Optimal. Leaf size=306 \[ -\frac{2 c^3 d \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{b^{2/3}}-\frac{4 c^3 d \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3}}+\frac{a d^4 \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{3 b^{5/3}}+\frac{2 a d^4 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3}}+\frac{6 c^2 d^2 \sqrt [3]{a+b x^3}}{b}+\frac{c^4 x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac{c d^3 x^4 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac{d^4 x^2 \sqrt [3]{a+b x^3}}{3 b} \]
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Rubi [A] time = 0.267671, antiderivative size = 416, normalized size of antiderivative = 1.36, number of steps used = 22, number of rules used = 14, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.737, Rules used = {1893, 246, 245, 331, 292, 31, 634, 617, 204, 628, 261, 365, 364, 321} \[ -\frac{4 c^3 d \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}+\frac{2 c^3 d \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3}}-\frac{4 c^3 d \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3}}+\frac{2 a d^4 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}-\frac{a d^4 \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{9 b^{5/3}}+\frac{2 a d^4 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3}}+\frac{6 c^2 d^2 \sqrt [3]{a+b x^3}}{b}+\frac{c^4 x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac{c d^3 x^4 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac{d^4 x^2 \sqrt [3]{a+b x^3}}{3 b} \]
Antiderivative was successfully verified.
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Rule 1893
Rule 246
Rule 245
Rule 331
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 261
Rule 365
Rule 364
Rule 321
Rubi steps
\begin{align*} \int \frac{(c+d x)^4}{\left (a+b x^3\right )^{2/3}} \, dx &=\int \left (\frac{c^4}{\left (a+b x^3\right )^{2/3}}+\frac{4 c^3 d x}{\left (a+b x^3\right )^{2/3}}+\frac{6 c^2 d^2 x^2}{\left (a+b x^3\right )^{2/3}}+\frac{4 c d^3 x^3}{\left (a+b x^3\right )^{2/3}}+\frac{d^4 x^4}{\left (a+b x^3\right )^{2/3}}\right ) \, dx\\ &=c^4 \int \frac{1}{\left (a+b x^3\right )^{2/3}} \, dx+\left (4 c^3 d\right ) \int \frac{x}{\left (a+b x^3\right )^{2/3}} \, dx+\left (6 c^2 d^2\right ) \int \frac{x^2}{\left (a+b x^3\right )^{2/3}} \, dx+\left (4 c d^3\right ) \int \frac{x^3}{\left (a+b x^3\right )^{2/3}} \, dx+d^4 \int \frac{x^4}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=\frac{6 c^2 d^2 \sqrt [3]{a+b x^3}}{b}+\frac{d^4 x^2 \sqrt [3]{a+b x^3}}{3 b}+\left (4 c^3 d\right ) \operatorname{Subst}\left (\int \frac{x}{1-b x^3} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )-\frac{\left (2 a d^4\right ) \int \frac{x}{\left (a+b x^3\right )^{2/3}} \, dx}{3 b}+\frac{\left (c^4 \left (1+\frac{b x^3}{a}\right )^{2/3}\right ) \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}+\frac{\left (4 c d^3 \left (1+\frac{b x^3}{a}\right )^{2/3}\right ) \int \frac{x^3}{\left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=\frac{6 c^2 d^2 \sqrt [3]{a+b x^3}}{b}+\frac{d^4 x^2 \sqrt [3]{a+b x^3}}{3 b}+\frac{c^4 x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac{c d^3 x^4 \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac{\left (4 c^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt [3]{b} x} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}-\frac{\left (4 c^3 d\right ) \operatorname{Subst}\left (\int \frac{1-\sqrt [3]{b} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}-\frac{\left (2 a d^4\right ) \operatorname{Subst}\left (\int \frac{x}{1-b x^3} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 b}\\ &=\frac{6 c^2 d^2 \sqrt [3]{a+b x^3}}{b}+\frac{d^4 x^2 \sqrt [3]{a+b x^3}}{3 b}+\frac{c^4 x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac{c d^3 x^4 \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}-\frac{4 c^3 d \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}+\frac{\left (2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{\sqrt [3]{b}+2 b^{2/3} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}-\frac{\left (2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{b}}-\frac{\left (2 a d^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt [3]{b} x} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{4/3}}+\frac{\left (2 a d^4\right ) \operatorname{Subst}\left (\int \frac{1-\sqrt [3]{b} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{4/3}}\\ &=\frac{6 c^2 d^2 \sqrt [3]{a+b x^3}}{b}+\frac{d^4 x^2 \sqrt [3]{a+b x^3}}{3 b}+\frac{c^4 x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac{c d^3 x^4 \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}-\frac{4 c^3 d \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}+\frac{2 a d^4 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}+\frac{2 c^3 d \log \left (1+\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}}\right )}{3 b^{2/3}}+\frac{\left (4 c^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{b^{2/3}}-\frac{\left (a d^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt [3]{b}+2 b^{2/3} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}+\frac{\left (a d^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{4/3}}\\ &=\frac{6 c^2 d^2 \sqrt [3]{a+b x^3}}{b}+\frac{d^4 x^2 \sqrt [3]{a+b x^3}}{3 b}-\frac{4 c^3 d \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3}}+\frac{c^4 x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac{c d^3 x^4 \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}-\frac{4 c^3 d \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}+\frac{2 a d^4 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}+\frac{2 c^3 d \log \left (1+\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}}\right )}{3 b^{2/3}}-\frac{a d^4 \log \left (1+\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}}\right )}{9 b^{5/3}}-\frac{\left (2 a d^4\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{5/3}}\\ &=\frac{6 c^2 d^2 \sqrt [3]{a+b x^3}}{b}+\frac{d^4 x^2 \sqrt [3]{a+b x^3}}{3 b}-\frac{4 c^3 d \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3}}+\frac{2 a d^4 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3}}+\frac{c^4 x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac{c d^3 x^4 \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}-\frac{4 c^3 d \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}+\frac{2 a d^4 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}+\frac{2 c^3 d \log \left (1+\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}}\right )}{3 b^{2/3}}-\frac{a d^4 \log \left (1+\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}}\right )}{9 b^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.144735, size = 166, normalized size = 0.54 \[ \frac{d \left (x^2 \left (6 b c^3-a d^3\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{b x^3}{b x^3+a}\right )+d \left (\left (a+b x^3\right ) \left (18 c^2+d^2 x^2\right )+3 b c d x^4 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )\right )\right )+3 b c^4 x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{3 b \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx+c \right ) ^{4} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{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 (d x + c\right )}^{4}}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{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 [A] time = 4.32191, size = 204, normalized size = 0.67 \begin{align*} 6 c^{2} d^{2} \left (\begin{cases} \frac{x^{3}}{3 a^{\frac{2}{3}}} & \text{for}\: b = 0 \\\frac{\sqrt [3]{a + b x^{3}}}{b} & \text{otherwise} \end{cases}\right ) + \frac{c^{4} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} + \frac{4 c^{3} d x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{5}{3}\right )} + \frac{4 c d^{3} x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{7}{3}\right )} + \frac{d^{4} x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{8}{3}\right )} \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 + c\right )}^{4}}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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