Optimal. Leaf size=304 \[ -\frac{d x^2 \sqrt{\frac{b x^3}{a}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{2 \sqrt{a+b x^3} \left (a c^2-d^2\right )}+\frac{\log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}-\frac{\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}} \]
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Rubi [A] time = 0.299333, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2156, 292, 31, 634, 617, 204, 628, 511, 510} \[ -\frac{d x^2 \sqrt{\frac{b x^3}{a}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{2 \sqrt{a+b x^3} \left (a c^2-d^2\right )}+\frac{\log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}-\frac{\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}} \]
Antiderivative was successfully verified.
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Rule 2156
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx &=(a c) \int \frac{x}{a^2 c^2-a d^2+a b c^2 x^3} \, dx-(a d) \int \frac{x}{\sqrt{a+b x^3} \left (a^2 c^2-a d^2+a b c^2 x^3\right )} \, dx\\ &=-\frac{\left (\sqrt [3]{a} \sqrt [3]{c}\right ) \int \frac{1}{\sqrt [3]{a} \sqrt [3]{a c^2-d^2}+\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x} \, dx}{3 \sqrt [3]{b} \sqrt [3]{a c^2-d^2}}+\frac{\left (\sqrt [3]{a} \sqrt [3]{c}\right ) \int \frac{\sqrt [3]{a} \sqrt [3]{a c^2-d^2}+\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{3 \sqrt [3]{b} \sqrt [3]{a c^2-d^2}}-\frac{\left (a d \sqrt{1+\frac{b x^3}{a}}\right ) \int \frac{x}{\sqrt{1+\frac{b x^3}{a}} \left (a^2 c^2-a d^2+a b c^2 x^3\right )} \, dx}{\sqrt{a+b x^3}}\\ &=-\frac{d x^2 \sqrt{1+\frac{b x^3}{a}} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{2 \left (a c^2-d^2\right ) \sqrt{a+b x^3}}-\frac{\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}+\frac{\left (a^{2/3} \sqrt [3]{c}\right ) \int \frac{1}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{2 \sqrt [3]{b}}+\frac{\int \frac{-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}+2 a^{2/3} b^{2/3} c^{4/3} x}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}\\ &=-\frac{d x^2 \sqrt{1+\frac{b x^3}{a}} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{2 \left (a c^2-d^2\right ) \sqrt{a+b x^3}}-\frac{\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}+\frac{\log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )}{b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}\\ &=-\frac{d x^2 \sqrt{1+\frac{b x^3}{a}} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{2 \left (a c^2-d^2\right ) \sqrt{a+b x^3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}-\frac{\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}+\frac{\log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}\\ \end{align*}
Mathematica [F] time = 0.165597, size = 0, normalized size = 0. \[ \int \frac{x}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 0.046, size = 619, normalized size = 2. \begin{align*} \text{result too large to display} \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{x}{b c x^{3} + a c + \sqrt{b x^{3} + a} d}\,{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{x}{a c + b c x^{3} + d \sqrt{a + b 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{x}{b c x^{3} + a c + \sqrt{b x^{3} + a} d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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