3.1.79 \(\int \frac {(a^2+2 a b x^3+b^2 x^6)^{5/2}}{x^{16}} \, dx\)

Optimal. Leaf size=251 \[ \frac {b^5 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 x^{15} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 x^{12} \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 266, 43} \begin {gather*} -\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 x^{15} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 x^{12} \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {b^5 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^16,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(15*x^15*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(12*x
^12*(a + b*x^3)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(9*x^9*(a + b*x^3)) - (5*a^2*b^3*Sqrt[a^2 + 2*
a*b*x^3 + b^2*x^6])/(3*x^6*(a + b*x^3)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*x^3*(a + b*x^3)) + (b^5
*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(a + b*x^3)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{16}} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^5}{x^{16}} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x^6} \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int \left (\frac {a^5 b^5}{x^6}+\frac {5 a^4 b^6}{x^5}+\frac {10 a^3 b^7}{x^4}+\frac {10 a^2 b^8}{x^3}+\frac {5 a b^9}{x^2}+\frac {b^{10}}{x}\right ) \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{15 x^{15} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 x^{12} \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 85, normalized size = 0.34 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (a \left (12 a^4+75 a^3 b x^3+200 a^2 b^2 x^6+300 a b^3 x^9+300 b^4 x^{12}\right )-180 b^5 x^{15} \log (x)\right )}{180 x^{15} \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^16,x]

[Out]

-1/180*(Sqrt[(a + b*x^3)^2]*(a*(12*a^4 + 75*a^3*b*x^3 + 200*a^2*b^2*x^6 + 300*a*b^3*x^9 + 300*b^4*x^12) - 180*
b^5*x^15*Log[x]))/(x^15*(a + b*x^3))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 3.58, size = 2386, normalized size = 9.51 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^16,x]

[Out]

(4*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*(-12*a^8*b - 123*a^7*b^2*x^3 - 572*a^6*b^3*x^6 - 1598*a^5*b^4*x^9 - 3
012*a^4*b^5*x^12 - 3875*a^3*b^6*x^15 - 3200*a^2*b^7*x^18 - 1500*a*b^8*x^21 - 300*b^9*x^24) + 4*a*b^4*Sqrt[b^2]
*(12*a^9 + 135*a^8*b*x^3 + 695*a^7*b^2*x^6 + 2170*a^6*b^3*x^9 + 4610*a^5*b^4*x^12 + 6887*a^4*b^5*x^15 + 7075*a
^3*b^6*x^18 + 4700*a^2*b^7*x^21 + 1800*a*b^8*x^24 + 300*b^9*x^27))/(45*Sqrt[b^2]*x^15*Sqrt[a^2 + 2*a*b*x^3 + b
^2*x^6]*(-16*a^4*b^4 - 64*a^3*b^5*x^3 - 96*a^2*b^6*x^6 - 64*a*b^7*x^9 - 16*b^8*x^12) + 45*x^15*(16*a^5*b^5 + 8
0*a^4*b^6*x^3 + 160*a^3*b^7*x^6 + 160*a^2*b^8*x^9 + 80*a*b^9*x^12 + 16*b^10*x^15)) + (b^5*Log[-a - Sqrt[b^2]*x
^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/6 - (b^4*Sqrt[b^2]*Log[-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*
x^6]])/6 + (a^10*b^5*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(6*(-a - Sqrt[b^2]*x^3 + Sqrt[a
^2 + 2*a*b*x^3 + b^2*x^6])^5*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5) + (a^10*b^4*Sqrt[b^2]*Lo
g[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(6*(-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^
6])^5*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5) - (5*a^8*b^5*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a
*b*x^3 + b^2*x^6])^2*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(6*(-a - Sqrt[b^2]*x^3 + Sqrt[a
^2 + 2*a*b*x^3 + b^2*x^6])^5*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5) - (5*a^8*b^4*Sqrt[b^2]*(
-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^2*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]]
)/(6*(-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*
x^6])^5) + (5*a^6*b^5*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^4*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6]])/(3*(-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5*(a - Sqrt[b^2]*x^3 + Sqrt
[a^2 + 2*a*b*x^3 + b^2*x^6])^5) + (5*a^6*b^4*Sqrt[b^2]*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^4*
Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(3*(-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*
x^6])^5*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5) - (5*a^4*b^5*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2
*a*b*x^3 + b^2*x^6])^6*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(3*(-a - Sqrt[b^2]*x^3 + Sqrt
[a^2 + 2*a*b*x^3 + b^2*x^6])^5*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5) - (5*a^4*b^4*Sqrt[b^2]
*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^6*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6
]])/(3*(-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^
2*x^6])^5) + (5*a^2*b^5*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^8*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^
2 + 2*a*b*x^3 + b^2*x^6]])/(6*(-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5*(a - Sqrt[b^2]*x^3 + Sq
rt[a^2 + 2*a*b*x^3 + b^2*x^6])^5) + (5*a^2*b^4*Sqrt[b^2]*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^
8*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(6*(-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^
2*x^6])^5*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5) - (b^5*(-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6])^10*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(6*(-a - Sqrt[b^2]*x^3 + Sqrt[a^
2 + 2*a*b*x^3 + b^2*x^6])^5*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5) - (b^4*Sqrt[b^2]*(-(Sqrt[
b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^10*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(6*(
-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^5*(a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])^
5)

