \,e^{7} \left (e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}\right )} \] Input:

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

Output:

(60*log(d + e*x)*a**2*b**4*d**5*e**2 + 240*log(d + e*x)*a**2*b**4*d**4*e** 
3*x + 360*log(d + e*x)*a**2*b**4*d**3*e**4*x**2 + 240*log(d + e*x)*a**2*b* 
*4*d**2*e**5*x**3 + 60*log(d + e*x)*a**2*b**4*d*e**6*x**4 - 120*log(d + e* 
x)*a*b**5*d**6*e - 480*log(d + e*x)*a*b**5*d**5*e**2*x - 720*log(d + e*x)* 
a*b**5*d**4*e**3*x**2 - 480*log(d + e*x)*a*b**5*d**3*e**4*x**3 - 120*log(d 
 + e*x)*a*b**5*d**2*e**5*x**4 + 60*log(d + e*x)*b**6*d**7 + 240*log(d + e* 
x)*b**6*d**6*e*x + 360*log(d + e*x)*b**6*d**5*e**2*x**2 + 240*log(d + e*x) 
*b**6*d**4*e**3*x**3 + 60*log(d + e*x)*b**6*d**3*e**4*x**4 - a**6*d*e**6 - 
 2*a**5*b*d**2*e**5 - 8*a**5*b*d*e**6*x - 5*a**4*b**2*d**3*e**4 - 20*a**4* 
b**2*d**2*e**5*x - 30*a**4*b**2*d*e**6*x**2 + 20*a**3*b**3*e**7*x**4 + 65* 
a**2*b**4*d**5*e**2 + 200*a**2*b**4*d**4*e**3*x + 180*a**2*b**4*d**3*e**4* 
x**2 - 60*a**2*b**4*d*e**6*x**4 - 130*a*b**5*d**6*e - 400*a*b**5*d**5*e**2 
*x - 360*a*b**5*d**4*e**3*x**2 + 120*a*b**5*d**2*e**5*x**4 + 24*a*b**5*d*e 
**6*x**5 + 65*b**6*d**7 + 200*b**6*d**6*e*x + 180*b**6*d**5*e**2*x**2 - 60 
*b**6*d**3*e**4*x**4 - 12*b**6*d**2*e**5*x**5 + 2*b**6*d*e**6*x**6)/(4*d*e 
**7*(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4))