Optimal. Leaf size=61 \[ \frac {(c x)^{1+m} \left (a+b c^3 x^3\right )^{5/3} \, _2F_1\left (1,\frac {6+m}{3};\frac {4+m}{3};-\frac {b c^3 x^3}{a}\right )}{a c (1+m)} \]
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Rubi [A]
time = 0.01, 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, Rules used = {372, 371}
\begin {gather*} \frac {\left (a+b x^3\right )^{2/3} (c x)^{m+1} \, _2F_1\left (-\frac {2}{3},\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{c (m+1) \left (\frac {b x^3}{a}+1\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rubi steps
\begin {align*} \int (c x)^m \left (a+b x^3\right )^{2/3} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int (c x)^m \left (1+\frac {b x^3}{a}\right )^{2/3} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=\frac {(c x)^{1+m} \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{c (1+m) \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 66, normalized size = 1.08 \begin {gather*} \frac {x (c x)^m \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {1+m}{3};1+\frac {1+m}{3};-\frac {b x^3}{a}\right )}{(1+m) \left (1+\frac {b x^3}{a}\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (c x \right )^{m} \left (b \,x^{3}+a \right )^{\frac {2}{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.37, size = 17, normalized size = 0.28 \begin {gather*} {\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (c x\right )^{m}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.04, size = 58, normalized size = 0.95 \begin {gather*} \frac {a^{\frac {2}{3}} c^{m} x x^{m} \Gamma \left (\frac {m}{3} + \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (c\,x\right )}^m\,{\left (b\,x^3+a\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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