Optimal. Leaf size=351 \[ \frac{b^{5/3} (3 b c-4 a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 d^3}-\frac{2 b^{5/3} (3 b c-4 a d) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} d^3}+\frac{(b c-a d)^{5/3} (a d+3 b c) \log \left (c+d x^3\right )}{9 c^{5/3} d^3}-\frac{(b c-a d)^{5/3} (a d+3 b c) \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{3 c^{5/3} d^3}+\frac{2 (b c-a d)^{5/3} (a d+3 b c) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} c^{5/3} d^3}+\frac{b x \left (a+b x^3\right )^{2/3} (2 b c-a d)}{3 c d^2}-\frac{x \left (a+b x^3\right )^{5/3} (b c-a d)}{3 c d \left (c+d x^3\right )} \]
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Rubi [C] time = 0.0286094, antiderivative size = 62, normalized size of antiderivative = 0.18, 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} \[ \frac{a^2 x \left (a+b x^3\right )^{2/3} F_1\left (\frac{1}{3};-\frac{8}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c^2 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^2} \, dx &=\frac{\left (a^2 \left (a+b x^3\right )^{2/3}\right ) \int \frac{\left (1+\frac{b x^3}{a}\right )^{8/3}}{\left (c+d x^3\right )^2} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=\frac{a^2 x \left (a+b x^3\right )^{2/3} F_1\left (\frac{1}{3};-\frac{8}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c^2 \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.890721, size = 698, normalized size = 1.99 \[ \frac{1}{18} \left (\frac{2 a^2 b \left (\log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )-2 \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )}{c^{2/3} d \sqrt [3]{b c-a d}}+\frac{2 a^3 \left (\log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )-2 \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )}{c^{5/3} \sqrt [3]{b c-a d}}-\frac{9 b^3 x^4 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{d^2 \sqrt [3]{a+b x^3}}+\frac{12 a b^2 x^4 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c d \sqrt [3]{a+b x^3}}-\frac{2 a b^2 \sqrt [3]{c} \left (\log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )-2 \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )}{d^2 \sqrt [3]{b c-a d}}+\frac{6 x \left (a+b x^3\right )^{2/3} \left (\frac{(b c-a d)^2}{c \left (c+d x^3\right )}+b^2\right )}{d^2}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.248, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{8}{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{8}{3}}}{{\left (d x^{3} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 40.0811, size = 1810, normalized size = 5.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{8}{3}}}{{\left (d x^{3} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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