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

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

[Out]

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

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Rubi [A]  time = 0.0084455, 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} \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^m/(a + b*x^3),x]

[Out]

(x^(1 + m)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(a*(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}{a+b x^3} \, dx &=\frac{x^{1+m} \, _2F_1\left (1,\frac{1+m}{3};\frac{4+m}{3};-\frac{b x^3}{a}\right )}{a (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0069612, size = 41, normalized size = 1.05 \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+1}{3}+1;-\frac{b x^3}{a}\right )}{a (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m/(a + b*x^3),x]

[Out]

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

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{b{x}^{3}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/(b*x^3 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m/(b*x^3 + a), x)

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Sympy [C]  time = 8.72115, size = 88, normalized size = 2.26 \begin{align*} \frac{m x x^{m} \Phi \left (\frac{b x^{3} e^{i \pi }}{a}, 1, \frac{m}{3} + \frac{1}{3}\right ) \Gamma \left (\frac{m}{3} + \frac{1}{3}\right )}{9 a \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} + \frac{x x^{m} \Phi \left (\frac{b x^{3} e^{i \pi }}{a}, 1, \frac{m}{3} + \frac{1}{3}\right ) \Gamma \left (\frac{m}{3} + \frac{1}{3}\right )}{9 a \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

m*x*x**m*lerchphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(9*a*gamma(m/3 + 4/3)) + x*x**m*ler
chphi(b*x**3*exp_polar(I*pi)/a, 1, m/3 + 1/3)*gamma(m/3 + 1/3)/(9*a*gamma(m/3 + 4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/(b*x^3 + a), x)