3.1874 \(\int \frac{x^5}{(a+\frac{b}{x^2})^3} \, dx\)

Optimal. Leaf size=87 \[ \frac{3 b^2 x^2}{a^5}-\frac{5 b^4}{2 a^6 \left (a x^2+b\right )}+\frac{b^5}{4 a^6 \left (a x^2+b\right )^2}-\frac{5 b^3 \log \left (a x^2+b\right )}{a^6}-\frac{3 b x^4}{4 a^4}+\frac{x^6}{6 a^3} \]

[Out]

(3*b^2*x^2)/a^5 - (3*b*x^4)/(4*a^4) + x^6/(6*a^3) + b^5/(4*a^6*(b + a*x^2)^2) - (5*b^4)/(2*a^6*(b + a*x^2)) -
(5*b^3*Log[b + a*x^2])/a^6

________________________________________________________________________________________

Rubi [A]  time = 0.0639311, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac{3 b^2 x^2}{a^5}-\frac{5 b^4}{2 a^6 \left (a x^2+b\right )}+\frac{b^5}{4 a^6 \left (a x^2+b\right )^2}-\frac{5 b^3 \log \left (a x^2+b\right )}{a^6}-\frac{3 b x^4}{4 a^4}+\frac{x^6}{6 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b/x^2)^3,x]

[Out]

(3*b^2*x^2)/a^5 - (3*b*x^4)/(4*a^4) + x^6/(6*a^3) + b^5/(4*a^6*(b + a*x^2)^2) - (5*b^4)/(2*a^6*(b + a*x^2)) -
(5*b^3*Log[b + a*x^2])/a^6

Rule 263

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

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 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])

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a+\frac{b}{x^2}\right )^3} \, dx &=\int \frac{x^{11}}{\left (b+a x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^5}{(b+a x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{6 b^2}{a^5}-\frac{3 b x}{a^4}+\frac{x^2}{a^3}-\frac{b^5}{a^5 (b+a x)^3}+\frac{5 b^4}{a^5 (b+a x)^2}-\frac{10 b^3}{a^5 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac{3 b^2 x^2}{a^5}-\frac{3 b x^4}{4 a^4}+\frac{x^6}{6 a^3}+\frac{b^5}{4 a^6 \left (b+a x^2\right )^2}-\frac{5 b^4}{2 a^6 \left (b+a x^2\right )}-\frac{5 b^3 \log \left (b+a x^2\right )}{a^6}\\ \end{align*}

Mathematica [A]  time = 0.0466435, size = 71, normalized size = 0.82 \[ \frac{-9 a^2 b x^4+2 a^3 x^6+36 a b^2 x^2-\frac{3 b^4 \left (10 a x^2+9 b\right )}{\left (a x^2+b\right )^2}-60 b^3 \log \left (a x^2+b\right )}{12 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b/x^2)^3,x]

[Out]

(36*a*b^2*x^2 - 9*a^2*b*x^4 + 2*a^3*x^6 - (3*b^4*(9*b + 10*a*x^2))/(b + a*x^2)^2 - 60*b^3*Log[b + a*x^2])/(12*
a^6)

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 80, normalized size = 0.9 \begin{align*} 3\,{\frac{{b}^{2}{x}^{2}}{{a}^{5}}}-{\frac{3\,b{x}^{4}}{4\,{a}^{4}}}+{\frac{{x}^{6}}{6\,{a}^{3}}}+{\frac{{b}^{5}}{4\,{a}^{6} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{5\,{b}^{4}}{2\,{a}^{6} \left ( a{x}^{2}+b \right ) }}-5\,{\frac{{b}^{3}\ln \left ( a{x}^{2}+b \right ) }{{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a+1/x^2*b)^3,x)

[Out]

3*b^2*x^2/a^5-3/4*b*x^4/a^4+1/6*x^6/a^3+1/4*b^5/a^6/(a*x^2+b)^2-5/2*b^4/a^6/(a*x^2+b)-5*b^3*ln(a*x^2+b)/a^6

________________________________________________________________________________________

Maxima [A]  time = 1.01067, size = 120, normalized size = 1.38 \begin{align*} -\frac{10 \, a b^{4} x^{2} + 9 \, b^{5}}{4 \,{\left (a^{8} x^{4} + 2 \, a^{7} b x^{2} + a^{6} b^{2}\right )}} - \frac{5 \, b^{3} \log \left (a x^{2} + b\right )}{a^{6}} + \frac{2 \, a^{2} x^{6} - 9 \, a b x^{4} + 36 \, b^{2} x^{2}}{12 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(10*a*b^4*x^2 + 9*b^5)/(a^8*x^4 + 2*a^7*b*x^2 + a^6*b^2) - 5*b^3*log(a*x^2 + b)/a^6 + 1/12*(2*a^2*x^6 - 9
*a*b*x^4 + 36*b^2*x^2)/a^5

________________________________________________________________________________________

Fricas [A]  time = 1.42239, size = 240, normalized size = 2.76 \begin{align*} \frac{2 \, a^{5} x^{10} - 5 \, a^{4} b x^{8} + 20 \, a^{3} b^{2} x^{6} + 63 \, a^{2} b^{3} x^{4} + 6 \, a b^{4} x^{2} - 27 \, b^{5} - 60 \,{\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )} \log \left (a x^{2} + b\right )}{12 \,{\left (a^{8} x^{4} + 2 \, a^{7} b x^{2} + a^{6} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*a^5*x^10 - 5*a^4*b*x^8 + 20*a^3*b^2*x^6 + 63*a^2*b^3*x^4 + 6*a*b^4*x^2 - 27*b^5 - 60*(a^2*b^3*x^4 + 2*
a*b^4*x^2 + b^5)*log(a*x^2 + b))/(a^8*x^4 + 2*a^7*b*x^2 + a^6*b^2)

________________________________________________________________________________________

Sympy [A]  time = 0.796733, size = 90, normalized size = 1.03 \begin{align*} - \frac{10 a b^{4} x^{2} + 9 b^{5}}{4 a^{8} x^{4} + 8 a^{7} b x^{2} + 4 a^{6} b^{2}} + \frac{x^{6}}{6 a^{3}} - \frac{3 b x^{4}}{4 a^{4}} + \frac{3 b^{2} x^{2}}{a^{5}} - \frac{5 b^{3} \log{\left (a x^{2} + b \right )}}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(10*a*b**4*x**2 + 9*b**5)/(4*a**8*x**4 + 8*a**7*b*x**2 + 4*a**6*b**2) + x**6/(6*a**3) - 3*b*x**4/(4*a**4) + 3
*b**2*x**2/a**5 - 5*b**3*log(a*x**2 + b)/a**6

________________________________________________________________________________________

Giac [A]  time = 1.18754, size = 124, normalized size = 1.43 \begin{align*} -\frac{5 \, b^{3} \log \left ({\left | a x^{2} + b \right |}\right )}{a^{6}} + \frac{30 \, a^{2} b^{3} x^{4} + 50 \, a b^{4} x^{2} + 21 \, b^{5}}{4 \,{\left (a x^{2} + b\right )}^{2} a^{6}} + \frac{2 \, a^{6} x^{6} - 9 \, a^{5} b x^{4} + 36 \, a^{4} b^{2} x^{2}}{12 \, a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*b^3*log(abs(a*x^2 + b))/a^6 + 1/4*(30*a^2*b^3*x^4 + 50*a*b^4*x^2 + 21*b^5)/((a*x^2 + b)^2*a^6) + 1/12*(2*a^
6*x^6 - 9*a^5*b*x^4 + 36*a^4*b^2*x^2)/a^9