Optimal. Leaf size=270 \[ \frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{18 a^{11/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{9 a^{11/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{3 \sqrt{3} a^{11/3} b^{4/3}}+\frac{2 b c-a d}{2 a^3 x^2}-\frac{c}{5 a^2 x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27243, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1829, 1488, 200, 31, 634, 617, 204, 628} \[ \frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{18 a^{11/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{9 a^{11/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{3 \sqrt{3} a^{11/3} b^{4/3}}+\frac{2 b c-a d}{2 a^3 x^2}-\frac{c}{5 a^2 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1829
Rule 1488
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^6 \left (a+b x^3\right )^2} \, dx &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}-\frac{\int \frac{-3 b^3 c+3 b^3 \left (\frac{b c}{a}-d\right ) x^3-b^2 \left (\frac{2 b^3 c}{a^2}-\frac{2 b^2 d}{a}+2 b e+a f\right ) x^6}{x^6 \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^3 c}{a x^6}-\frac{3 b^3 (-2 b c+a d)}{a^2 x^3}-\frac{b^2 \left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right )}{a^2 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{5 a^2 x^5}+\frac{2 b c-a d}{2 a^3 x^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{3 a^3 b}\\ &=-\frac{c}{5 a^2 x^5}+\frac{2 b c-a d}{2 a^3 x^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{11/3} b}+\frac{\left (8 b^3 c-5 a b^2 d+2 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}{9 a^{11/3} b}\\ &=-\frac{c}{5 a^2 x^5}+\frac{2 b c-a d}{2 a^3 x^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3} b^{4/3}}-\frac{\left (8 b^3 c-5 a b^2 d+2 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}{18 a^{11/3} b^{4/3}}+\frac{\left (8 b^3 c-5 a b^2 d+2 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}{6 a^{10/3} b}\\ &=-\frac{c}{5 a^2 x^5}+\frac{2 b c-a d}{2 a^3 x^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3} b^{4/3}}-\frac{\left (8 b^3 c-5 a b^2 d+2 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 )}{18 a^{11/3} b^{4/3}}+\frac{\left (8 b^3 c-5 a b^2 d+2 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 )}{3 a^{11/3} b^{4/3}}\\ &=-\frac{c}{5 a^2 x^5}+\frac{2 b c-a d}{2 a^3 x^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}-\frac{\left (8 b^3 c-5 a b^2 d+2 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 )}{3 \sqrt{3} a^{11/3} b^{4/3}}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3} b^{4/3}}-\frac{\left (8 b^3 c-5 a b^2 d+2 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 )}{18 a^{11/3} b^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.162183, size = 253, normalized size = 0.94 \[ \frac{-\frac{30 a^{2/3} x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )}-\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{b^{4/3}}+\frac{10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{b^{4/3}}-\frac{10 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{b^{4/3}}-\frac{45 a^{2/3} (a d-2 b c)}{x^2}-\frac{18 a^{5/3} c}{x^5}}{90 a^{11/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.011, size = 477, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.55885, size = 1991, normalized size = 7.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.08144, size = 429, normalized size = 1.59 \begin{align*} -\frac{{\left (8 \, b^{3} c - 5 \, a b^{2} d + a^{3} f + 2 \, 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 )}{9 \, a^{4} b} + \frac{\sqrt{3}{\left (8 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 2 \, \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 )}{9 \, a^{4} b^{2}} + \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} a^{3} b} + \frac{{\left (8 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 2 \, \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 )}{18 \, a^{4} b^{2}} + \frac{10 \, b c x^{3} - 5 \, a d x^{3} - 2 \, a c}{10 \, a^{3} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]