\(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^{10}} \, dx\) [328]

Optimal result

Integrand size = 29, antiderivative size = 157 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {b^2 (A b-a B) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 a^4 x^6}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{9 a^2 x^9}+\frac {(11 A b-9 a B) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{72 a^3 x^8}-\frac {b (83 A b-81 a B) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{504 a^4 x^7} \] Output:

1/6*b^2*(A*b-B*a)*(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/a^4/x^6-1/9*A*(b^2*x 
^2+2*a*b*x+a^2)^(7/2)/a^2/x^9+1/72*(11*A*b-9*B*a)*(b^2*x^2+2*a*b*x+a^2)^(7 
/2)/a^3/x^8-1/504*b*(83*A*b-81*B*a)*(b^2*x^2+2*a*b*x+a^2)^(7/2)/a^4/x^7
 

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.80 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (42 b^5 x^5 (3 A+4 B x)+126 a b^4 x^4 (4 A+5 B x)+168 a^2 b^3 x^3 (5 A+6 B x)+120 a^3 b^2 x^2 (6 A+7 B x)+45 a^4 b x (7 A+8 B x)+7 a^5 (8 A+9 B x)\right )}{504 x^9 (a+b x)} \] Input:

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^10,x]
 

Output:

-1/504*(Sqrt[(a + b*x)^2]*(42*b^5*x^5*(3*A + 4*B*x) + 126*a*b^4*x^4*(4*A + 
 5*B*x) + 168*a^2*b^3*x^3*(5*A + 6*B*x) + 120*a^3*b^2*x^2*(6*A + 7*B*x) + 
45*a^4*b*x*(7*A + 8*B*x) + 7*a^5*(8*A + 9*B*x)))/(x^9*(a + b*x))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1187, 27, 85, 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.

\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{x^{10}} \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{x^{10}}dx}{b^5 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{x^{10}}dx}{a+b x}\)

\(\Big \downarrow \) 85

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

\(\Big \downarrow \) 2009

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

Input:

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^10,x]
 

Output:

((-1/9*(a^5*A)/x^9 - (a^4*(5*A*b + a*B))/(8*x^8) - (5*a^3*b*(2*A*b + a*B)) 
/(7*x^7) - (5*a^2*b^2*(A*b + a*B))/(3*x^6) - (a*b^3*(A*b + 2*a*B))/x^5 - ( 
b^4*(A*b + 5*a*B))/(4*x^4) - (b^5*B)/(3*x^3))*Sqrt[a^2 + 2*a*b*x + b^2*x^2 
])/(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 85

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : 
> Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, 
 d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* 
f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n 
 + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 
1])
 

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 = 1.63 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.87

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {B \,b^{5} x^{6}}{3}+\left (-\frac {1}{4} A \,b^{5}-\frac {5}{4} B a \,b^{4}\right ) x^{5}+\left (-A a \,b^{4}-2 B \,a^{2} b^{3}\right ) x^{4}+\left (-\frac {5}{3} A \,a^{2} b^{3}-\frac {5}{3} B \,a^{3} b^{2}\right ) x^{3}+\left (-\frac {10}{7} a^{3} A \,b^{2}-\frac {5}{7} B \,a^{4} b \right ) x^{2}+\left (-\frac {5}{8} A \,a^{4} b -\frac {1}{8} B \,a^{5}\right ) x -\frac {A \,a^{5}}{9}\right )}{\left (b x +a \right ) x^{9}}\)	\(136\)
gosper	\(-\frac {\left (168 B \,b^{5} x^{6}+126 A \,b^{5} x^{5}+630 B a \,b^{4} x^{5}+504 A a \,b^{4} x^{4}+1008 B \,a^{2} b^{3} x^{4}+840 A \,a^{2} b^{3} x^{3}+840 B \,a^{3} b^{2} x^{3}+720 A \,a^{3} b^{2} x^{2}+360 B \,a^{4} b \,x^{2}+315 A \,a^{4} b x +63 B \,a^{5} x +56 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 x^{9} \left (b x +a \right )^{5}}\)	\(140\)
default	\(-\frac {\left (168 B \,b^{5} x^{6}+126 A \,b^{5} x^{5}+630 B a \,b^{4} x^{5}+504 A a \,b^{4} x^{4}+1008 B \,a^{2} b^{3} x^{4}+840 A \,a^{2} b^{3} x^{3}+840 B \,a^{3} b^{2} x^{3}+720 A \,a^{3} b^{2} x^{2}+360 B \,a^{4} b \,x^{2}+315 A \,a^{4} b x +63 B \,a^{5} x +56 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 x^{9} \left (b x +a \right )^{5}}\)	\(140\)
orering	\(-\frac {\left (168 B \,b^{5} x^{6}+126 A \,b^{5} x^{5}+630 B a \,b^{4} x^{5}+504 A a \,b^{4} x^{4}+1008 B \,a^{2} b^{3} x^{4}+840 A \,a^{2} b^{3} x^{3}+840 B \,a^{3} b^{2} x^{3}+720 A \,a^{3} b^{2} x^{2}+360 B \,a^{4} b \,x^{2}+315 A \,a^{4} b x +63 B \,a^{5} x +56 A \,a^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{504 x^{9} \left (b x +a \right )^{5}}\)	\(149\)

Input:

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^10,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.76 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {168 \, B b^{5} x^{6} + 56 \, A a^{5} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 504 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 360 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{504 \, x^{9}} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^10,x, algorithm="fricas")
 

Output:

-1/504*(168*B*b^5*x^6 + 56*A*a^5 + 126*(5*B*a*b^4 + A*b^5)*x^5 + 504*(2*B* 
a^2*b^3 + A*a*b^4)*x^4 + 840*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 360*(B*a^4*b + 
2*A*a^3*b^2)*x^2 + 63*(B*a^5 + 5*A*a^4*b)*x)/x^9
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{10}}\, dx \] Input:

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/x**10,x)
 

Output:

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**10, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (141) = 282\).

