Optimal. Leaf size=240 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{2/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{2/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{2/3} b^{10/3}}+\frac{x \left (a^2 f-a b e+b^2 d\right )}{b^3}+\frac{x^4 (b e-a f)}{4 b^2}+\frac{f x^7}{7 b} \]
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Rubi [A] time = 0.152689, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {1887, 200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{2/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{2/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{2/3} b^{10/3}}+\frac{x \left (a^2 f-a b e+b^2 d\right )}{b^3}+\frac{x^4 (b e-a f)}{4 b^2}+\frac{f x^7}{7 b} \]
Antiderivative was successfully verified.
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Rule 1887
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{c+d x^3+e x^6+f x^9}{a+b x^3} \, dx &=\int \left (\frac{b^2 d-a b e+a^2 f}{b^3}+\frac{(b e-a f) x^3}{b^2}+\frac{f x^6}{b}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{b^3 \left (a+b x^3\right )}\right ) \, dx\\ &=\frac{\left (b^2 d-a b e+a^2 f\right ) x}{b^3}+\frac{(b e-a f) x^4}{4 b^2}+\frac{f x^7}{7 b}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{b^3}\\ &=\frac{\left (b^2 d-a b e+a^2 f\right ) x}{b^3}+\frac{(b e-a f) x^4}{4 b^2}+\frac{f x^7}{7 b}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} b^3}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} b^3}\\ &=\frac{\left (b^2 d-a b e+a^2 f\right ) x}{b^3}+\frac{(b e-a f) x^4}{4 b^2}+\frac{f x^7}{7 b}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\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^{10/3}}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\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^3}\\ &=\frac{\left (b^2 d-a b e+a^2 f\right ) x}{b^3}+\frac{(b e-a f) x^4}{4 b^2}+\frac{f x^7}{7 b}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\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^{10/3}}\\ &=\frac{\left (b^2 d-a b e+a^2 f\right ) x}{b^3}+\frac{(b e-a f) x^4}{4 b^2}+\frac{f x^7}{7 b}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\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^{10/3}}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}\\ \end{align*}
Mathematica [A] time = 0.147161, size = 229, normalized size = 0.95 \[ \frac{\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^{2/3}}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^{2/3}}+84 \sqrt [3]{b} x \left (a^2 f-a b e+b^2 d\right )+21 b^{4/3} x^4 (b e-a f)+12 b^{7/3} f x^7}{84 b^{10/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 442, normalized size = 1.8 \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 [A] time = 1.36121, size = 1350, normalized size = 5.62 \begin{align*} \left [\frac{12 \, a^{2} b^{3} f x^{7} + 21 \,{\left (a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{4} - 42 \, \sqrt{\frac{1}{3}}{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} \sqrt{\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x^{3} + 3 \, \left (-a^{2} b\right )^{\frac{1}{3}} a x - a^{2} - 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{2} + \left (-a^{2} b\right )^{\frac{2}{3}} x + \left (-a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}}}{b x^{3} + a}\right ) - 14 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (-a^{2} b\right )^{\frac{2}{3}} x - \left (-a^{2} b\right )^{\frac{1}{3}} a\right ) + 28 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac{2}{3}}\right ) + 84 \,{\left (a^{2} b^{3} d - a^{3} b^{2} e + a^{4} b f\right )} x}{84 \, a^{2} b^{4}}, \frac{12 \, a^{2} b^{3} f x^{7} + 21 \,{\left (a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{4} + 84 \, \sqrt{\frac{1}{3}}{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} \sqrt{-\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (-a^{2} b\right )^{\frac{2}{3}} x + \left (-a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) - 14 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (-a^{2} b\right )^{\frac{2}{3}} x - \left (-a^{2} b\right )^{\frac{1}{3}} a\right ) + 28 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac{2}{3}}\right ) + 84 \,{\left (a^{2} b^{3} d - a^{3} b^{2} e + a^{4} b f\right )} x}{84 \, a^{2} b^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.89, size = 340, normalized size = 1.42 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{10} + a^{9} f^{3} - 3 a^{8} b e f^{2} + 3 a^{7} b^{2} d f^{2} + 3 a^{7} b^{2} e^{2} f - 3 a^{6} b^{3} c f^{2} - 6 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} + 6 a^{5} b^{4} c e f + 3 a^{5} b^{4} d^{2} f + 3 a^{5} b^{4} d e^{2} - 6 a^{4} b^{5} c d f - 3 a^{4} b^{5} c e^{2} - 3 a^{4} b^{5} d^{2} e + 3 a^{3} b^{6} c^{2} f + 6 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} - 3 a^{2} b^{7} c^{2} e - 3 a^{2} b^{7} c d^{2} + 3 a b^{8} c^{2} d - b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{3 t a b^{3}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x^{7}}{7 b} - \frac{x^{4} \left (a f - b e\right )}{4 b^{2}} + \frac{x \left (a^{2} f - a b e + b^{2} d\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07513, size = 414, normalized size = 1.72 \begin{align*} \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} 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^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} 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^{4}} - \frac{{\left (b^{7} c - a b^{6} d - a^{3} b^{4} f + a^{2} b^{5} 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^{7}} + \frac{4 \, b^{6} f x^{7} - 7 \, a b^{5} f x^{4} + 7 \, b^{6} x^{4} e + 28 \, b^{6} d x + 28 \, a^{2} b^{4} f x - 28 \, a b^{5} x e}{28 \, b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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