3.1861 \(\int \frac{x^3}{(a+\frac{b}{x^2})^2} \, dx\)

Optimal. Leaf size=57 \[ \frac{b^3}{2 a^4 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^2}{a^3}+\frac{x^4}{4 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0420627, antiderivative size = 57, 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} \[ \frac{b^3}{2 a^4 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^2}{a^3}+\frac{x^4}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b/x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0154264, size = 49, normalized size = 0.86 \[ \frac{a^2 x^4+\frac{2 b^3}{a x^2+b}+6 b^2 \log \left (a x^2+b\right )-4 a b x^2}{4 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b/x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.01, size = 52, normalized size = 0.9 \begin{align*} -{\frac{b{x}^{2}}{{a}^{3}}}+{\frac{{x}^{4}}{4\,{a}^{2}}}+{\frac{{b}^{3}}{2\,{a}^{4} \left ( a{x}^{2}+b \right ) }}+{\frac{3\,{b}^{2}\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^3/(a+1/x^2*b)^2,x)

[Out]

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

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Maxima [A]  time = 1.05457, size = 73, normalized size = 1.28 \begin{align*} \frac{b^{3}}{2 \,{\left (a^{5} x^{2} + a^{4} b\right )}} + \frac{3 \, b^{2} \log \left (a x^{2} + b\right )}{2 \, a^{4}} + \frac{a x^{4} - 4 \, b x^{2}}{4 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.39442, size = 143, normalized size = 2.51 \begin{align*} \frac{a^{3} x^{6} - 3 \, a^{2} b x^{4} - 4 \, a b^{2} x^{2} + 2 \, b^{3} + 6 \,{\left (a b^{2} x^{2} + b^{3}\right )} \log \left (a x^{2} + b\right )}{4 \,{\left (a^{5} x^{2} + a^{4} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.532448, size = 53, normalized size = 0.93 \begin{align*} \frac{b^{3}}{2 a^{5} x^{2} + 2 a^{4} b} + \frac{x^{4}}{4 a^{2}} - \frac{b x^{2}}{a^{3}} + \frac{3 b^{2} \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**3/(a+b/x**2)**2,x)

[Out]

b**3/(2*a**5*x**2 + 2*a**4*b) + x**4/(4*a**2) - b*x**2/a**3 + 3*b**2*log(a*x**2 + b)/(2*a**4)

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Giac [A]  time = 1.18324, size = 74, normalized size = 1.3 \begin{align*} \frac{3 \, b^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} + \frac{b^{3}}{2 \,{\left (a x^{2} + b\right )} a^{4}} + \frac{a^{2} x^{4} - 4 \, a b x^{2}}{4 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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