Optimal. Leaf size=44 \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right ) \]
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Rubi [A] time = 0.0267679, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^3)^m,x]
[Out]
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Rubi in Sympy [A] time = 3.92162, size = 34, normalized size = 0.77 \[ x \left (1 + \frac{b x^{3}}{a}\right )^{- m} \left (a + b x^{3}\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a)**m,x)
[Out]
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Mathematica [C] time = 0.287412, size = 196, normalized size = 4.45 \[ \frac{2^{-m} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{-m} \left (\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}\right )^{-m} \left (a+b x^3\right )^m F_1\left (m+1;-m,-m;m+2;-\frac{i \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{\sqrt [3]{b} (m+1)} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^3)^m,x]
[Out]
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Maple [F] time = 0.035, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) ^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)^m,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^m,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{3} + a\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 54.2667, size = 34, normalized size = 0.77 \[ \frac{a^{m} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, - m \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a)**m,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^m,x, algorithm="giac")
[Out]