Optimal. Leaf size=60 \[ \frac{a x \sqrt [3]{a+b x^3} F_1\left (\frac{1}{3};-\frac{4}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c \sqrt [3]{\frac{b x^3}{a}+1}} \]
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Rubi [A] time = 0.0273729, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {430, 429} \[ \frac{a x \sqrt [3]{a+b x^3} F_1\left (\frac{1}{3};-\frac{4}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c \sqrt [3]{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx &=\frac{\left (a \sqrt [3]{a+b x^3}\right ) \int \frac{\left (1+\frac{b x^3}{a}\right )^{4/3}}{c+d x^3} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{a x \sqrt [3]{a+b x^3} F_1\left (\frac{1}{3};-\frac{4}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [B] time = 0.303343, size = 346, normalized size = 5.77 \[ \frac{x \left (\frac{4 \left (b x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (3 a d F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 b c F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-4 a c \left (2 a^2 d+a b d x^3+b^2 x^3 \left (c+d x^3\right )\right ) F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )}{\left (c+d x^3\right ) \left (x^3 \left (3 a d F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 b c F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-4 a c F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )}+\frac{b x^3 \left (\frac{b x^3}{a}+1\right )^{2/3} (3 a d-2 b c) F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c}\right )}{8 d \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}}\, 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{4}{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{\left (a + b x^{3}\right )^{\frac{4}{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{4}{3}}}{d x^{3} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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