3.2693 \(\int \frac{x^m}{\sqrt [3]{a+b x^{3 (1+m)}}} \, dx\)

Optimal. Leaf size=97 \[ \frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{m+1}}{\sqrt [3]{a+b x^{3 (m+1)}}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b} (m+1)}-\frac{\log \left (\sqrt [3]{b} x^{m+1}-\sqrt [3]{a+b x^{3 (m+1)}}\right )}{2 \sqrt [3]{b} (m+1)} \]

[Out]

ArcTan[(1 + (2*b^(1/3)*x^(1 + m))/(a + b*x^(3*(1 + m)))^(1/3))/Sqrt[3]]/(Sqrt[3]
*b^(1/3)*(1 + m)) - Log[b^(1/3)*x^(1 + m) - (a + b*x^(3*(1 + m)))^(1/3)]/(2*b^(1
/3)*(1 + m))

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Rubi [A]  time = 0.115021, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{m+1}}{\sqrt [3]{a+b x^{3 (m+1)}}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b} (m+1)}-\frac{\log \left (\sqrt [3]{b} x^{m+1}-\sqrt [3]{a+b x^{3 (m+1)}}\right )}{2 \sqrt [3]{b} (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(1 + (2*b^(1/3)*x^(1 + m))/(a + b*x^(3*(1 + m)))^(1/3))/Sqrt[3]]/(Sqrt[3]
*b^(1/3)*(1 + m)) - Log[b^(1/3)*x^(1 + m) - (a + b*x^(3*(1 + m)))^(1/3)]/(2*b^(1
/3)*(1 + m))

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Rubi in Sympy [A]  time = 7.52415, size = 56, normalized size = 0.58 \[ \frac{x^{m + 1} \left (a + b x^{3 m + 3}\right )^{\frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3 m + 3}}{a}} \right )}}{a \left (1 + \frac{b x^{3 m + 3}}{a}\right )^{\frac{2}{3}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.080677, size = 68, normalized size = 0.7 \[ \frac{x^{m+1} \sqrt [3]{\frac{a+b x^{3 m+3}}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^{3 m+3}}{a}\right )}{(m+1) \sqrt [3]{a+b x^{3 m+3}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{{x}^{m}{\frac{1}{\sqrt [3]{a+b{x}^{3+3\,m}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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