________________________________________________________________________________________

fricas [A]  time = 1.23, size = 61, normalized size = 0.24 \begin {gather*} \frac {180 \, b^{5} x^{15} \log \relax (x) - 300 \, a b^{4} x^{12} - 300 \, a^{2} b^{3} x^{9} - 200 \, a^{3} b^{2} x^{6} - 75 \, a^{4} b x^{3} - 12 \, a^{5}}{180 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^16,x, algorithm="fricas")

[Out]

1/180*(180*b^5*x^15*log(x) - 300*a*b^4*x^12 - 300*a^2*b^3*x^9 - 200*a^3*b^2*x^6 - 75*a^4*b*x^3 - 12*a^5)/x^15

________________________________________________________________________________________

giac [A]  time = 0.37, size = 123, normalized size = 0.49 \begin {gather*} b^{5} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {137 \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + 300 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 300 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 200 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 75 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 12 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{180 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^16,x, algorithm="giac")

[Out]

b^5*log(abs(x))*sgn(b*x^3 + a) - 1/180*(137*b^5*x^15*sgn(b*x^3 + a) + 300*a*b^4*x^12*sgn(b*x^3 + a) + 300*a^2*
b^3*x^9*sgn(b*x^3 + a) + 200*a^3*b^2*x^6*sgn(b*x^3 + a) + 75*a^4*b*x^3*sgn(b*x^3 + a) + 12*a^5*sgn(b*x^3 + a))
/x^15

________________________________________________________________________________________

maple [A]  time = 0.01, size = 82, normalized size = 0.33 \begin {gather*} \frac {\left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} \left (180 b^{5} x^{15} \ln \relax (x )-300 a \,b^{4} x^{12}-300 a^{2} b^{3} x^{9}-200 a^{3} b^{2} x^{6}-75 a^{4} b \,x^{3}-12 a^{5}\right )}{180 \left (b \,x^{3}+a \right )^{5} x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^16,x)

[Out]

1/180*((b*x^3+a)^2)^(5/2)*(180*b^5*ln(x)*x^15-300*a*b^4*x^12-300*a^2*b^3*x^9-200*a^3*b^2*x^6-75*a^4*b*x^3-12*a
^5)/(b*x^3+a)^5/x^15

________________________________________________________________________________________

maxima [B]  time = 1.10, size = 374, normalized size = 1.49 \begin {gather*} \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{6} x^{3}}{6 \, a^{2}} + \frac {1}{3} \, \left (-1\right )^{2 \, b^{2} x^{3} + 2 \, a b} b^{5} \log \left (2 \, b^{2} x^{3} + 2 \, a b\right ) - \frac {1}{3} \, \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} b^{5} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right ) + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{6} x^{3}}{12 \, a^{4}} + \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{5}}{2 \, a} + \frac {7 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{5}}{36 \, a^{3}} - \frac {2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{5}}{45 \, a^{5}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{4}}{9 \, a^{4} x^{3}} + \frac {2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{3}}{45 \, a^{5} x^{6}} - \frac {11 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{2}}{180 \, a^{4} x^{9}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b}{20 \, a^{3} x^{12}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}}}{15 \, a^{2} x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^16,x, algorithm="maxima")

[Out]

1/6*sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*b^6*x^3/a^2 + 1/3*(-1)^(2*b^2*x^3 + 2*a*b)*b^5*log(2*b^2*x^3 + 2*a*b) - 1/
3*(-1)^(2*a*b*x^3 + 2*a^2)*b^5*log(2*a*b*x/abs(x) + 2*a^2/(x^2*abs(x))) + 1/12*(b^2*x^6 + 2*a*b*x^3 + a^2)^(3/
2)*b^6*x^3/a^4 + 1/2*sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*b^5/a + 7/36*(b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*b^5/a^3 -
2/45*(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*b^5/a^5 - 1/9*(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*b^4/(a^4*x^3) + 2/45*(b
^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)*b^3/(a^5*x^6) - 11/180*(b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)*b^2/(a^4*x^9) + 1/20*
(b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)*b/(a^3*x^12) - 1/15*(b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)/(a^2*x^15)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}}{x^{16}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)/x^16,x)

[Out]

int((a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)/x^16, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{16}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x**16,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**16, x)

________________________________________________________________________________________