Optimal. Leaf size=93 \[ \frac {d x \left (a+b x^3\right )^{1+m}}{b (4+3 m)}-\frac {(a d-b c (4+3 m)) x \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )}{b (4+3 m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {396, 252, 251}
\begin {gather*} x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} \left (c-\frac {a d}{3 b m+4 b}\right ) \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )+\frac {d x \left (a+b x^3\right )^{m+1}}{b (3 m+4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 396
Rubi steps
\begin {align*} \int \left (a+b x^3\right )^m \left (c+d x^3\right ) \, dx &=\frac {d x \left (a+b x^3\right )^{1+m}}{b (4+3 m)}-\left (-c+\frac {a d}{4 b+3 b m}\right ) \int \left (a+b x^3\right )^m \, dx\\ &=\frac {d x \left (a+b x^3\right )^{1+m}}{b (4+3 m)}-\left (\left (-c+\frac {a d}{4 b+3 b m}\right ) \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m}\right ) \int \left (1+\frac {b x^3}{a}\right )^m \, dx\\ &=\frac {d x \left (a+b x^3\right )^{1+m}}{b (4+3 m)}+\left (c-\frac {a d}{4 b+3 b m}\right ) x \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 90, normalized size = 0.97 \begin {gather*} \frac {x \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} \left (d \left (a+b x^3\right ) \left (1+\frac {b x^3}{a}\right )^m+(-a d+b c (4+3 m)) \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )\right )}{b (4+3 m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (b \,x^{3}+a \right )^{m} \left (d \,x^{3}+c \right )\, dx\]
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 = 0.00, 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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Sympy [C] Result contains complex when optimal does not.
time = 40.02, size = 75, normalized size = 0.81 \begin {gather*} \frac {a^{m} c 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 )} + \frac {a^{m} d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, - m \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{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 = -0.01 \begin {gather*} \int {\left (b\,x^3+a\right )}^m\,\left (d\,x^3+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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