Optimal. Leaf size=64 \[ \frac {x^4 \sqrt [3]{a+b x^3} F_1\left (\frac {4}{3};-\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 c \sqrt [3]{\frac {b x^3}{a}+1}} \]
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Rubi [A] time = 0.05, antiderivative size = 64, 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} \[ \frac {x^4 \sqrt [3]{a+b x^3} F_1\left (\frac {4}{3};-\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 c \sqrt [3]{\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx &=\frac {\sqrt [3]{a+b x^3} \int \frac {x^3 \sqrt [3]{1+\frac {b x^3}{a}}}{c+d x^3} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {x^4 \sqrt [3]{a+b x^3} F_1\left (\frac {4}{3};-\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 c \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}
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Mathematica [B] time = 0.37, size = 240, normalized size = 3.75 \[ \frac {x \left (4 \left (\frac {4 a^2 c^2 F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\left (c+d x^3\right ) \left (x^3 \left (3 a d F_1\left (\frac {4}{3};\frac {2}{3},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c F_1\left (\frac {4}{3};\frac {5}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )-4 a c F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}+a+b x^3\right )+\frac {x^3 \left (\frac {b x^3}{a}+1\right )^{2/3} (a d-2 b c) F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c}\right )}{8 d \left (a+b x^3\right )^{2/3}} \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{3}}{d x^{3} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{3}}{d \,x^{3}+c}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{3}}{d x^{3} + c}\,{d x} \]
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 \[ \int \frac {x^3\,{\left (b\,x^3+a\right )}^{1/3}}{d\,x^3+c} \,d x \]
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 \[ \int \frac {x^{3} \sqrt [3]{a + b x^{3}}}{c + d x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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