Optimal. Leaf size=133 \[ \frac {a x \sqrt [3]{a+b x^3} \left (2 a^2 d^2-11 a b c d+44 b^2 c^2\right ) \, _2F_1\left (-\frac {4}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{44 b^2 \sqrt [3]{\frac {b x^3}{a}+1}}+\frac {d x \left (a+b x^3\right )^{7/3} (7 b c-2 a d)}{44 b^2}+\frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b} \]
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Rubi [A] time = 0.08, antiderivative size = 133, 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 = {416, 388, 246, 245} \[ \frac {a x \sqrt [3]{a+b x^3} \left (2 a^2 d^2-11 a b c d+44 b^2 c^2\right ) \, _2F_1\left (-\frac {4}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{44 b^2 \sqrt [3]{\frac {b x^3}{a}+1}}+\frac {d x \left (a+b x^3\right )^{7/3} (7 b c-2 a d)}{44 b^2}+\frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 388
Rule 416
Rubi steps
\begin {align*} \int \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 \, dx &=\frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b}+\frac {\int \left (a+b x^3\right )^{4/3} \left (c (11 b c-a d)+2 d (7 b c-2 a d) x^3\right ) \, dx}{11 b}\\ &=\frac {d (7 b c-2 a d) x \left (a+b x^3\right )^{7/3}}{44 b^2}+\frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b}-\frac {(2 a d (7 b c-2 a d)-8 b c (11 b c-a d)) \int \left (a+b x^3\right )^{4/3} \, dx}{88 b^2}\\ &=\frac {d (7 b c-2 a d) x \left (a+b x^3\right )^{7/3}}{44 b^2}+\frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b}-\frac {\left (a (2 a d (7 b c-2 a d)-8 b c (11 b c-a d)) \sqrt [3]{a+b x^3}\right ) \int \left (1+\frac {b x^3}{a}\right )^{4/3} \, dx}{88 b^2 \sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {d (7 b c-2 a d) x \left (a+b x^3\right )^{7/3}}{44 b^2}+\frac {d x \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )}{11 b}+\frac {a \left (44 b^2 c^2-11 a b c d+2 a^2 d^2\right ) 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 )}{44 b^2 \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}
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Mathematica [A] time = 5.19, size = 171, normalized size = 1.29 \[ \frac {x \left (2 a^2 \left (\frac {b x^3}{a}+1\right )^{2/3} \left (2 a^2 d^2-11 a b c d+44 b^2 c^2\right ) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )-\left (a+b x^3\right ) \left (4 a^3 d^2-2 a^2 b d \left (11 c+d x^3\right )-3 a b^2 \left (44 c^2+33 c d x^3+10 d^2 x^6\right )-b^3 x^3 \left (44 c^2+55 c d x^3+20 d^2 x^6\right )\right )\right )}{220 b^2 \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b d^{2} x^{9} + {\left (2 \, b c d + a d^{2}\right )} x^{6} + {\left (b c^{2} + 2 \, a c d\right )} x^{3} + a c^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}, x\right ) \]
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 \[ \int {\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )^{2}\, dx \]
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 \[ \int {\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{2}\,{d x} \]
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 \[ \int {\left (b\,x^3+a\right )}^{4/3}\,{\left (d\,x^3+c\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.06, size = 270, normalized size = 2.03 \[ \frac {a^{\frac {4}{3}} c^{2} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {2 a^{\frac {4}{3}} c d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {a^{\frac {4}{3}} d^{2} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {\sqrt [3]{a} b c^{2} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {2 \sqrt [3]{a} b c d x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {\sqrt [3]{a} b d^{2} x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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