Optimal. Leaf size=248 \[ -\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.242314, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {1823, 1855, 1860, 31, 634, 617, 204, 628} \[ -\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 1823
Rule 1855
Rule 1860
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^4} \, dx &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{\int \frac{d+2 e x}{\left (a+b x^3\right )^3} \, dx}{9 b}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}-\frac{\int \frac{-5 d-8 e x}{\left (a+b x^3\right )^2} \, dx}{54 a b}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}+\frac{\int \frac{10 d+8 e x}{a+b x^3} \, dx}{162 a^2 b}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}+\frac{\int \frac{\sqrt [3]{a} \left (20 \sqrt [3]{b} d+8 \sqrt [3]{a} e\right )+\sqrt [3]{b} \left (-10 \sqrt [3]{b} d+8 \sqrt [3]{a} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{486 a^{8/3} b^{4/3}}+\frac{\left (5 d-\frac{4 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{8/3} b}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\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}{486 a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d+4 \sqrt [3]{a} e\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{162 a^{7/3} b^{4/3}}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d+4 \sqrt [3]{a} e\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^{8/3} b^{5/3}}\\ &=-\frac{c+d x+e x^2}{9 b \left (a+b x^3\right )^3}+\frac{x (d+2 e x)}{54 a b \left (a+b x^3\right )^2}+\frac{x (5 d+8 e x)}{162 a^2 b \left (a+b x^3\right )}-\frac{\left (5 \sqrt [3]{b} d+4 \sqrt [3]{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{8/3} b^{5/3}}+\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{8/3} b^{5/3}}-\frac{\left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{486 a^{8/3} b^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.232971, size = 230, normalized size = 0.93 \[ \frac{\frac{3 b^{2/3} x (5 d+8 e x)}{a^2 \left (a+b x^3\right )}+\frac{\left (4 \sqrt [3]{a} e-5 \sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{8/3}}+\frac{2 \left (5 \sqrt [3]{b} d-4 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{8/3}}-\frac{2 \sqrt{3} \left (4 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{8/3}}-\frac{54 b^{2/3} (c+x (d+e x))}{\left (a+b x^3\right )^3}+\frac{9 b^{2/3} x (d+2 e x)}{a \left (a+b x^3\right )^2}}{486 b^{5/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 275, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{3}} \left ({\frac{4\,be{x}^{8}}{81\,{a}^{2}}}+{\frac{5\,bd{x}^{7}}{162\,{a}^{2}}}+{\frac{11\,e{x}^{5}}{81\,a}}+{\frac{13\,d{x}^{4}}{162\,a}}-{\frac{2\,e{x}^{2}}{81\,b}}-{\frac{5\,dx}{81\,b}}-{\frac{c}{9\,b}} \right ) }+{\frac{5\,d}{243\,{b}^{2}{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,d}{486\,{b}^{2}{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\,d\sqrt{3}}{243\,{b}^{2}{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{4\,e}{243\,{b}^{2}{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,e}{243\,{b}^{2}{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{4\,e\sqrt{3}}{243\,{b}^{2}{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 = 8.78528, size = 5952, normalized size = 24. \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 = 13.5946, size = 201, normalized size = 0.81 \begin{align*} \operatorname{RootSum}{\left (14348907 t^{3} a^{8} b^{5} + 14580 t a^{3} b^{2} d e + 64 a e^{3} - 125 b d^{3}, \left ( t \mapsto t \log{\left (x + \frac{236196 t^{2} a^{6} b^{3} e + 6075 t a^{3} b^{2} d^{2} + 160 a d e^{2}}{64 a e^{3} + 125 b d^{3}} \right )} \right )\right )} + \frac{- 18 a^{2} c - 10 a^{2} d x - 4 a^{2} e x^{2} + 13 a b d x^{4} + 22 a b e x^{5} + 5 b^{2} d x^{7} + 8 b^{2} e x^{8}}{162 a^{5} b + 486 a^{4} b^{2} x^{3} + 486 a^{3} b^{3} x^{6} + 162 a^{2} b^{4} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07474, size = 333, normalized size = 1.34 \begin{align*} -\frac{{\left (4 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 5 \, d\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^{3} b} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} e\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^{3} b^{3}} + \frac{8 \, b^{2} x^{8} e + 5 \, b^{2} d x^{7} + 22 \, a b x^{5} e + 13 \, a b d x^{4} - 4 \, a^{2} x^{2} e - 10 \, a^{2} d x - 18 \, a^{2} c}{162 \,{\left (b x^{3} + a\right )}^{3} a^{2} b} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b e\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{486 \, a^{4} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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