Optimal. Leaf size=341 \[ \frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{8 a^3 x^8}-\frac{a^2 e-3 a b d+6 b^2 c}{2 a^5 x^2}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{9 \sqrt{3} a^{17/3} \sqrt [3]{b}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac{x \left (-5 a^3 f+11 a^2 b e-17 a b^2 d+23 b^3 c\right )}{18 a^5 \left (a+b x^3\right )}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 1.12027, 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 \[ \frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{8 a^3 x^8}-\frac{a^2 e-3 a b d+6 b^2 c}{2 a^5 x^2}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{9 \sqrt{3} a^{17/3} \sqrt [3]{b}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac{x \left (-5 a^3 f+11 a^2 b e-17 a b^2 d+23 b^3 c\right )}{18 a^5 \left (a+b x^3\right )}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^9*(a + b*x^3)^3),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**9/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.481443, size = 324, normalized size = 0.95 \[ \frac{-\frac{216 a^{5/3} (a d-3 b c)}{x^5}-\frac{135 a^{8/3} c}{x^8}-\frac{540 a^{2/3} \left (a^2 e-3 a b d+6 b^2 c\right )}{x^2}+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f-20 a^2 b e+44 a b^2 d-77 b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{\sqrt [3]{b}}+\frac{180 a^{5/3} x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{60 a^{2/3} x \left (5 a^3 f-11 a^2 b e+17 a b^2 d-23 b^3 c\right )}{a+b x^3}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{\sqrt [3]{b}}}{1080 a^{17/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^9*(a + b*x^3)^3),x]
[Out]
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Maple [B] time = 0.025, size = 603, 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^9/(b*x^3+a)^3,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)^3*x^9),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224288, size = 794, normalized size = 2.33 \[ \frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left ({\left (77 \, b^{5} c - 44 \, a b^{4} d + 20 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{14} + 2 \,{\left (77 \, a b^{4} c - 44 \, a^{2} b^{3} d + 20 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{11} +{\left (77 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 20 \, a^{4} b e - 5 \, a^{5} f\right )} x^{8}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 40 \, \sqrt{3}{\left ({\left (77 \, b^{5} c - 44 \, a b^{4} d + 20 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{14} + 2 \,{\left (77 \, a b^{4} c - 44 \, a^{2} b^{3} d + 20 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{11} +{\left (77 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 20 \, a^{4} b e - 5 \, a^{5} f\right )} x^{8}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 120 \,{\left ({\left (77 \, b^{5} c - 44 \, a b^{4} d + 20 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{14} + 2 \,{\left (77 \, a b^{4} c - 44 \, a^{2} b^{3} d + 20 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{11} +{\left (77 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 20 \, a^{4} b e - 5 \, a^{5} f\right )} x^{8}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (20 \,{\left (77 \, b^{4} c - 44 \, a b^{3} d + 20 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 32 \,{\left (77 \, a b^{3} c - 44 \, a^{2} b^{2} d + 20 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} + 9 \,{\left (77 \, a^{2} b^{2} c - 44 \, a^{3} b d + 20 \, a^{4} e\right )} x^{6} + 45 \, a^{4} c - 18 \,{\left (7 \, a^{3} b c - 4 \, a^{4} d\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{3240 \,{\left (a^{5} b^{2} x^{14} + 2 \, a^{6} b x^{11} + a^{7} x^{8}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]
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)^3*x^9),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**9+e*x**6+d*x**3+c)/x**9/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.219569, size = 532, normalized size = 1.56 \[ \frac{{\left (77 \, b^{3} c - 44 \, a b^{2} d - 5 \, a^{3} f + 20 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{6}} - \frac{\sqrt{3}{\left (77 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{6} b} - \frac{{\left (77 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{6} b} - \frac{23 \, b^{4} c x^{4} - 17 \, a b^{3} d x^{4} - 5 \, a^{3} b f x^{4} + 11 \, a^{2} b^{2} x^{4} e + 26 \, a b^{3} c x - 20 \, a^{2} b^{2} d x - 8 \, a^{4} f x + 14 \, a^{3} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{5}} - \frac{120 \, b^{2} c x^{6} - 60 \, a b d x^{6} + 20 \, a^{2} x^{6} e - 24 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{5} x^{8}} \]
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)^3*x^9),x, algorithm="giac")
[Out]