Optimal. Leaf size=64 \[ \frac {x^2 \left (a+b x^3\right )^{2/3} F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c \left (\frac {b x^3}{a}+1\right )^{2/3}} \]
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Rubi [A] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {511, 510} \[ \frac {x^2 \left (a+b x^3\right )^{2/3} F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c \left (\frac {b x^3}{a}+1\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {x \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int \frac {x \left (1+\frac {b x^3}{a}\right )^{2/3}}{c+d x^3} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=\frac {x^2 \left (a+b x^3\right )^{2/3} F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 65, normalized size = 1.02 \[ \frac {x^2 \left (a+b x^3\right )^{2/3} F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c \left (\frac {a+b x^3}{a}\right )^{2/3}} \]
Antiderivative was successfully verified.
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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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x}{d x^{3} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.57, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x}{d \,x^{3}+c}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x}{d x^{3} + c}\,{d x} \]
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 \[ \int \frac {x\,{\left (b\,x^3+a\right )}^{2/3}}{d\,x^3+c} \,d x \]
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 \[ \int \frac {x \left (a + b x^{3}\right )^{\frac {2}{3}}}{c + d x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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