Optimal. Leaf size=335 \[ \frac{2 b c-a d}{8 a^3 x^8}-\frac{c}{11 a^2 x^{11}}-\frac{a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{18 a^{17/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{9 a^{17/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{3 \sqrt{3} a^{17/3}}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{2 a^5 x^2} \]
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Rubi [A] time = 0.965721, antiderivative size = 335, 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 \[ \frac{2 b c-a d}{8 a^3 x^8}-\frac{c}{11 a^2 x^{11}}-\frac{a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{18 a^{17/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{9 a^{17/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{3 \sqrt{3} a^{17/3}}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{2 a^5 x^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^2),x]
[Out]
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Rubi in Sympy [A] time = 172.126, size = 330, normalized size = 0.99 \[ - \frac{x \left (\frac{a^{3} f}{x^{12}} - \frac{a^{2} b e}{x^{12}} + \frac{a b^{2} d}{x^{12}} - \frac{b^{3} c}{x^{12}}\right )}{3 a b^{3} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{11 a b^{3} x^{11}} + \frac{2 a^{2} f - 2 a b e + b^{2} d}{8 a^{2} b^{2} x^{8}} - \frac{3 a^{2} f - 2 a b e + b^{2} d}{5 a^{3} b x^{5}} + \frac{3 a^{2} f - 2 a b e + b^{2} d}{2 a^{4} x^{2}} + \frac{b^{\frac{2}{3}} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{14}{3}}} - \frac{b^{\frac{2}{3}} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{14}{3}}} - \frac{\sqrt{3} b^{\frac{2}{3}} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{14}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**12/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.345357, size = 317, normalized size = 0.95 \[ \frac{-\frac{495 a^{8/3} (a d-2 b c)}{x^8}-\frac{360 a^{11/3} c}{x^{11}}-\frac{792 a^{5/3} \left (a^2 e-2 a b d+3 b^2 c\right )}{x^5}+440 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )-440 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )-\frac{1320 a^{2/3} b x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}-\frac{1980 a^{2/3} \left (a^3 f-2 a^2 b e+3 a b^2 d-4 b^3 c\right )}{x^2}+220 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f-8 a^2 b e+11 a b^2 d-14 b^3 c\right )}{3960 a^{17/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^2),x]
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Maple [A] time = 0.023, size = 566, normalized size = 1.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^9+e*x^6+d*x^3+c)/x^12/(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^12),x, algorithm="maxima")
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Fricas [A] time = 0.23277, size = 670, normalized size = 2. \[ \frac{\sqrt{3}{\left (220 \, \sqrt{3}{\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} +{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 440 \, \sqrt{3}{\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} +{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 1320 \,{\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} +{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (220 \,{\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 132 \,{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} - 33 \,{\left (14 \, a^{2} b^{2} c - 11 \, a^{3} b d + 8 \, a^{4} e\right )} x^{6} - 120 \, a^{4} c + 15 \,{\left (14 \, a^{3} b c - 11 \, a^{4} d\right )} x^{3}\right )}\right )}}{11880 \,{\left (a^{5} b x^{14} + a^{6} x^{11}\right )}} \]
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^12),x, algorithm="fricas")
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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**12/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.217836, size = 528, normalized size = 1.58 \[ \frac{\sqrt{3}{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 11 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{6}} - \frac{{\left (14 \, b^{4} c - 11 \, a b^{3} d - 5 \, a^{3} b f + 8 \, a^{2} b^{2} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{6}} + \frac{{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 11 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{6}} + \frac{b^{4} c x - a b^{3} d x - a^{3} b f x + a^{2} b^{2} x e}{3 \,{\left (b x^{3} + a\right )} a^{5}} + \frac{880 \, b^{3} c x^{9} - 660 \, a b^{2} d x^{9} - 220 \, a^{3} f x^{9} + 440 \, a^{2} b x^{9} e - 264 \, a b^{2} c x^{6} + 176 \, a^{2} b d x^{6} - 88 \, a^{3} x^{6} e + 110 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{5} x^{11}} \]
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^12),x, algorithm="giac")
[Out]