Optimal. Leaf size=62 \[ -\frac {\left (a+b x^3\right )^{2/3} F_1\left (-\frac {1}{3};-\frac {2}{3},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c x \left (\frac {b x^3}{a}+1\right )^{2/3}} \]
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Rubi [A] time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac {\left (a+b x^3\right )^{2/3} F_1\left (-\frac {1}{3};-\frac {2}{3},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c x \left (\frac {b x^3}{a}+1\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^2 \left (c+d x^3\right )} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int \frac {\left (1+\frac {b x^3}{a}\right )^{2/3}}{x^2 \left (c+d x^3\right )} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=-\frac {\left (a+b x^3\right )^{2/3} F_1\left (-\frac {1}{3};-\frac {2}{3},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c x \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}
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Mathematica [B] time = 0.14, size = 138, normalized size = 2.23 \[ \frac {-5 x^3 \sqrt [3]{\frac {b x^3}{a}+1} (a d-2 b c) F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b d x^6 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-10 c \left (a+b x^3\right )}{10 c^2 x \sqrt [3]{a+b x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.69, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{\left (d \,x^{3}+c \right ) x^{2}}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,x^3+a\right )}^{2/3}}{x^2\,\left (d\,x^3+c\right )} \,d x \]
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 \[ \int \frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{x^{2} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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