Optimal. Leaf size=258 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{6 a^{2/3} b^{5/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{2/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} a^{2/3} b^{5/3}}-\frac{(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac{c \log (x)}{a}+\frac{g x}{b}+\frac{h x^2}{2 b} \]
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Rubi [A] time = 0.470611, antiderivative size = 256, normalized size of antiderivative = 0.99, 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} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}-a g+b d\right )}{6 a^{2/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{2/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt{3} a^{2/3} b^{5/3}}-\frac{(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac{c \log (x)}{a}+\frac{g x}{b}+\frac{h x^2}{2 b} \]
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 \left (a+b x^3\right )} \, dx &=\int \left (\frac{g}{b}+\frac{c}{a x}+\frac{h x}{b}+\frac{a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2}{a b \left (a+b x^3\right )}\right ) \, dx\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{c \log (x)}{a}+\frac{\int \frac{a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2}{a+b x^3} \, dx}{a b}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{c \log (x)}{a}+\frac{\int \frac{a (b d-a g)+a (b e-a h) x}{a+b x^3} \, dx}{a b}-\frac{(b c-a f) \int \frac{x^2}{a+b x^3} \, dx}{a}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{c \log (x)}{a}-\frac{(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac{\int \frac{\sqrt [3]{a} \left (2 a \sqrt [3]{b} (b d-a g)+a^{4/3} (b e-a h)\right )+\sqrt [3]{b} \left (-a \sqrt [3]{b} (b d-a g)+a^{4/3} (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 d-a g-\frac{\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} b}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{c \log (x)}{a}+\frac{\left (b d-a g-\frac{\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac{\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{a} b^{4/3}}-\frac{\left (b d-a g-\frac{\sqrt [3]{a} (b e-a h)}{\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}{6 a^{2/3} b^{4/3}}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{c \log (x)}{a}+\frac{\left (b d-a g-\frac{\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{\left (b d-a g-\frac{\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{4/3}}-\frac{(b c-a f) \log \left (a+b x^3\right )}{3 a b}+\frac{\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} 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^{2/3} b^{5/3}}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}-\frac{\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{5/3}}+\frac{c \log (x)}{a}+\frac{\left (b d-a g-\frac{\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{\left (b d-a g-\frac{\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{4/3}}-\frac{(b c-a f) \log \left (a+b x^3\right )}{3 a b}\\ \end{align*}
Mathematica [A] time = 0.212392, size = 258, normalized size = 1. \[ \frac{-\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )+2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} h-\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )+2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h-\sqrt [3]{a} b e+a \sqrt [3]{b} g-b^{4/3} d\right )-2 b^{2/3} (b c-a f) \log \left (a+b x^3\right )+6 a b^{2/3} g x+3 a b^{2/3} h x^2+6 b^{5/3} c \log (x)}{6 a b^{5/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 426, normalized size = 1.7 \begin{align*} \text{result too large to display} \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 = 1.06272, size = 409, normalized size = 1.59 \begin{align*} \frac{c \log \left ({\left | x \right |}\right )}{a} - \frac{{\left (b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a b} + \frac{b h x^{2} + 2 \, b g x}{2 \, b^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h - \left (-a b^{2}\right )^{\frac{2}{3}} b 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 )}{3 \, a b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h + \left (-a b^{2}\right )^{\frac{2}{3}} 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 )}{6 \, a b^{3}} + \frac{{\left (a^{3} b^{2} h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - a^{2} b^{3} d + a^{3} b^{2} g\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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