Optimal. Leaf size=341 \[ -\frac{x \left (11 a^2 b e-5 a^3 f-17 a b^2 d+23 b^3 c\right )}{18 a^5 \left (a+b x^3\right )}-\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (20 a^2 b e-5 a^3 f-44 a b^2 d+77 b^3 c\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (20 a^2 b e-5 a^3 f-44 a b^2 d+77 b^3 c\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (20 a^2 b e-5 a^3 f-44 a b^2 d+77 b^3 c\right )}{9 \sqrt{3} a^{17/3} \sqrt [3]{b}}-\frac{a^2 e-3 a b d+6 b^2 c}{2 a^5 x^2}+\frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{8 a^3 x^8} \]
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Rubi [A] time = 0.54676, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1829, 1834, 200, 31, 634, 617, 204, 628} \[ -\frac{x \left (11 a^2 b e-5 a^3 f-17 a b^2 d+23 b^3 c\right )}{18 a^5 \left (a+b x^3\right )}-\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (20 a^2 b e-5 a^3 f-44 a b^2 d+77 b^3 c\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (20 a^2 b e-5 a^3 f-44 a b^2 d+77 b^3 c\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (20 a^2 b e-5 a^3 f-44 a b^2 d+77 b^3 c\right )}{9 \sqrt{3} a^{17/3} \sqrt [3]{b}}-\frac{a^2 e-3 a b d+6 b^2 c}{2 a^5 x^2}+\frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{8 a^3 x^8} \]
Antiderivative was successfully verified.
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Rule 1829
Rule 1834
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}{x^9 \left (a+b x^3\right )^3} \, dx &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b^3 c+6 b^3 \left (\frac{b c}{a}-d\right ) x^3-\frac{6 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac{5 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}}{x^9 \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}+\frac{\int \frac{18 b^6 c-18 b^6 \left (\frac{2 b c}{a}-d\right ) x^3+18 b^6 \left (\frac{3 b^2 c}{a^2}-\frac{2 b d}{a}+e\right ) x^6-\frac{2 b^6 \left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x^9}{a^3}}{x^9 \left (a+b x^3\right )} \, dx}{18 a^2 b^6}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}+\frac{\int \left (\frac{18 b^6 c}{a x^9}+\frac{18 b^6 (-3 b c+a d)}{a^2 x^6}+\frac{18 b^6 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^3}+\frac{2 b^6 \left (-77 b^3 c+44 a b^2 d-20 a^2 b e+5 a^3 f\right )}{a^3 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^6}\\ &=-\frac{c}{8 a^3 x^8}+\frac{3 b c-a d}{5 a^4 x^5}-\frac{6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{9 a^5}\\ &=-\frac{c}{8 a^3 x^8}+\frac{3 b c-a d}{5 a^4 x^5}-\frac{6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{17/3}}-\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 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}{27 a^{17/3}}\\ &=-\frac{c}{8 a^3 x^8}+\frac{3 b c-a d}{5 a^4 x^5}-\frac{6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{17/3} \sqrt [3]{b}}-\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{16/3}}+\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 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}{54 a^{17/3} \sqrt [3]{b}}\\ &=-\frac{c}{8 a^3 x^8}+\frac{3 b c-a d}{5 a^4 x^5}-\frac{6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}-\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 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 )}{9 a^{17/3} \sqrt [3]{b}}\\ &=-\frac{c}{8 a^3 x^8}+\frac{3 b c-a d}{5 a^4 x^5}-\frac{6 b^2 c-3 a b d+a^2 e}{2 a^5 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^4 \left (a+b x^3\right )^2}-\frac{\left (23 b^3 c-17 a b^2 d+11 a^2 b e-5 a^3 f\right ) x}{18 a^5 \left (a+b x^3\right )}+\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{17/3} \sqrt [3]{b}}-\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac{\left (77 b^3 c-44 a b^2 d+20 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{17/3} \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.235911, size = 324, normalized size = 0.95 \[ \frac{\frac{180 a^{5/3} x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{60 a^{2/3} x \left (-11 a^2 b e+5 a^3 f+17 a b^2 d-23 b^3 c\right )}{a+b x^3}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (20 a^2 b e-5 a^3 f-44 a b^2 d+77 b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^2 b e+5 a^3 f+44 a b^2 d-77 b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (20 a^2 b e-5 a^3 f-44 a b^2 d+77 b^3 c\right )}{\sqrt [3]{b}}-\frac{540 a^{2/3} \left (a^2 e-3 a b d+6 b^2 c\right )}{x^2}-\frac{216 a^{5/3} (a d-3 b c)}{x^5}-\frac{135 a^{8/3} c}{x^8}}{1080 a^{17/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 603, 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 [B] time = 1.77692, size = 3013, normalized size = 8.84 \begin{align*} \text{result too large to display} \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 = 1.09785, size = 532, normalized size = 1.56 \begin{align*} \frac{{\left (77 \, b^{3} c - 44 \, a b^{2} d - 5 \, a^{3} f + 20 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{6}} - \frac{\sqrt{3}{\left (77 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 20 \, \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 )}{27 \, a^{6} b} - \frac{{\left (77 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 20 \, \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 )}{54 \, a^{6} b} - \frac{23 \, b^{4} c x^{4} - 17 \, a b^{3} d x^{4} - 5 \, a^{3} b f x^{4} + 11 \, a^{2} b^{2} x^{4} e + 26 \, a b^{3} c x - 20 \, a^{2} b^{2} d x - 8 \, a^{4} f x + 14 \, a^{3} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{5}} - \frac{120 \, b^{2} c x^{6} - 60 \, a b d x^{6} + 20 \, a^{2} x^{6} e - 24 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{5} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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