Optimal. Leaf size=61 \[ \frac{(c x)^{m+1} \left (a+b c^3 x^3\right )^{4/3} \, _2F_1\left (1,\frac{m+5}{3};\frac{m+4}{3};-\frac{b c^3 x^3}{a}\right )}{a c (m+1)} \]
[Out]
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Rubi [A] time = 0.0633637, antiderivative size = 68, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\sqrt [3]{a+b x^3} (c x)^{m+1} \, _2F_1\left (-\frac{1}{3},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{c (m+1) \sqrt [3]{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^m*(a + b*x^3)^(1/3),x]
[Out]
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Rubi in Sympy [A] time = 7.66497, size = 56, normalized size = 0.92 \[ \frac{\left (c x\right )^{m + 1} \sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{c \sqrt [3]{1 + \frac{b x^{3}}{a}} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**m*(b*x**3+a)**(1/3),x)
[Out]
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Mathematica [A] time = 0.0295587, size = 64, normalized size = 1.05 \[ \frac{x \sqrt [3]{a+b x^3} (c x)^m \, _2F_1\left (-\frac{1}{3},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{(m+1) \sqrt [3]{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^m*(a + b*x^3)^(1/3),x]
[Out]
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Maple [F] time = 0.029, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{m}\sqrt [3]{b{x}^{3}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^m*(b*x^3+a)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (c x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(1/3)*(c*x)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (c x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(1/3)*(c*x)^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.0469, size = 58, normalized size = 0.95 \[ \frac{\sqrt [3]{a} c^{m} x x^{m} \Gamma \left (\frac{m}{3} + \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**m*(b*x**3+a)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (c x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(1/3)*(c*x)^m,x, algorithm="giac")
[Out]