Optimal. Leaf size=338 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{18 a^{8/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{3 a^2 \left (a+b x^3\right )}+\frac{(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac{\log (x) (2 b c-a f)}{a^3}-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]
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Rubi [A] time = 0.727317, antiderivative size = 336, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (4 b e-a h)}{\sqrt [3]{b}}-2 a g+5 b d\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b d-2 a g)-\sqrt [3]{a} (4 b e-a h)\right )}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-h)+4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{3 a^2 \left (a+b x^3\right )}+\frac{(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac{\log (x) (2 b c-a f)}{a^3}-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]
Antiderivative was successfully verified.
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Rule 1829
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^4 \left (a+b x^3\right )^2} \, dx &=-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{\int \frac{-3 b^2 c-3 b^2 d x-3 b^2 e x^2+3 b^2 \left (\frac{b c}{a}-f\right ) x^3+2 b^2 \left (\frac{b d}{a}-g\right ) x^4+b^2 \left (\frac{b e}{a}-h\right ) x^5}{x^4 \left (a+b x^3\right )} \, dx}{3 a b^2}\\ &=-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^2 c}{a x^4}-\frac{3 b^2 d}{a x^3}-\frac{3 b^2 e}{a x^2}-\frac{3 b^2 (-2 b c+a f)}{a^2 x}+\frac{b^2 \left (a (5 b d-2 a g)+a (4 b e-a h) x-3 b (2 b c-a f) x^2\right )}{a^2 \left (a+b x^3\right )}\right ) \, dx}{3 a b^2}\\ &=-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{(2 b c-a f) \log (x)}{a^3}-\frac{\int \frac{a (5 b d-2 a g)+a (4 b e-a h) x-3 b (2 b c-a f) x^2}{a+b x^3} \, dx}{3 a^3}\\ &=-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{(2 b c-a f) \log (x)}{a^3}-\frac{\int \frac{a (5 b d-2 a g)+a (4 b e-a h) x}{a+b x^3} \, dx}{3 a^3}+\frac{(b (2 b c-a f)) \int \frac{x^2}{a+b x^3} \, dx}{a^3}\\ &=-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{(2 b c-a f) \log (x)}{a^3}+\frac{(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac{\int \frac{\sqrt [3]{a} \left (2 a \sqrt [3]{b} (5 b d-2 a g)+a^{4/3} (4 b e-a h)\right )+\sqrt [3]{b} \left (-a \sqrt [3]{b} (5 b d-2 a g)+a^{4/3} (4 b e-a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{11/3} \sqrt [3]{b}}-\frac{\left (5 b d-2 a g-\frac{\sqrt [3]{a} (4 b e-a h)}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{8/3}}\\ &=-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{(2 b c-a f) \log (x)}{a^3}-\frac{\left (5 b d-2 a g-\frac{\sqrt [3]{a} (4 b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac{(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac{\left (5 b^{4/3} d+4 \sqrt [3]{a} b e-2 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}{6 a^{7/3} \sqrt [3]{b}}+\frac{\left (5 b d-2 a g-\frac{\sqrt [3]{a} (4 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}{18 a^{8/3} \sqrt [3]{b}}\\ &=-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{(2 b c-a f) \log (x)}{a^3}-\frac{\left (5 b d-2 a g-\frac{\sqrt [3]{a} (4 b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac{\left (5 b d-2 a g-\frac{\sqrt [3]{a} (4 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 )}{18 a^{8/3} \sqrt [3]{b}}+\frac{(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}-\frac{\left (5 b^{4/3} d+4 \sqrt [3]{a} b e-2 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 )}{3 a^{8/3} b^{2/3}}\\ &=-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x}-\frac{x \left (b d-a g+(b e-a h) x-b \left (\frac{b c}{a}-f\right ) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac{\left (5 b^{4/3} d+4 \sqrt [3]{a} b e-2 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 )}{3 \sqrt{3} a^{8/3} b^{2/3}}-\frac{(2 b c-a f) \log (x)}{a^3}-\frac{\left (5 b d-2 a g-\frac{\sqrt [3]{a} (4 b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac{\left (5 b d-2 a g-\frac{\sqrt [3]{a} (4 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 )}{18 a^{8/3} \sqrt [3]{b}}+\frac{(2 b c-a f) \log \left (a+b x^3\right )}{3 a^3}\\ \end{align*}
Mathematica [A] time = 0.454282, size = 303, normalized size = 0.9 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{2/3}}-\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 b^{4/3} d\right )}{b^{2/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} h-4 \sqrt [3]{a} b e+2 a \sqrt [3]{b} g-5 b^{4/3} d\right )}{b^{2/3}}+\frac{a (6 a (f+x (g+h x))-6 b (c+x (d+e x)))}{a+b x^3}+6 (2 b c-a f) \log \left (a+b x^3\right )+18 \log (x) (a f-2 b c)-\frac{6 a c}{x^3}-\frac{9 a d}{x^2}-\frac{18 a e}{x}}{18 a^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 561, 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 = 1.09594, size = 520, normalized size = 1.54 \begin{align*} \frac{{\left (2 \, b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} - \frac{{\left (2 \, b c - a f\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g + \left (-a b^{2}\right )^{\frac{2}{3}} a h - 4 \, \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 )}{9 \, a^{3} b^{2}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g - \left (-a b^{2}\right )^{\frac{2}{3}} a h + 4 \, \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 )}{18 \, a^{3} b^{2}} - \frac{{\left (a^{5} b h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a^{4} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - 5 \, a^{4} b^{2} d + 2 \, a^{5} b g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{7} b} + \frac{2 \,{\left (a^{2} h - 4 \, a b e\right )} x^{5} -{\left (5 \, a b d - 2 \, a^{2} g\right )} x^{4} - 6 \, a^{2} x^{2} e - 3 \, a^{2} d x - 2 \,{\left (2 \, a b c - a^{2} f\right )} x^{3} - 2 \, a^{2} c}{6 \,{\left (b x^{3} + a\right )} a^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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