Optimal. Leaf size=159 \[ -\frac{d x \left (a+b x^3\right )^{2/3} (3 b c-4 a d)}{3 a b^2}-\frac{d (3 b c-2 a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 b^{7/3}}+\frac{2 d (3 b c-2 a d) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{7/3}}+\frac{x \left (c+d x^3\right ) (b c-a d)}{a b \sqrt [3]{a+b x^3}} \]
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Rubi [A] time = 0.102334, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {413, 388, 239} \[ -\frac{d x \left (a+b x^3\right )^{2/3} (3 b c-4 a d)}{3 a b^2}-\frac{d (3 b c-2 a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 b^{7/3}}+\frac{2 d (3 b c-2 a d) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{7/3}}+\frac{x \left (c+d x^3\right ) (b c-a d)}{a b \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 413
Rule 388
Rule 239
Rubi steps
\begin{align*} \int \frac{\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx &=\frac{(b c-a d) x \left (c+d x^3\right )}{a b \sqrt [3]{a+b x^3}}+\frac{\int \frac{a c d-d (3 b c-4 a d) x^3}{\sqrt [3]{a+b x^3}} \, dx}{a b}\\ &=-\frac{d (3 b c-4 a d) x \left (a+b x^3\right )^{2/3}}{3 a b^2}+\frac{(b c-a d) x \left (c+d x^3\right )}{a b \sqrt [3]{a+b x^3}}+\frac{(2 d (3 b c-2 a d)) \int \frac{1}{\sqrt [3]{a+b x^3}} \, dx}{3 b^2}\\ &=-\frac{d (3 b c-4 a d) x \left (a+b x^3\right )^{2/3}}{3 a b^2}+\frac{(b c-a d) x \left (c+d x^3\right )}{a b \sqrt [3]{a+b x^3}}+\frac{2 d (3 b c-2 a d) \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3} b^{7/3}}-\frac{d (3 b c-2 a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 b^{7/3}}\\ \end{align*}
Mathematica [A] time = 5.14927, size = 168, normalized size = 1.06 \[ \frac{x \left (a+b x^3\right )^{2/3} \left (\frac{3 (b c-a d)^2}{a \left (a+b x^3\right )}+d^2\right )}{3 b^2}+\frac{d (3 b c-2 a d) \left (\log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )\right )}{9 b^{7/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.416, size = 0, normalized size = 0. \begin{align*} \int{ \left ( d{x}^{3}+c \right ) ^{2} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73604, size = 1538, normalized size = 9.67 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac{4}{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 (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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