Optimal. Leaf size=189 \[ -\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{2/3}}+\frac{x (c+d x)}{3 a \left (a+b x^3\right )} \]
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Rubi [A] time = 0.139398, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{2/3}}+\frac{x (c+d x)}{3 a \left (a+b x^3\right )} \]
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 )^2} \, dx &=\frac{x (c+d x)}{3 a \left (a+b x^3\right )}-\frac{\int \frac{-2 c-d x}{a+b x^3} \, dx}{3 a}\\ &=\frac{x (c+d x)}{3 a \left (a+b x^3\right )}-\frac{\int \frac{\sqrt [3]{a} \left (-4 \sqrt [3]{b} c-\sqrt [3]{a} d\right )+\sqrt [3]{b} \left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{5/3} \sqrt [3]{b}}+\frac{\left (2 c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3}}\\ &=\frac{x (c+d x)}{3 a \left (a+b x^3\right )}+\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (2 \sqrt [3]{b} c-\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}{18 a^{5/3} b^{2/3}}+\frac{\left (2 c+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{4/3}}\\ &=\frac{x (c+d x)}{3 a \left (a+b x^3\right )}+\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} c+\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 )}{3 a^{5/3} b^{2/3}}\\ &=\frac{x (c+d x)}{3 a \left (a+b x^3\right )}-\frac{\left (2 \sqrt [3]{b} c+\sqrt [3]{a} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.150732, size = 180, normalized size = 0.95 \[ \frac{\frac{\left (a^{2/3} d-2 \sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{2 \left (2 \sqrt [3]{a} \sqrt [3]{b} c-a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{6 a x (c+d x)}{a+b x^3}}{18 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 238, normalized size = 1.3 \begin{align*}{\frac{cx}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{2\,c}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{9\,ab}\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{2\,c\sqrt{3}}{9\,ab}\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}}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{d}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d}{18\,ab}\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{d\sqrt{3}}{9\,ab}\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 = 7.97279, size = 4961, normalized size = 26.25 \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 = 1.10274, size = 105, normalized size = 0.56 \begin{align*} \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{2} + 54 t a^{2} b c d + a d^{3} - 8 b c^{3}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{2} a^{4} b d + 36 t a^{2} b c^{2} + 4 a c d^{2}}{a d^{3} + 8 b c^{3}} \right )} \right )\right )} + \frac{c x + d x^{2}}{3 a^{2} + 3 a b x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10932, size = 252, normalized size = 1.33 \begin{align*} -\frac{{\left (d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2}} + \frac{d x^{2} + c x}{3 \,{\left (b x^{3} + a\right )} a} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c - \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 )}{9 \, a^{2} b^{2}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c + \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 )}{18 \, a^{3} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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