Optimal. Leaf size=391 \[ -\frac{(b c-a d)^{2/3} \left (5 a^2 d^2+6 a b c d+9 b^2 c^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^3}+\frac{(b c-a d)^{2/3} \left (5 a^2 d^2+6 a b c d+9 b^2 c^2\right ) \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^3}-\frac{(b c-a d)^{2/3} \left (5 a^2 d^2+6 a b c d+9 b^2 c^2\right ) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} c^{8/3} d^3}-\frac{b^{8/3} \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 d^3}+\frac{b^{8/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d^3}-\frac{x \left (a+b x^3\right )^{2/3} (b c-a d) (5 a d+6 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac{x \left (a+b x^3\right )^{5/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]
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Rubi [C] time = 0.0275823, antiderivative size = 62, normalized size of antiderivative = 0.16, 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} \[ \frac{a^2 x \left (a+b x^3\right )^{2/3} F_1\left (\frac{1}{3};-\frac{8}{3},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c^3 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]
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 )^3} \, 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 )^3} \, 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},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c^3 \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.902278, size = 651, normalized size = 1.66 \[ \frac{\frac{10 a^3 \left (\log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )-2 \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )}{\sqrt [3]{b c-a d}}+\frac{2 a^2 b c \left (\log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )-2 \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )}{d \sqrt [3]{b c-a d}}+\frac{27 b^3 c^{5/3} x^4 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{d^2 \sqrt [3]{a+b x^3}}+\frac{6 a b^2 c^2 \left (\log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )-2 \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )}{d^2 \sqrt [3]{b c-a d}}+\frac{6 c^{2/3} x \left (a+b x^3\right )^{2/3} (a d-b c) \left (a d \left (8 c+5 d x^3\right )+3 b c \left (2 c+3 d x^3\right )\right )}{d^2 \left (c+d x^3\right )^2}}{108 c^{8/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.423, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{3}} \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 16.0328, size = 2080, normalized size = 5.32 \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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