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 = 0.0530495, antiderivative size = 64, normalized size of antiderivative = 1., 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 511
Rule 510
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*}
Mathematica [B] time = 0.280696, 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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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{d{x}^{3}+c}\sqrt [3]{b{x}^{3}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{3}}{d x^{3} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \sqrt [3]{a + b x^{3}}}{c + d x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{3}}{d x^{3} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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