Optimal. Leaf size=146 \[ -\frac {d (b c-2 a d) x \sqrt [3]{a+b x^3}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+\frac {\left (b^2 c^2+2 a b c d-2 a^2 d^2\right ) x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a b^2 \left (a+b x^3\right )^{2/3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 146, 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 = {424, 396, 252,
251} \begin {gather*} \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \left (-2 a^2 d^2+2 a b c d+b^2 c^2\right ) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a b^2 \left (a+b x^3\right )^{2/3}}-\frac {d x \sqrt [3]{a+b x^3} (b c-2 a d)}{2 a b^2}+\frac {x \left (c+d x^3\right ) (b c-a d)}{2 a b \left (a+b x^3\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 396
Rule 424
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{5/3}} \, dx &=\frac {(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+\frac {\int \frac {c (b c+a d)-2 d (b c-2 a d) x^3}{\left (a+b x^3\right )^{2/3}} \, dx}{2 a b}\\ &=-\frac {d (b c-2 a d) x \sqrt [3]{a+b x^3}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+-\frac {(-2 a d (b c-2 a d)-2 b c (b c+a d)) \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx}{4 a b^2}\\ &=-\frac {d (b c-2 a d) x \sqrt [3]{a+b x^3}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+-\frac {\left ((-2 a d (b c-2 a d)-2 b c (b c+a d)) \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{4 a b^2 \left (a+b x^3\right )^{2/3}}\\ &=-\frac {d (b c-2 a d) x \sqrt [3]{a+b x^3}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+\frac {\left (b^2 c^2+2 a b c d-2 a^2 d^2\right ) x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a b^2 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 11.93, size = 171, normalized size = 1.17 \begin {gather*} \frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \Gamma \left (\frac {2}{3}\right ) \left (4 a \left (14 c^2+7 c d x^3+2 d^2 x^6\right ) \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {10}{3};-\frac {b x^3}{a}\right )-b x^3 \left (11 c^2+16 c d x^3+5 d^2 x^6\right ) \, _2F_1\left (\frac {4}{3},\frac {8}{3};\frac {13}{3};-\frac {b x^3}{a}\right )-3 b x^3 \left (c+d x^3\right )^2 \, _3F_2\left (\frac {4}{3},2,\frac {8}{3};1,\frac {13}{3};-\frac {b x^3}{a}\right )\right )}{84 a^2 \left (a+b x^3\right )^{2/3} \Gamma \left (\frac {5}{3}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (d \,x^{3}+c \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {5}{3}}}\, dx\]
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 = 0.00, 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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac {5}{3}}}\, dx \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 \frac {{\left (d\,x^3+c\right )}^2}{{\left (b\,x^3+a\right )}^{5/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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