Optimal. Leaf size=83 \[ \frac {d x \left (a+b x^3\right )^{7/3}}{8 b}+\frac {a (8 b c-a d) x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {4}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{8 b \sqrt [3]{1+\frac {b x^3}{a}}} \]
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Rubi [A]
time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {396, 252, 251}
\begin {gather*} \frac {a x \sqrt [3]{a+b x^3} (8 b c-a d) \, _2F_1\left (-\frac {4}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{8 b \sqrt [3]{\frac {b x^3}{a}+1}}+\frac {d x \left (a+b x^3\right )^{7/3}}{8 b} \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 )^{4/3} \left (c+d x^3\right ) \, dx &=\frac {d x \left (a+b x^3\right )^{7/3}}{8 b}-\frac {(-8 b c+a d) \int \left (a+b x^3\right )^{4/3} \, dx}{8 b}\\ &=\frac {d x \left (a+b x^3\right )^{7/3}}{8 b}-\frac {\left (a (-8 b c+a d) \sqrt [3]{a+b x^3}\right ) \int \left (1+\frac {b x^3}{a}\right )^{4/3} \, dx}{8 b \sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {d x \left (a+b x^3\right )^{7/3}}{8 b}+\frac {a (8 b c-a d) x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {4}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{8 b \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}
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Mathematica [A]
time = 7.49, size = 75, normalized size = 0.90 \begin {gather*} \frac {x \sqrt [3]{a+b x^3} \left (d \left (a+b x^3\right )^2-\frac {a (-8 b c+a d) \, _2F_1\left (-\frac {4}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\sqrt [3]{1+\frac {b x^3}{a}}}\right )}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )\, dx\]
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 = 2.23, size = 170, normalized size = 2.05 \begin {gather*} \frac {a^{\frac {4}{3}} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{\frac {4}{3}} d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt [3]{a} b c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt [3]{a} b d x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{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 )}^{4/3}\,\left (d\,x^3+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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