3.1846 \(\int \frac{x^3}{a+\frac{b}{x^2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac{b x^2}{2 a^2}+\frac{x^4}{4 a} \]

[Out]

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

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Rubi [A]  time = 0.0256779, antiderivative size = 40, 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} \[ \frac{b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac{b x^2}{2 a^2}+\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b/x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0050959, size = 40, normalized size = 1. \[ \frac{b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac{b x^2}{2 a^2}+\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b/x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 35, normalized size = 0.9 \begin{align*} -{\frac{b{x}^{2}}{2\,{a}^{2}}}+{\frac{{x}^{4}}{4\,a}}+{\frac{{b}^{2}\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02035, size = 46, normalized size = 1.15 \begin{align*} \frac{b^{2} \log \left (a x^{2} + b\right )}{2 \, a^{3}} + \frac{a x^{4} - 2 \, b x^{2}}{4 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.40931, size = 73, normalized size = 1.82 \begin{align*} \frac{a^{2} x^{4} - 2 \, a b x^{2} + 2 \, b^{2} \log \left (a x^{2} + b\right )}{4 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.29354, size = 32, normalized size = 0.8 \begin{align*} \frac{x^{4}}{4 a} - \frac{b x^{2}}{2 a^{2}} + \frac{b^{2} \log{\left (a x^{2} + b \right )}}{2 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17355, size = 47, normalized size = 1.18 \begin{align*} \frac{b^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{3}} + \frac{a x^{4} - 2 \, b x^{2}}{4 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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