3.342 \(\int \frac{1}{(\frac{b}{x^3}+a x^2)^3} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^{10}}{10 b \left (a x^5+b\right )^2} \]

[Out]

x^10/(10*b*(b + a*x^5)^2)

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Rubi [A]  time = 0.006112, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1593, 264} \[ \frac{x^{10}}{10 b \left (a x^5+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(b/x^3 + a*x^2)^(-3),x]

[Out]

x^10/(10*b*(b + a*x^5)^2)

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{\left (\frac{b}{x^3}+a x^2\right )^3} \, dx &=\int \frac{x^9}{\left (b+a x^5\right )^3} \, dx\\ &=\frac{x^{10}}{10 b \left (b+a x^5\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0087693, size = 24, normalized size = 1.26 \[ -\frac{2 a x^5+b}{10 a^2 \left (a x^5+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b/x^3 + a*x^2)^(-3),x]

[Out]

-(b + 2*a*x^5)/(10*a^2*(b + a*x^5)^2)

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Maple [A]  time = 0.004, size = 31, normalized size = 1.6 \begin{align*}{\frac{b}{10\,{a}^{2} \left ( a{x}^{5}+b \right ) ^{2}}}-{\frac{1}{5\,{a}^{2} \left ( a{x}^{5}+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b/a^2/(a*x^5+b)^2-1/5/a^2/(a*x^5+b)

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Maxima [B]  time = 1.0571, size = 49, normalized size = 2.58 \begin{align*} -\frac{2 \, a x^{5} + b}{10 \,{\left (a^{4} x^{10} + 2 \, a^{3} b x^{5} + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/x^3+a*x^2)^3,x, algorithm="maxima")

[Out]

-1/10*(2*a*x^5 + b)/(a^4*x^10 + 2*a^3*b*x^5 + a^2*b^2)

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Fricas [B]  time = 0.631051, size = 76, normalized size = 4. \begin{align*} -\frac{2 \, a x^{5} + b}{10 \,{\left (a^{4} x^{10} + 2 \, a^{3} b x^{5} + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/x^3+a*x^2)^3,x, algorithm="fricas")

[Out]

-1/10*(2*a*x^5 + b)/(a^4*x^10 + 2*a^3*b*x^5 + a^2*b^2)

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Sympy [B]  time = 5.42171, size = 36, normalized size = 1.89 \begin{align*} - \frac{2 a x^{5} + b}{10 a^{4} x^{10} + 20 a^{3} b x^{5} + 10 a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*x**5 + b)/(10*a**4*x**10 + 20*a**3*b*x**5 + 10*a**2*b**2)

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Giac [A]  time = 1.1578, size = 30, normalized size = 1.58 \begin{align*} -\frac{2 \, a x^{5} + b}{10 \,{\left (a x^{5} + b\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/x^3+a*x^2)^3,x, algorithm="giac")

[Out]

-1/10*(2*a*x^5 + b)/((a*x^5 + b)^2*a^2)