Optimal. Leaf size=275 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 \sqrt [3]{a} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 \sqrt [3]{a} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac{x (b e-a h)}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b} \]
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Rubi [A] time = 0.921478, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {1836, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 \sqrt [3]{a} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 \sqrt [3]{a} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\frac{x (b e-a h)}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 1836
Rule 1887
Rule 1871
Rule 1860
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 260
Rubi steps
\begin{align*} \int \frac{x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx &=\frac{h x^4}{4 b}+\frac{\int \frac{x \left (4 b c+4 b d x+4 (b e-a h) x^2+4 b f x^3+4 b g x^4\right )}{a+b x^3} \, dx}{4 b}\\ &=\frac{g x^3}{3 b}+\frac{h x^4}{4 b}+\frac{\int \frac{x \left (12 b^2 c+12 b (b d-a g) x+12 b (b e-a h) x^2+12 b^2 f x^3\right )}{a+b x^3} \, dx}{12 b^2}\\ &=\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}+\frac{\int \frac{x \left (24 b^2 (b c-a f)+24 b^2 (b d-a g) x+24 b^2 (b e-a h) x^2\right )}{a+b x^3} \, dx}{24 b^3}\\ &=\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}+\frac{\int \left (24 b (b e-a h)-\frac{24 \left (a b (b e-a h)-b^2 (b c-a f) x-b^2 (b d-a g) x^2\right )}{a+b x^3}\right ) \, dx}{24 b^3}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}-\frac{\int \frac{a b (b e-a h)-b^2 (b c-a f) x-b^2 (b d-a g) x^2}{a+b x^3} \, dx}{b^3}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}-\frac{\int \frac{a b (b e-a h)-b^2 (b c-a f) x}{a+b x^3} \, dx}{b^3}+\frac{(b d-a g) \int \frac{x^2}{a+b x^3} \, dx}{b}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}-\frac{\int \frac{\sqrt [3]{a} \left (-\sqrt [3]{a} b^2 (b c-a f)+2 a b^{4/3} (b e-a h)\right )+\sqrt [3]{b} \left (-\sqrt [3]{a} b^2 (b c-a f)-a b^{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^{2/3} b^{10/3}}-\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 \sqrt [3]{a} b^2}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}-\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 \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\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 b^2}+\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 \sqrt [3]{a} b^{7/3}}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}-\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 \sqrt [3]{a} b^{7/3}}+\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 \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}+\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 )}{\sqrt [3]{a} b^{7/3}}\\ &=\frac{(b e-a h) x}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 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} \sqrt [3]{a} b^{7/3}}-\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 \sqrt [3]{a} b^{7/3}}+\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 \sqrt [3]{a} b^{7/3}}+\frac{(b d-a g) \log \left (a+b x^3\right )}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.372748, size = 272, normalized size = 0.99 \[ \frac{\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+b^{5/3} c\right )}{\sqrt [3]{a}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^{2/3} b e+a^{5/3} h+a b^{2/3} f-b^{5/3} c\right )}{\sqrt [3]{a}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt [3]{a}}+4 \sqrt [3]{b} (b d-a g) \log \left (a+b x^3\right )+12 \sqrt [3]{b} x (b e-a h)+6 b^{4/3} f x^2+4 b^{4/3} g x^3+3 b^{4/3} h x^4}{12 b^{7/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 455, 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 [B] time = 33.0156, size = 811, normalized size = 2.95 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a b^{7} + t^{2} \left (27 a^{2} b^{5} g - 27 a b^{6} d\right ) + t \left (- 9 a^{3} b^{3} f h + 9 a^{3} b^{3} g^{2} + 9 a^{2} b^{4} c h - 18 a^{2} b^{4} d g + 9 a^{2} b^{4} e f - 9 a b^{5} c e + 9 a b^{5} d^{2}\right ) - a^{5} h^{3} + 3 a^{4} b e h^{2} - 3 a^{4} b f g h + a^{4} b g^{3} + 3 a^{3} b^{2} c g h + 3 a^{3} b^{2} d f h - 3 a^{3} b^{2} d g^{2} - 3 a^{3} b^{2} e^{2} h + 3 a^{3} b^{2} e f g - a^{3} b^{2} f^{3} - 3 a^{2} b^{3} c d h - 3 a^{2} b^{3} c e g + 3 a^{2} b^{3} c f^{2} + 3 a^{2} b^{3} d^{2} g - 3 a^{2} b^{3} d e f + a^{2} b^{3} e^{3} - 3 a b^{4} c^{2} f + 3 a b^{4} c d e - a b^{4} d^{3} + b^{5} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 9 t^{2} a^{2} b^{5} f + 9 t^{2} a b^{6} c + 3 t a^{4} b^{2} h^{2} - 6 t a^{3} b^{3} e h - 6 t a^{3} b^{3} f g + 6 t a^{2} b^{4} c g + 6 t a^{2} b^{4} d f + 3 t a^{2} b^{4} e^{2} - 6 t a b^{5} c d + a^{5} g h^{2} - a^{4} b d h^{2} - 2 a^{4} b e g h + 2 a^{4} b f^{2} h - a^{4} b f g^{2} - 4 a^{3} b^{2} c f h + a^{3} b^{2} c g^{2} + 2 a^{3} b^{2} d e h + 2 a^{3} b^{2} d f g + a^{3} b^{2} e^{2} g - 2 a^{3} b^{2} e f^{2} + 2 a^{2} b^{3} c^{2} h - 2 a^{2} b^{3} c d g + 4 a^{2} b^{3} c e f - a^{2} b^{3} d^{2} f - a^{2} b^{3} d e^{2} - 2 a b^{4} c^{2} e + a b^{4} c d^{2}}{a^{5} h^{3} - 3 a^{4} b e h^{2} + 3 a^{3} b^{2} e^{2} h - a^{3} b^{2} f^{3} + 3 a^{2} b^{3} c f^{2} - a^{2} b^{3} e^{3} - 3 a b^{4} c^{2} f + b^{5} c^{3}} \right )} \right )\right )} + \frac{f x^{2}}{2 b} + \frac{g x^{3}}{3 b} + \frac{h x^{4}}{4 b} - \frac{x \left (a h - b e\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08277, size = 428, normalized size = 1.56 \begin{align*} \frac{{\left (b d - a g\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \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 b^{3}} + \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 b^{3}} + \frac{3 \, b^{3} h x^{4} + 4 \, b^{3} g x^{3} + 6 \, b^{3} f x^{2} - 12 \, a b^{2} h x + 12 \, b^{3} x e}{12 \, b^{4}} - \frac{{\left (b^{9} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{8} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b^{7} h - a b^{8} 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 b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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