Optimal. Leaf size=215 \[ -\frac{\left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}}+\frac{\left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{5/3}}-\frac{\left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{5/3}}-\frac{c+d x+e x^2}{6 b \left (a+b x^3\right )^2}+\frac{x (d+2 e x)}{18 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.196885, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{\left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}}+\frac{\left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{5/3}}-\frac{\left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{5/3}}-\frac{c+d x+e x^2}{6 b \left (a+b x^3\right )^2}+\frac{x (d+2 e x)}{18 a b \left (a+b x^3\right )} \]
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 )^3} \, dx &=-\frac{c+d x+e x^2}{6 b \left (a+b x^3\right )^2}+\frac{\int \frac{d+2 e x}{\left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac{c+d x+e x^2}{6 b \left (a+b x^3\right )^2}+\frac{x (d+2 e x)}{18 a b \left (a+b x^3\right )}-\frac{\int \frac{-2 d-2 e x}{a+b x^3} \, dx}{18 a b}\\ &=-\frac{c+d x+e x^2}{6 b \left (a+b x^3\right )^2}+\frac{x (d+2 e x)}{18 a b \left (a+b x^3\right )}-\frac{\int \frac{\sqrt [3]{a} \left (-4 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right )+\sqrt [3]{b} \left (2 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{5/3} b^{4/3}}+\frac{\left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{5/3} b}\\ &=-\frac{c+d x+e x^2}{6 b \left (a+b x^3\right )^2}+\frac{x (d+2 e x)}{18 a b \left (a+b x^3\right )}+\frac{\left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3}}+\frac{\left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{4/3} b^{4/3}}-\frac{\left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\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^{5/3} b^{4/3}}\\ &=-\frac{c+d x+e x^2}{6 b \left (a+b x^3\right )^2}+\frac{x (d+2 e x)}{18 a b \left (a+b x^3\right )}+\frac{\left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3}}-\frac{\left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}}+\frac{\left (\sqrt [3]{b} d+\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 )}{9 a^{5/3} b^{5/3}}\\ &=-\frac{c+d x+e x^2}{6 b \left (a+b x^3\right )^2}+\frac{x (d+2 e x)}{18 a b \left (a+b x^3\right )}-\frac{\left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{5/3}}+\frac{\left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3}}-\frac{\left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.179103, size = 198, normalized size = 0.92 \[ \frac{\frac{\left (\sqrt [3]{a} e-\sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac{2 \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac{2 \sqrt{3} \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{5/3}}-\frac{9 b^{2/3} (c+x (d+e x))}{\left (a+b x^3\right )^2}+\frac{3 b^{2/3} x (d+2 e x)}{a \left (a+b x^3\right )}}{54 b^{5/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 255, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{e{x}^{5}}{9\,a}}+{\frac{d{x}^{4}}{18\,a}}-{\frac{e{x}^{2}}{18\,b}}-{\frac{dx}{9\,b}}-{\frac{c}{6\,b}} \right ) }+{\frac{d}{27\,{b}^{2}a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{d}{54\,{b}^{2}a}\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{d\sqrt{3}}{27\,{b}^{2}a}\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{e}{27\,{b}^{2}a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{e}{54\,{b}^{2}a}\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{\sqrt{3}e}{27\,{b}^{2}a}\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 = 9.74419, size = 5122, normalized size = 23.82 \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 = 5.12363, size = 148, normalized size = 0.69 \begin{align*} \operatorname{RootSum}{\left (19683 t^{3} a^{5} b^{5} + 81 t a^{2} b^{2} d e + a e^{3} - b d^{3}, \left ( t \mapsto t \log{\left (x + \frac{729 t^{2} a^{4} b^{3} e + 27 t a^{2} b^{2} d^{2} + 2 a d e^{2}}{a e^{3} + b d^{3}} \right )} \right )\right )} + \frac{- 3 a c - 2 a d x - a e x^{2} + b d x^{4} + 2 b e x^{5}}{18 a^{3} b + 36 a^{2} b^{2} x^{3} + 18 a b^{3} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11349, size = 288, normalized size = 1.34 \begin{align*} -\frac{{\left (\left (-\frac{a}{b}\right )^{\frac{1}{3}} e + d\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^{2} b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b d - \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 )}{27 \, a^{2} b^{3}} + \frac{2 \, b x^{5} e + b d x^{4} - a x^{2} e - 2 \, a d x - 3 \, a c}{18 \,{\left (b x^{3} + a\right )}^{2} a b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + \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 )}{54 \, a^{3} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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