Optimal. Leaf size=95 \[ \frac{b c-a d}{3 a^2 x^3}+\frac{\log (x) \left (a^2 e-a b d+b^2 c\right )}{a^3}-\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b}-\frac{c}{6 a x^6} \]
[Out]
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Rubi [A] time = 0.243478, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{b c-a d}{3 a^2 x^3}+\frac{\log (x) \left (a^2 e-a b d+b^2 c\right )}{a^3}-\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b}-\frac{c}{6 a x^6} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^7*(a + b*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 44.4511, size = 87, normalized size = 0.92 \[ - \frac{c}{6 a x^{6}} - \frac{a d - b c}{3 a^{2} x^{3}} + \frac{\left (a^{2} e - a b d + b^{2} c\right ) \log{\left (x^{3} \right )}}{3 a^{3}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**7/(b*x**3+a),x)
[Out]
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Mathematica [A] time = 0.138683, size = 88, normalized size = 0.93 \[ \frac{6 \log (x) \left (a^2 e-a b d+b^2 c\right )+\log \left (a+b x^3\right ) \left (\frac{2 a^3 f}{b}-2 a^2 e+2 a b d-2 b^2 c\right )-\frac{a \left (a c+2 a d x^3-2 b c x^3\right )}{x^6}}{6 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^7*(a + b*x^3)),x]
[Out]
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Maple [A] time = 0.011, size = 116, normalized size = 1.2 \[ -{\frac{c}{6\,a{x}^{6}}}-{\frac{d}{3\,a{x}^{3}}}+{\frac{bc}{3\,{a}^{2}{x}^{3}}}+{\frac{e\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( x \right ) bd}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}c}{{a}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) f}{3\,b}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,a}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) d}{3\,{a}^{2}}}-{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^9+e*x^6+d*x^3+c)/x^7/(b*x^3+a),x)
[Out]
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Maxima [A] time = 1.38527, size = 126, normalized size = 1.33 \[ \frac{{\left (b^{2} c - a b d + a^{2} e\right )} \log \left (x^{3}\right )}{3 \, a^{3}} - \frac{{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{3} b} + \frac{2 \,{\left (b c - a d\right )} x^{3} - a c}{6 \, a^{2} x^{6}} \]
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)*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.248078, size = 136, normalized size = 1.43 \[ -\frac{2 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} \log \left (b x^{3} + a\right ) - 6 \,{\left (b^{3} c - a b^{2} d + a^{2} b e\right )} x^{6} \log \left (x\right ) + a^{2} b c - 2 \,{\left (a b^{2} c - a^{2} b d\right )} x^{3}}{6 \, a^{3} b x^{6}} \]
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)*x^7),x, algorithm="fricas")
[Out]
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Sympy [A] time = 92.4216, size = 85, normalized size = 0.89 \[ - \frac{a c + x^{3} \left (2 a d - 2 b c\right )}{6 a^{2} x^{6}} + \frac{\left (a^{2} e - a b d + b^{2} c\right ) \log{\left (x \right )}}{a^{3}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**9+e*x**6+d*x**3+c)/x**7/(b*x**3+a),x)
[Out]
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GIAC/XCAS [A] time = 0.215596, size = 170, normalized size = 1.79 \[ \frac{{\left (b^{2} c - a b d + a^{2} e\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3} b} - \frac{3 \, b^{2} c x^{6} - 3 \, a b d x^{6} + 3 \, a^{2} x^{6} e - 2 \, a b c x^{3} + 2 \, a^{2} d x^{3} + a^{2} c}{6 \, a^{3} x^{6}} \]
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)*x^7),x, algorithm="giac")
[Out]