3.590 \(\int \frac{x^m}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=39 \[ \frac{x^{m+1} \, _2F_1\left (3,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a^3 (m+1)} \]

[Out]

(x^(1 + m)*Hypergeometric2F1[3, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(a^3*(1 + m))

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Rubi [A]  time = 0.0081129, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {364} \[ \frac{x^{m+1} \, _2F_1\left (3,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a^3 (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^m/(a + b*x^3)^3,x]

[Out]

(x^(1 + m)*Hypergeometric2F1[3, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(a^3*(1 + m))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{x^m}{\left (a+b x^3\right )^3} \, dx &=\frac{x^{1+m} \, _2F_1\left (3,\frac{1+m}{3};\frac{4+m}{3};-\frac{b x^3}{a}\right )}{a^3 (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0076304, size = 41, normalized size = 1.05 \[ \frac{x^{m+1} \, _2F_1\left (3,\frac{m+1}{3};\frac{m+1}{3}+1;-\frac{b x^3}{a}\right )}{a^3 (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m/(a + b*x^3)^3,x]

[Out]

(x^(1 + m)*Hypergeometric2F1[3, (1 + m)/3, 1 + (1 + m)/3, -((b*x^3)/a)])/(a^3*(1 + m))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( b{x}^{3}+a \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/(b*x^3 + a)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m/(b^3*x^9 + 3*a*b^2*x^6 + 3*a^2*b*x^3 + a^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/(b*x^3 + a)^3, x)