3.1876 \(\int \frac{x^3}{(a+\frac{b}{x^2})^3} \, dx\)

Optimal. Leaf size=74 \[ -\frac{b^4}{4 a^5 \left (a x^2+b\right )^2}+\frac{2 b^3}{a^5 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{a^5}-\frac{3 b x^2}{2 a^4}+\frac{x^4}{4 a^3} \]

[Out]

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

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Rubi [A]  time = 0.0558214, antiderivative size = 74, 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} \[ -\frac{b^4}{4 a^5 \left (a x^2+b\right )^2}+\frac{2 b^3}{a^5 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{a^5}-\frac{3 b x^2}{2 a^4}+\frac{x^4}{4 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b/x^2)^3,x]

[Out]

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

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 )^3} \, dx &=\int \frac{x^9}{\left (b+a x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{(b+a x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{3 b}{a^4}+\frac{x}{a^3}+\frac{b^4}{a^4 (b+a x)^3}-\frac{4 b^3}{a^4 (b+a x)^2}+\frac{6 b^2}{a^4 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 b x^2}{2 a^4}+\frac{x^4}{4 a^3}-\frac{b^4}{4 a^5 \left (b+a x^2\right )^2}+\frac{2 b^3}{a^5 \left (b+a x^2\right )}+\frac{3 b^2 \log \left (b+a x^2\right )}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.0420365, size = 58, normalized size = 0.78 \[ \frac{a^2 x^4+\frac{b^3 \left (8 a x^2+7 b\right )}{\left (a x^2+b\right )^2}+12 b^2 \log \left (a x^2+b\right )-6 a b x^2}{4 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b/x^2)^3,x]

[Out]

(-6*a*b*x^2 + a^2*x^4 + (b^3*(7*b + 8*a*x^2))/(b + a*x^2)^2 + 12*b^2*Log[b + a*x^2])/(4*a^5)

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Maple [A]  time = 0.01, size = 69, normalized size = 0.9 \begin{align*} -{\frac{3\,b{x}^{2}}{2\,{a}^{4}}}+{\frac{{x}^{4}}{4\,{a}^{3}}}-{\frac{{b}^{4}}{4\,{a}^{5} \left ( a{x}^{2}+b \right ) ^{2}}}+2\,{\frac{{b}^{3}}{{a}^{5} \left ( a{x}^{2}+b \right ) }}+3\,{\frac{{b}^{2}\ln \left ( a{x}^{2}+b \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.31239, size = 104, normalized size = 1.41 \begin{align*} \frac{8 \, a b^{3} x^{2} + 7 \, b^{4}}{4 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}} + \frac{3 \, b^{2} \log \left (a x^{2} + b\right )}{a^{5}} + \frac{a x^{4} - 6 \, 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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.38963, size = 211, normalized size = 2.85 \begin{align*} \frac{a^{4} x^{8} - 4 \, a^{3} b x^{6} - 11 \, a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + 7 \, b^{4} + 12 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \log \left (a x^{2} + b\right )}{4 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a^4*x^8 - 4*a^3*b*x^6 - 11*a^2*b^2*x^4 + 2*a*b^3*x^2 + 7*b^4 + 12*(a^2*b^2*x^4 + 2*a*b^3*x^2 + b^4)*log(a
*x^2 + b))/(a^7*x^4 + 2*a^6*b*x^2 + a^5*b^2)

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Sympy [A]  time = 0.728944, size = 78, normalized size = 1.05 \begin{align*} \frac{8 a b^{3} x^{2} + 7 b^{4}}{4 a^{7} x^{4} + 8 a^{6} b x^{2} + 4 a^{5} b^{2}} + \frac{x^{4}}{4 a^{3}} - \frac{3 b x^{2}}{2 a^{4}} + \frac{3 b^{2} \log{\left (a x^{2} + b \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15127, size = 93, normalized size = 1.26 \begin{align*} \frac{3 \, b^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{a^{5}} + \frac{a^{3} x^{4} - 6 \, a^{2} b x^{2}}{4 \, a^{6}} + \frac{8 \, a b^{3} x^{2} + 7 \, b^{4}}{4 \,{\left (a x^{2} + b\right )}^{2} a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b^2*log(abs(a*x^2 + b))/a^5 + 1/4*(a^3*x^4 - 6*a^2*b*x^2)/a^6 + 1/4*(8*a*b^3*x^2 + 7*b^4)/((a*x^2 + b)^2*a^5
)