Optimal. Leaf size=131 \[ \frac {d (11 b c-4 a d) x \left (a+b x^3\right )^{4/3}}{40 b^2}+\frac {d x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}{8 b}+\frac {\left (10 b^2 c^2-4 a b c d+a^2 d^2\right ) x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{10 b^2 \sqrt [3]{1+\frac {b x^3}{a}}} \]
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Rubi [A]
time = 0.05, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {427, 396, 252,
251} \begin {gather*} \frac {x \sqrt [3]{a+b x^3} \left (a^2 d^2-4 a b c d+10 b^2 c^2\right ) \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{10 b^2 \sqrt [3]{\frac {b x^3}{a}+1}}+\frac {d x \left (a+b x^3\right )^{4/3} (11 b c-4 a d)}{40 b^2}+\frac {d x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 396
Rule 427
Rubi steps
\begin {align*} \int \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2 \, dx &=\frac {d x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}{8 b}+\frac {\int \sqrt [3]{a+b x^3} \left (c (8 b c-a d)+d (11 b c-4 a d) x^3\right ) \, dx}{8 b}\\ &=\frac {d (11 b c-4 a d) x \left (a+b x^3\right )^{4/3}}{40 b^2}+\frac {d x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}{8 b}+\frac {\left (10 b^2 c^2-4 a b c d+a^2 d^2\right ) \int \sqrt [3]{a+b x^3} \, dx}{10 b^2}\\ &=\frac {d (11 b c-4 a d) x \left (a+b x^3\right )^{4/3}}{40 b^2}+\frac {d x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}{8 b}+\frac {\left (\left (10 b^2 c^2-4 a b c d+a^2 d^2\right ) \sqrt [3]{a+b x^3}\right ) \int \sqrt [3]{1+\frac {b x^3}{a}} \, dx}{10 b^2 \sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {d (11 b c-4 a d) x \left (a+b x^3\right )^{4/3}}{40 b^2}+\frac {d x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}{8 b}+\frac {\left (10 b^2 c^2-4 a b c d+a^2 d^2\right ) x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{10 b^2 \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}
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Mathematica [A]
time = 7.69, size = 179, normalized size = 1.37 \begin {gather*} \frac {x \sqrt [3]{a+b x^3} \left (20 a \left (14 c^2+7 c d x^3+2 d^2 x^6\right ) \Gamma \left (-\frac {1}{3}\right ) \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {10}{3};-\frac {b x^3}{a}\right )-3 b x^3 \left (11 c^2+16 c d x^3+5 d^2 x^6\right ) \Gamma \left (\frac {2}{3}\right ) \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {13}{3};-\frac {b x^3}{a}\right )-9 b x^3 \left (c+d x^3\right )^2 \Gamma \left (\frac {2}{3}\right ) \, _3F_2\left (\frac {2}{3},\frac {4}{3},2;1,\frac {13}{3};-\frac {b x^3}{a}\right )\right )}{280 a \sqrt [3]{1+\frac {b x^3}{a}} \Gamma \left (-\frac {1}{3}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2}\, dx\]
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 = 1.81, size = 131, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{a} c^{2} 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 {2 \sqrt [3]{a} c 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} d^{2} 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 )}^{1/3}\,{\left (d\,x^3+c\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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