Optimal. Leaf size=215 \[ -\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}+\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac{\left (2 \sqrt [3]{a} d+5 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} b^{2/3}}+\frac{x (5 c+4 d x)}{18 a^2 \left (a+b x^3\right )}+\frac{x (c+d x)}{6 a \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.187046, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {1855, 1860, 31, 634, 617, 204, 628} \[ -\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}+\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac{\left (2 \sqrt [3]{a} d+5 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} b^{2/3}}+\frac{x (5 c+4 d x)}{18 a^2 \left (a+b x^3\right )}+\frac{x (c+d x)}{6 a \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 1855
Rule 1860
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{c+d x}{\left (a+b x^3\right )^3} \, dx &=\frac{x (c+d x)}{6 a \left (a+b x^3\right )^2}-\frac{\int \frac{-5 c-4 d x}{\left (a+b x^3\right )^2} \, dx}{6 a}\\ &=\frac{x (c+d x)}{6 a \left (a+b x^3\right )^2}+\frac{x (5 c+4 d x)}{18 a^2 \left (a+b x^3\right )}+\frac{\int \frac{10 c+4 d x}{a+b x^3} \, dx}{18 a^2}\\ &=\frac{x (c+d x)}{6 a \left (a+b x^3\right )^2}+\frac{x (5 c+4 d x)}{18 a^2 \left (a+b x^3\right )}+\frac{\int \frac{\sqrt [3]{a} \left (20 \sqrt [3]{b} c+4 \sqrt [3]{a} d\right )+\sqrt [3]{b} \left (-10 \sqrt [3]{b} c+4 \sqrt [3]{a} d\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{8/3} \sqrt [3]{b}}+\frac{\left (5 c-\frac{2 \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{8/3}}\\ &=\frac{x (c+d x)}{6 a \left (a+b x^3\right )^2}+\frac{x (5 c+4 d x)}{18 a^2 \left (a+b x^3\right )}+\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{8/3} b^{2/3}}+\frac{\left (5 \sqrt [3]{b} c+2 \sqrt [3]{a} d\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{7/3} \sqrt [3]{b}}\\ &=\frac{x (c+d x)}{6 a \left (a+b x^3\right )^2}+\frac{x (5 c+4 d x)}{18 a^2 \left (a+b x^3\right )}+\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}+\frac{\left (5 \sqrt [3]{b} c+2 \sqrt [3]{a} d\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{8/3} b^{2/3}}\\ &=\frac{x (c+d x)}{6 a \left (a+b x^3\right )^2}+\frac{x (5 c+4 d x)}{18 a^2 \left (a+b x^3\right )}-\frac{\left (5 \sqrt [3]{b} c+2 \sqrt [3]{a} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} b^{2/3}}+\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.153778, size = 205, normalized size = 0.95 \[ \frac{\frac{\left (2 a^{2/3} d-5 \sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{2 \left (5 \sqrt [3]{a} \sqrt [3]{b} c-2 a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac{9 a^2 x (c+d x)}{\left (a+b x^3\right )^2}-\frac{2 \sqrt{3} \sqrt [3]{a} \left (2 \sqrt [3]{a} d+5 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{3 a x (5 c+4 d x)}{a+b x^3}}{54 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 272, normalized size = 1.3 \begin{align*}{\frac{cx}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,cx}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{5\,c}{27\,b{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,c}{54\,b{a}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,c\sqrt{3}}{27\,b{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d{x}^{2}}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{2\,d{x}^{2}}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,d}{27\,b{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d}{27\,b{a}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,d\sqrt{3}}{27\,b{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \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 [C] time = 8.43691, size = 5554, normalized size = 25.83 \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 [A] time = 1.90385, size = 146, normalized size = 0.68 \begin{align*} \operatorname{RootSum}{\left (19683 t^{3} a^{8} b^{2} + 810 t a^{3} b c d + 8 a d^{3} - 125 b c^{3}, \left ( t \mapsto t \log{\left (x + \frac{1458 t^{2} a^{6} b d + 675 t a^{3} b c^{2} + 40 a c d^{2}}{8 a d^{3} + 125 b c^{3}} \right )} \right )\right )} + \frac{8 a c x + 7 a d x^{2} + 5 b c x^{4} + 4 b d x^{5}}{18 a^{4} + 36 a^{3} b x^{3} + 18 a^{2} b^{2} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12425, size = 279, normalized size = 1.3 \begin{align*} -\frac{{\left (2 \, d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{2}} + \frac{4 \, b d x^{5} + 5 \, b c x^{4} + 7 \, a d x^{2} + 8 \, a c x}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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