3.126 \(\int \frac{x^{-m}}{a^3+x^3} \, dx\)

Optimal. Leaf size=46 \[ \frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1-m}{3},\frac{4-m}{3},-\frac{x^3}{a^3}\right )}{a^3 (1-m)} \]

[Out]

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

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Rubi [A]  time = 0.0295284, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{x^{1-m} \, _2F_1\left (1,\frac{1-m}{3};\frac{4-m}{3};-\frac{x^3}{a^3}\right )}{a^3 (1-m)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^m*(a^3 + x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.07274, size = 31, normalized size = 0.67 \[ \frac{x^{- m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{m}{3} + \frac{1}{3} \\ - \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{x^{3}}{a^{3}}} \right )}}{a^{3} \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**m)/(a**3+x**3),x)

[Out]

x**(-m + 1)*hyper((1, -m/3 + 1/3), (-m/3 + 4/3,), -x**3/a**3)/(a**3*(-m + 1))

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Mathematica [A]  time = 0.0257698, size = 47, normalized size = 1.02 \[ -\frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1-m}{3},\frac{1-m}{3}+1,-\frac{x^3}{a^3}\right )}{a^3 (m-1)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^m*(a^3 + x^3)),x]

[Out]

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

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{m} \left ({a}^{3}+{x}^{3} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^m)/(a^3+x^3),x)

[Out]

int(1/(x^m)/(a^3+x^3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-m}}{a^{3} + x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^3 + x^3)*x^m),x, algorithm="maxima")

[Out]

integrate(x^(-m)/(a^3 + x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (a^{3} + x^{3}\right )} x^{m}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^3 + x^3)*x^m),x, algorithm="fricas")

[Out]

integral(1/((a^3 + x^3)*x^m), x)

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Sympy [A]  time = 7.5076, size = 92, normalized size = 2. \[ - \frac{m x x^{- m} \Phi \left (\frac{x^{3} e^{i \pi }}{a^{3}}, 1, - \frac{m}{3} + \frac{1}{3}\right ) \Gamma \left (- \frac{m}{3} + \frac{1}{3}\right )}{9 a^{3} \Gamma \left (- \frac{m}{3} + \frac{4}{3}\right )} + \frac{x x^{- m} \Phi \left (\frac{x^{3} e^{i \pi }}{a^{3}}, 1, - \frac{m}{3} + \frac{1}{3}\right ) \Gamma \left (- \frac{m}{3} + \frac{1}{3}\right )}{9 a^{3} \Gamma \left (- \frac{m}{3} + \frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**m)/(a**3+x**3),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a^{3} + x^{3}\right )} x^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^3 + x^3)*x^m),x, algorithm="giac")

[Out]

integrate(1/((a^3 + x^3)*x^m), x)