Optimal. Leaf size=140 \[ \frac{x^3 \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac{a \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right ) \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^6 (b e-2 a f)}{6 b^3}+\frac{f x^9}{9 b^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.369875, antiderivative size = 140, 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 \[ \frac{x^3 \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac{a \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right ) \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^6 (b e-2 a f)}{6 b^3}+\frac{f x^9}{9 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 b^{5} \left (a + b x^{3}\right )} + \left (a^{2} f - \frac{2 a b e}{3} + \frac{b^{2} d}{3}\right ) \int ^{x^{3}} \frac{1}{b^{4}}\, dx + \frac{f x^{9}}{9 b^{2}} - \frac{\left (2 a f - b e\right ) \int ^{x^{3}} x\, dx}{3 b^{3}} - \frac{\left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.180594, size = 129, normalized size = 0.92 \[ \frac{6 b x^3 \left (3 a^2 f-2 a b e+b^2 d\right )+\frac{6 a \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a+b x^3}+6 \log \left (a+b x^3\right ) \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )+3 b^2 x^6 (b e-2 a f)+2 b^3 f x^9}{18 b^5} \]
Antiderivative was successfully verified.
[In] Integrate[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.016, size = 192, normalized size = 1.4 \[{\frac{f{x}^{9}}{9\,{b}^{2}}}-{\frac{{x}^{6}af}{3\,{b}^{3}}}+{\frac{e{x}^{6}}{6\,{b}^{2}}}+{\frac{{a}^{2}f{x}^{3}}{{b}^{4}}}-{\frac{2\,ae{x}^{3}}{3\,{b}^{3}}}+{\frac{d{x}^{3}}{3\,{b}^{2}}}-{\frac{4\,\ln \left ( b{x}^{3}+a \right ){a}^{3}f}{3\,{b}^{5}}}+{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}e}{{b}^{4}}}-{\frac{2\,\ln \left ( b{x}^{3}+a \right ) ad}{3\,{b}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) c}{3\,{b}^{2}}}-{\frac{{a}^{4}f}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}+{\frac{{a}^{3}e}{3\,{b}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}d}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{ac}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.37511, size = 186, normalized size = 1.33 \[ \frac{a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f}{3 \,{\left (b^{6} x^{3} + a b^{5}\right )}} + \frac{2 \, b^{2} f x^{9} + 3 \,{\left (b^{2} e - 2 \, a b f\right )} x^{6} + 6 \,{\left (b^{2} d - 2 \, a b e + 3 \, a^{2} f\right )} x^{3}}{18 \, b^{4}} + \frac{{\left (b^{3} c - 2 \, a b^{2} d + 3 \, a^{2} b e - 4 \, a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{5}} \]
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, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.232821, size = 273, normalized size = 1.95 \[ \frac{2 \, b^{4} f x^{12} +{\left (3 \, b^{4} e - 4 \, a b^{3} f\right )} x^{9} + 3 \,{\left (2 \, b^{4} d - 3 \, a b^{3} e + 4 \, a^{2} b^{2} f\right )} x^{6} + 6 \, a b^{3} c - 6 \, a^{2} b^{2} d + 6 \, a^{3} b e - 6 \, a^{4} f + 6 \,{\left (a b^{3} d - 2 \, a^{2} b^{2} e + 3 \, a^{3} b f\right )} x^{3} + 6 \,{\left (a b^{3} c - 2 \, a^{2} b^{2} d + 3 \, a^{3} b e - 4 \, a^{4} f +{\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{18 \,{\left (b^{6} x^{3} + a b^{5}\right )}} \]
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, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 18.4291, size = 138, normalized size = 0.99 \[ - \frac{a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c}{3 a b^{5} + 3 b^{6} x^{3}} + \frac{f x^{9}}{9 b^{2}} - \frac{x^{6} \left (2 a f - b e\right )}{6 b^{3}} + \frac{x^{3} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{3 b^{4}} - \frac{\left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.216618, size = 293, normalized size = 2.09 \[ \frac{\frac{{\left (b x^{3} + a\right )}^{3}{\left (2 \, f - \frac{3 \,{\left (4 \, a b f - b^{2} e\right )}}{{\left (b x^{3} + a\right )} b} + \frac{6 \,{\left (b^{4} d + 6 \, a^{2} b^{2} f - 3 \, a b^{3} e\right )}}{{\left (b x^{3} + a\right )}^{2} b^{2}}\right )}}{b^{4}} - \frac{6 \,{\left (b^{3} c - 2 \, a b^{2} d - 4 \, a^{3} f + 3 \, a^{2} b e\right )}{\rm ln}\left (\frac{{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} + \frac{6 \,{\left (\frac{a b^{6} c}{b x^{3} + a} - \frac{a^{2} b^{5} d}{b x^{3} + a} - \frac{a^{4} b^{3} f}{b x^{3} + a} + \frac{a^{3} b^{4} e}{b x^{3} + a}\right )}}{b^{7}}}{18 \, b} \]
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, algorithm="giac")
[Out]