3.587 \(\int \frac{x^m}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=39 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a^2 (m+1)} \]

[Out]

(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(a^2*(1 + m
))

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Rubi [A]  time = 0.0283655, 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 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a^2 (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x^3)^2,x]

[Out]

(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(a^2*(1 + m
))

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Rubi in Sympy [A]  time = 3.87933, size = 31, normalized size = 0.79 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{a^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(b*x**3+a)**2,x)

[Out]

x**(m + 1)*hyper((2, m/3 + 1/3), (m/3 + 4/3,), -b*x**3/a)/(a**2*(m + 1))

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Mathematica [A]  time = 0.0302432, size = 41, normalized size = 1.05 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{3};\frac{m+1}{3}+1;-\frac{b x^3}{a}\right )}{a^2 (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a + b*x^3)^2,x]

[Out]

(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/3, 1 + (1 + m)/3, -((b*x^3)/a)])/(a^2*(1
+ m))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( b{x}^{3}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x^3+a)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

integrate(x^m/(b*x^3 + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

integrate(x^m/(b*x^3 + a)^2, x)