3.1844 \(\int \frac{x^5}{a+\frac{b}{x^2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^4}{4 a^2}+\frac{x^6}{6 a} \]

[Out]

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

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Rubi [A]  time = 0.0344523, antiderivative size = 53, 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} \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^4}{4 a^2}+\frac{x^6}{6 a} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b/x^2),x]

[Out]

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

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}{a+\frac{b}{x^2}} \, dx &=\int \frac{x^7}{b+a x^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{b+a x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{a^3}-\frac{b x}{a^2}+\frac{x^2}{a}-\frac{b^3}{a^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac{b^2 x^2}{2 a^3}-\frac{b x^4}{4 a^2}+\frac{x^6}{6 a}-\frac{b^3 \log \left (b+a x^2\right )}{2 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0052669, size = 53, normalized size = 1. \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^4}{4 a^2}+\frac{x^6}{6 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b/x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 46, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}{x}^{2}}{2\,{a}^{3}}}-{\frac{b{x}^{4}}{4\,{a}^{2}}}+{\frac{{x}^{6}}{6\,a}}-{\frac{{b}^{3}\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.992105, size = 62, normalized size = 1.17 \begin{align*} -\frac{b^{3} \log \left (a x^{2} + b\right )}{2 \, a^{4}} + \frac{2 \, a^{2} x^{6} - 3 \, a b x^{4} + 6 \, b^{2} x^{2}}{12 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^3*log(a*x^2 + b)/a^4 + 1/12*(2*a^2*x^6 - 3*a*b*x^4 + 6*b^2*x^2)/a^3

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Fricas [A]  time = 1.44927, size = 99, normalized size = 1.87 \begin{align*} \frac{2 \, a^{3} x^{6} - 3 \, a^{2} b x^{4} + 6 \, a b^{2} x^{2} - 6 \, b^{3} \log \left (a x^{2} + b\right )}{12 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*a^3*x^6 - 3*a^2*b*x^4 + 6*a*b^2*x^2 - 6*b^3*log(a*x^2 + b))/a^4

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Sympy [A]  time = 0.302942, size = 44, normalized size = 0.83 \begin{align*} \frac{x^{6}}{6 a} - \frac{b x^{4}}{4 a^{2}} + \frac{b^{2} x^{2}}{2 a^{3}} - \frac{b^{3} \log{\left (a x^{2} + b \right )}}{2 a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**6/(6*a) - b*x**4/(4*a**2) + b**2*x**2/(2*a**3) - b**3*log(a*x**2 + b)/(2*a**4)

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Giac [A]  time = 1.13512, size = 63, normalized size = 1.19 \begin{align*} -\frac{b^{3} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} + \frac{2 \, a^{2} x^{6} - 3 \, a b x^{4} + 6 \, b^{2} x^{2}}{12 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^3*log(abs(a*x^2 + b))/a^4 + 1/12*(2*a^2*x^6 - 3*a*b*x^4 + 6*b^2*x^2)/a^3