Optimal. Leaf size=240 \[ -\frac{\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{2 \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{x (c+d x)}{9 a \left (a+b x^3\right )^3} \]
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Rubi [A] time = 0.223105, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{2 \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{x (c+d x)}{9 a \left (a+b x^3\right )^3} \]
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 )^4} \, dx &=\frac{x (c+d x)}{9 a \left (a+b x^3\right )^3}-\frac{\int \frac{-8 c-7 d x}{\left (a+b x^3\right )^3} \, dx}{9 a}\\ &=\frac{x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{\int \frac{40 c+28 d x}{\left (a+b x^3\right )^2} \, dx}{54 a^2}\\ &=\frac{x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac{\int \frac{-80 c-28 d x}{a+b x^3} \, dx}{162 a^3}\\ &=\frac{x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac{\int \frac{\sqrt [3]{a} \left (-160 \sqrt [3]{b} c-28 \sqrt [3]{a} d\right )+\sqrt [3]{b} \left (80 \sqrt [3]{b} c-28 \sqrt [3]{a} d\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{486 a^{11/3} \sqrt [3]{b}}+\frac{\left (2 \left (20 c-\frac{7 \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{11/3}}\\ &=\frac{x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{\left (20 \sqrt [3]{b} c-7 \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}{243 a^{11/3} b^{2/3}}+\frac{\left (20 c+\frac{7 \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}{81 a^{10/3}}\\ &=\frac{x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{\left (2 \left (20 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{81 a^{11/3} b^{2/3}}\\ &=\frac{x (c+d x)}{9 a \left (a+b x^3\right )^3}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac{2 \left (20 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.192324, size = 229, normalized size = 0.95 \[ \frac{\frac{2 \left (7 a^{2/3} d-20 \sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{4 \left (20 \sqrt [3]{a} \sqrt [3]{b} c-7 a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac{54 a^3 x (c+d x)}{\left (a+b x^3\right )^3}+\frac{9 a^2 x (8 c+7 d x)}{\left (a+b x^3\right )^2}-\frac{4 \sqrt{3} \sqrt [3]{a} \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{12 a x (10 c+7 d x)}{a+b x^3}}{486 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 306, normalized size = 1.3 \begin{align*}{\frac{cx}{9\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{4\,cx}{27\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{20\,cx}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{40\,c}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,c}{243\,{a}^{3}b}\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{40\,c\sqrt{3}}{243\,{a}^{3}b}\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}}{9\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{7\,d{x}^{2}}{54\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{14\,d{x}^{2}}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{14\,d}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{7\,d}{243\,{a}^{3}b}\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{14\,d\sqrt{3}}{243\,{a}^{3}b}\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 = 7.25727, size = 5960, normalized size = 24.83 \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 = 3.36028, size = 185, normalized size = 0.77 \begin{align*} \operatorname{RootSum}{\left (14348907 t^{3} a^{11} b^{2} + 408240 t a^{4} b c d + 2744 a d^{3} - 64000 b c^{3}, \left ( t \mapsto t \log{\left (x + \frac{413343 t^{2} a^{8} b d + 194400 t a^{4} b c^{2} + 7840 a c d^{2}}{1372 a d^{3} + 32000 b c^{3}} \right )} \right )\right )} + \frac{82 a^{2} c x + 67 a^{2} d x^{2} + 104 a b c x^{4} + 77 a b d x^{5} + 40 b^{2} c x^{7} + 28 b^{2} d x^{8}}{162 a^{6} + 486 a^{5} b x^{3} + 486 a^{4} b^{2} x^{6} + 162 a^{3} b^{3} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10275, size = 312, normalized size = 1.3 \begin{align*} -\frac{2 \,{\left (7 \, d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 20 \, c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{243 \, a^{4}} + \frac{2 \, \sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c - 7 \, \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 )}{243 \, a^{4} b^{2}} + \frac{28 \, b^{2} d x^{8} + 40 \, b^{2} c x^{7} + 77 \, a b d x^{5} + 104 \, a b c x^{4} + 67 \, a^{2} d x^{2} + 82 \, a^{2} c x}{162 \,{\left (b x^{3} + a\right )}^{3} a^{3}} + \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c + 7 \, \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 )}{243 \, a^{5} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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