Time = 0.05 (sec) , antiderivative size = 555, normalized size of antiderivative = 3.54 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{8}}{6 \, a^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{9}}{6 \, a^{9}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{7}}{6 \, a^{7} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{8}}{6 \, a^{8} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{6}}{6 \, a^{8} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{7}}{6 \, a^{9} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{6 \, a^{7} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{6 \, a^{8} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{6 \, a^{6} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{6 \, a^{7} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{6 \, a^{5} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{6 \, a^{6} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{6 \, a^{4} x^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{6 \, a^{5} x^{6}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{56 \, a^{3} x^{7}} - \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{504 \, a^{4} x^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{8 \, a^{2} x^{8}} + \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{72 \, a^{3} x^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{9 \, a^{2} x^{9}} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^10,x, algorithm="maxima")
 

Output:

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^8/a^8 - 1/6*(b^2*x^2 + 2*a*b*x + a 
^2)^(5/2)*A*b^9/a^9 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^7/(a^7*x) - 
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^8/(a^8*x) - 1/6*(b^2*x^2 + 2*a*b*x 
 + a^2)^(7/2)*B*b^6/(a^8*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^7/ 
(a^9*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^5/(a^7*x^3) - 1/6*(b^2 
*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^6/(a^8*x^3) - 1/6*(b^2*x^2 + 2*a*b*x + a^2 
)^(7/2)*B*b^4/(a^6*x^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^5/(a^7*x 
^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^3/(a^5*x^5) - 1/6*(b^2*x^2 + 
 2*a*b*x + a^2)^(7/2)*A*b^4/(a^6*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2 
)*B*b^2/(a^4*x^6) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^3/(a^5*x^6) + 
9/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b/(a^3*x^7) - 83/504*(b^2*x^2 + 2*a 
*b*x + a^2)^(7/2)*A*b^2/(a^4*x^7) - 1/8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B/ 
(a^2*x^8) + 11/72*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b/(a^3*x^8) - 1/9*(b^2 
*x^2 + 2*a*b*x + a^2)^(7/2)*A/(a^2*x^9)
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {{\left (3 \, B a b^{8} - A b^{9}\right )} \mathrm {sgn}\left (b x + a\right )}{504 \, a^{4}} - \frac {168 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 630 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 126 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 1008 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 504 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 840 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 840 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 360 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 720 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 63 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 315 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 56 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{504 \, x^{9}} \] Input:

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^10,x, algorithm="giac")
 

Output:

-1/504*(3*B*a*b^8 - A*b^9)*sgn(b*x + a)/a^4 - 1/504*(168*B*b^5*x^6*sgn(b*x 
 + a) + 630*B*a*b^4*x^5*sgn(b*x + a) + 126*A*b^5*x^5*sgn(b*x + a) + 1008*B 
*a^2*b^3*x^4*sgn(b*x + a) + 504*A*a*b^4*x^4*sgn(b*x + a) + 840*B*a^3*b^2*x 
^3*sgn(b*x + a) + 840*A*a^2*b^3*x^3*sgn(b*x + a) + 360*B*a^4*b*x^2*sgn(b*x 
 + a) + 720*A*a^3*b^2*x^2*sgn(b*x + a) + 63*B*a^5*x*sgn(b*x + a) + 315*A*a 
^4*b*x*sgn(b*x + a) + 56*A*a^5*sgn(b*x + a))/x^9
 

Mupad [B] (verification not implemented)

Time = 10.94 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.81 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {\left (\frac {B\,a^5}{8}+\frac {5\,A\,b\,a^4}{8}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^8\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{4}+\frac {5\,B\,a\,b^4}{4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^4\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^3\,\left (a+b\,x\right )}-\frac {a\,b^3\,\left (A\,b+2\,B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {5\,a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^6\,\left (a+b\,x\right )} \] Input:

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/x^10,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.43 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {-84 b^{6} x^{6}-378 a \,b^{5} x^{5}-756 a^{2} b^{4} x^{4}-840 a^{3} b^{3} x^{3}-540 a^{4} b^{2} x^{2}-189 a^{5} b x -28 a^{6}}{252 x^{9}} \] Input:

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^10,x)
 

Output:

( - 28*a**6 - 189*a**5*b*x - 540*a**4*b**2*x**2 - 840*a**3*b**3*x**3 - 756 
*a**2*b**4*x**4 - 378*a*b**5*x**5 - 84*b**6*x**6)/(252*x**9)