Optimal. Leaf size=260 \[ -\frac{c}{2 a^2 x^2}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{18 a^{8/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{9 a^{8/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} a^{8/3} b^{7/3}}+\frac{f x}{b^2} \]
[Out]
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Rubi [A] time = 0.573303, antiderivative size = 260, 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 \[ -\frac{c}{2 a^2 x^2}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{18 a^{8/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{9 a^{8/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} a^{8/3} b^{7/3}}+\frac{f x}{b^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^3*(a + b*x^3)^2),x]
[Out]
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Rubi in Sympy [A] time = 132.498, size = 250, normalized size = 0.96 \[ \frac{f x}{b^{2}} - \frac{x \left (\frac{a^{3} f}{x^{3}} - \frac{a^{2} b e}{x^{3}} + \frac{a b^{2} d}{x^{3}} - \frac{b^{3} c}{x^{3}}\right )}{3 a b^{3} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{2 a b^{3} x^{2}} - \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}} b^{\frac{7}{3}}} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}} b^{\frac{7}{3}}} + \frac{\sqrt{3} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} b^{\frac{7}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**3/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.282511, size = 250, normalized size = 0.96 \[ \frac{1}{18} \left (-\frac{9 c}{a^2 x^2}+\frac{6 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^2 b^2 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{a^{8/3} b^{7/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{a^{8/3} b^{7/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{a^{8/3} b^{7/3}}+\frac{18 f x}{b^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^3*(a + b*x^3)^2),x]
[Out]
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Maple [B] time = 0.016, size = 463, normalized size = 1.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^9+e*x^6+d*x^3+c)/x^3/(b*x^3+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^2*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239142, size = 522, normalized size = 2.01 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (5 \, b^{4} c - 2 \, a b^{3} d - a^{2} b^{2} e + 4 \, a^{3} b f\right )} x^{5} +{\left (5 \, a b^{3} c - 2 \, a^{2} b^{2} d - a^{3} b e + 4 \, a^{4} f\right )} x^{2}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (5 \, b^{4} c - 2 \, a b^{3} d - a^{2} b^{2} e + 4 \, a^{3} b f\right )} x^{5} +{\left (5 \, a b^{3} c - 2 \, a^{2} b^{2} d - a^{3} b e + 4 \, a^{4} f\right )} x^{2}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 6 \,{\left ({\left (5 \, b^{4} c - 2 \, a b^{3} d - a^{2} b^{2} e + 4 \, a^{3} b f\right )} x^{5} +{\left (5 \, a b^{3} c - 2 \, a^{2} b^{2} d - a^{3} b e + 4 \, a^{4} f\right )} x^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (6 \, a^{2} b f x^{6} - 3 \, a b^{2} c -{\left (5 \, b^{3} c - 2 \, a b^{2} d + 2 \, a^{2} b e - 8 \, a^{3} f\right )} x^{3}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{54 \,{\left (a^{2} b^{3} x^{5} + a^{3} b^{2} x^{2}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^2*x^3),x, algorithm="fricas")
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Sympy [A] time = 107.283, size = 381, normalized size = 1.47 \[ \frac{- 3 a b^{2} c + x^{3} \left (2 a^{3} f - 2 a^{2} b e + 2 a b^{2} d - 5 b^{3} c\right )}{6 a^{3} b^{2} x^{2} + 6 a^{2} b^{3} x^{5}} + \operatorname{RootSum}{\left (729 t^{3} a^{8} b^{7} + 64 a^{9} f^{3} - 48 a^{8} b e f^{2} - 96 a^{7} b^{2} d f^{2} + 12 a^{7} b^{2} e^{2} f + 240 a^{6} b^{3} c f^{2} + 48 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} - 120 a^{5} b^{4} c e f + 48 a^{5} b^{4} d^{2} f - 6 a^{5} b^{4} d e^{2} - 240 a^{4} b^{5} c d f + 15 a^{4} b^{5} c e^{2} - 12 a^{4} b^{5} d^{2} e + 300 a^{3} b^{6} c^{2} f + 60 a^{3} b^{6} c d e - 8 a^{3} b^{6} d^{3} - 75 a^{2} b^{7} c^{2} e + 60 a^{2} b^{7} c d^{2} - 150 a b^{8} c^{2} d + 125 b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{9 t a^{3} b^{2}}{4 a^{3} f - a^{2} b e - 2 a b^{2} d + 5 b^{3} c} + x \right )} \right )\right )} + \frac{f x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**9+e*x**6+d*x**3+c)/x**3/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.217019, size = 417, normalized size = 1.6 \[ \frac{f x}{b^{2}} + \frac{{\left (5 \, b^{3} c - 2 \, a b^{2} d + 4 \, a^{3} f - a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3} b^{2}} - \frac{c}{2 \, a^{2} x^{2}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b^{3}} - \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} a^{2} b^{2}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^2*x^3),x, algorithm="giac")
[Out]