Optimal. Leaf size=269 \[ \frac{2 b c-a d}{a^3 x}-\frac{c}{4 a^2 x^4}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{18 a^{10/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{9 a^{10/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} a^{10/3} b^{5/3}} \]
[Out]
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Rubi [A] time = 0.656291, antiderivative size = 269, 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 \[ \frac{2 b c-a d}{a^3 x}-\frac{c}{4 a^2 x^4}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{18 a^{10/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{9 a^{10/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} a^{10/3} b^{5/3}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^5*(a + b*x^3)^2),x]
[Out]
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Rubi in Sympy [A] time = 139.282, size = 270, normalized size = 1. \[ - \frac{x \left (\frac{a^{3} f}{x^{5}} - \frac{a^{2} b e}{x^{5}} + \frac{a b^{2} d}{x^{5}} - \frac{b^{3} c}{x^{5}}\right )}{3 a b^{3} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{4 a b^{3} x^{4}} + \frac{2 a^{2} f - 2 a b e + b^{2} d}{a^{2} b^{2} x} - \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{7}{3}} b^{\frac{5}{3}}} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{7}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{7}{3}} b^{\frac{5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**5/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.301137, size = 255, normalized size = 0.95 \[ \frac{-\frac{9 a^{4/3} c}{x^4}-\frac{12 \sqrt [3]{a} x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}+\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac{36 \sqrt [3]{a} (a d-2 b c)}{x}}{36 a^{10/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^5*(a + b*x^3)^2),x]
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Maple [B] time = 0.022, size = 486, 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^5/(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^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23711, size = 532, normalized size = 1.98 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left ({\left (7 \, b^{4} c - 4 \, a b^{3} d + a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{7} +{\left (7 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{4}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 4 \, \sqrt{3}{\left ({\left (7 \, b^{4} c - 4 \, a b^{3} d + a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{7} +{\left (7 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{4}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 12 \,{\left ({\left (7 \, b^{4} c - 4 \, a b^{3} d + a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{7} +{\left (7 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{4}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (4 \,{\left (7 \, b^{3} c - 4 \, a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - 3 \, a^{2} b c + 3 \,{\left (7 \, a b^{2} c - 4 \, a^{2} b d\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{108 \,{\left (a^{3} b^{2} x^{7} + a^{4} b x^{4}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]
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^5),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**5/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.220432, size = 483, normalized size = 1.8 \[ -\frac{{\left (7 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{4} b} + \frac{b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{3} b} - \frac{\sqrt{3}{\left (7 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{4} b^{3}} + \frac{{\left (7 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{4} b^{3}} + \frac{8 \, b c x^{3} - 4 \, a d x^{3} - a c}{4 \, a^{3} x^{4}} \]
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^5),x, algorithm="giac")
[Out]