Optimal. Leaf size=146 \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \left (-2 a^2 d^2+2 a b c d+b^2 c^2\right ) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a b^2 \left (a+b x^3\right )^{2/3}}-\frac {d x \sqrt [3]{a+b x^3} (b c-2 a d)}{2 a b^2}+\frac {x \left (c+d x^3\right ) (b c-a d)}{2 a b \left (a+b x^3\right )^{2/3}} \]
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Rubi [A] time = 0.10, 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 = {413, 388, 246, 245} \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \left (-2 a^2 d^2+2 a b c d+b^2 c^2\right ) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 a b^2 \left (a+b x^3\right )^{2/3}}-\frac {d x \sqrt [3]{a+b x^3} (b c-2 a d)}{2 a b^2}+\frac {x \left (c+d x^3\right ) (b c-a d)}{2 a b \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 388
Rule 413
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 = 3.76, size = 171, normalized size = 1.17 \[ \frac {x \Gamma \left (\frac {2}{3}\right ) \left (\frac {b x^3}{a}+1\right )^{2/3} \left (-3 b x^3 \left (c+d x^3\right )^2 \, _3F_2\left (\frac {4}{3},2,\frac {8}{3};1,\frac {13}{3};-\frac {b x^3}{a}\right )-b x^3 \left (11 c^2+16 c d x^3+5 d^2 x^6\right ) \, _2F_1\left (\frac {4}{3},\frac {8}{3};\frac {13}{3};-\frac {b x^3}{a}\right )+4 a \left (14 c^2+7 c d x^3+2 d^2 x^6\right ) \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {10}{3};-\frac {b x^3}{a}\right )\right )}{84 a^2 \Gamma \left (\frac {5}{3}\right ) \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{2} x^{6} + 2 \, c d x^{3} + c^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, 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 \[ \int \frac {{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {5}{3}}}\,{d x} \]
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 \[ \int \frac {{\left (d\,x^3+c\right )}^2}{{\left (b\,x^3+a\right )}^{5/3}} \,d x \]
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 \[ \int \frac {\left (c + d x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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