Optimal. Leaf size=253 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 a^{4/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 a^{4/3} b^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} a^{4/3} b^{4/3}}-\frac{(b d-a g) \log \left (a+b x^3\right )}{3 a b}-\frac{c}{a x}+\frac{d \log (x)}{a}+\frac{h x}{b} \]
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Rubi [A] time = 0.453954, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237, Rules used = {1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 a^{4/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 a^{4/3} b^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} a^{4/3} b^{4/3}}-\frac{(b d-a g) \log \left (a+b x^3\right )}{3 a b}-\frac{c}{a x}+\frac{d \log (x)}{a}+\frac{h x}{b} \]
Antiderivative was successfully verified.
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Rule 1834
Rule 1871
Rule 1860
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 260
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^2 \left (a+b x^3\right )} \, dx &=\int \left (\frac{h}{b}+\frac{c}{a x^2}+\frac{d}{a x}+\frac{a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2}{a b \left (a+b x^3\right )}\right ) \, dx\\ &=-\frac{c}{a x}+\frac{h x}{b}+\frac{d \log (x)}{a}+\frac{\int \frac{a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2}{a+b x^3} \, dx}{a b}\\ &=-\frac{c}{a x}+\frac{h x}{b}+\frac{d \log (x)}{a}+\frac{\int \frac{a (b e-a h)-b (b c-a f) x}{a+b x^3} \, dx}{a b}-\frac{(b d-a g) \int \frac{x^2}{a+b x^3} \, dx}{a}\\ &=-\frac{c}{a x}+\frac{h x}{b}+\frac{d \log (x)}{a}-\frac{(b d-a g) \log \left (a+b x^3\right )}{3 a b}+\frac{\int \frac{\sqrt [3]{a} \left (-\sqrt [3]{a} b (b c-a f)+2 a \sqrt [3]{b} (b e-a h)\right )+\sqrt [3]{b} \left (-\sqrt [3]{a} b (b c-a f)-a \sqrt [3]{b} (b e-a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{5/3} b^{4/3}}+\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{4/3} b}\\ &=-\frac{c}{a x}+\frac{h x}{b}+\frac{d \log (x)}{a}+\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} b^{4/3}}-\frac{(b d-a g) \log \left (a+b x^3\right )}{3 a b}-\frac{\left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a b}-\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\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}{6 a^{4/3} b^{4/3}}\\ &=-\frac{c}{a x}+\frac{h x}{b}+\frac{d \log (x)}{a}+\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} b^{4/3}}-\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} b^{4/3}}-\frac{(b d-a g) \log \left (a+b x^3\right )}{3 a b}-\frac{\left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{4/3} b^{4/3}}\\ &=-\frac{c}{a x}+\frac{h x}{b}+\frac{\left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} b^{4/3}}+\frac{d \log (x)}{a}+\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} b^{4/3}}-\frac{\left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} b^{4/3}}-\frac{(b d-a g) \log \left (a+b x^3\right )}{3 a b}\\ \end{align*}
Mathematica [A] time = 0.280359, size = 257, normalized size = 1.02 \[ \frac{1}{6} \left (\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^{2/3} b e+a^{5/3} h+a b^{2/3} f-b^{5/3} c\right )}{a^{4/3} b^{4/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+b^{5/3} c\right )}{a^{4/3} b^{4/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{a^{4/3} b^{4/3}}+\frac{2 (a g-b d) \log \left (a+b x^3\right )}{a b}-\frac{6 c}{a x}+\frac{6 d \log (x)}{a}+\frac{6 h x}{b}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 423, normalized size = 1.7 \begin{align*}{\frac{hx}{b}}-{\frac{ah}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{e}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ah}{6\,{b}^{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{e}{6\,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{a\sqrt{3}h}{3\,{b}^{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{\sqrt{3}e}{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{f}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{f}{6\,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{c}{6\,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}f}{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}}}}}}-{\frac{c\sqrt{3}}{3\,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}}}}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) g}{3\,b}}-{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,a}}+{\frac{d\ln \left ( x \right ) }{a}}-{\frac{c}{ax}} \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09191, size = 404, normalized size = 1.6 \begin{align*} \frac{h x}{b} + \frac{d \log \left ({\left | x \right |}\right )}{a} - \frac{{\left (b d - a g\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a b} - \frac{c}{a x} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e - \left (-a b^{2}\right )^{\frac{2}{3}} b c + \left (-a b^{2}\right )^{\frac{2}{3}} a f\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 )}{3 \, a^{2} b^{2}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e + \left (-a b^{2}\right )^{\frac{2}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} a f\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b^{2}} + \frac{{\left (a b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{3} b^{2} h - a^{2} b^{3} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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