Optimal. Leaf size=88 \[ -\frac{a^2 (A b-a B)}{4 b^4 \left (a+b x^2\right )^2}+\frac{a (2 A b-3 a B)}{2 b^4 \left (a+b x^2\right )}+\frac{(A b-3 a B) \log \left (a+b x^2\right )}{2 b^4}+\frac{B x^2}{2 b^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.224423, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^2 (A b-a B)}{4 b^4 \left (a+b x^2\right )^2}+\frac{a (2 A b-3 a B)}{2 b^4 \left (a+b x^2\right )}+\frac{(A b-3 a B) \log \left (a+b x^2\right )}{2 b^4}+\frac{B x^2}{2 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^5*(A + B*x^2))/(a + b*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \left (A b - B a\right )}{4 b^{4} \left (a + b x^{2}\right )^{2}} + \frac{a \left (2 A b - 3 B a\right )}{2 b^{4} \left (a + b x^{2}\right )} + \frac{\int ^{x^{2}} B\, dx}{2 b^{3}} + \frac{\left (A b - 3 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(B*x**2+A)/(b*x**2+a)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0621631, size = 92, normalized size = 1.05 \[ \frac{2 a A b-3 a^2 B}{2 b^4 \left (a+b x^2\right )}+\frac{a^3 B-a^2 A b}{4 b^4 \left (a+b x^2\right )^2}+\frac{(A b-3 a B) \log \left (a+b x^2\right )}{2 b^4}+\frac{B x^2}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^5*(A + B*x^2))/(a + b*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.017, size = 109, normalized size = 1.2 \[{\frac{B{x}^{2}}{2\,{b}^{3}}}-{\frac{{a}^{2}A}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{B{a}^{3}}{4\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ) A}{2\,{b}^{3}}}-{\frac{3\,\ln \left ( b{x}^{2}+a \right ) Ba}{2\,{b}^{4}}}+{\frac{aA}{{b}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,{a}^{2}B}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(B*x^2+A)/(b*x^2+a)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.33848, size = 127, normalized size = 1.44 \[ -\frac{5 \, B a^{3} - 3 \, A a^{2} b + 2 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{2}}{4 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} + \frac{B x^{2}}{2 \, b^{3}} - \frac{{\left (3 \, B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^5/(b*x^2 + a)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.223146, size = 192, normalized size = 2.18 \[ \frac{2 \, B b^{3} x^{6} + 4 \, B a b^{2} x^{4} - 5 \, B a^{3} + 3 \, A a^{2} b - 4 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} - 2 \,{\left ({\left (3 \, B a b^{2} - A b^{3}\right )} x^{4} + 3 \, B a^{3} - A a^{2} b + 2 \,{\left (3 \, B a^{2} b - A a b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^5/(b*x^2 + a)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 5.53614, size = 94, normalized size = 1.07 \[ \frac{B x^{2}}{2 b^{3}} - \frac{- 3 A a^{2} b + 5 B a^{3} + x^{2} \left (- 4 A a b^{2} + 6 B a^{2} b\right )}{4 a^{2} b^{4} + 8 a b^{5} x^{2} + 4 b^{6} x^{4}} - \frac{\left (- A b + 3 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(B*x**2+A)/(b*x**2+a)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.226637, size = 126, normalized size = 1.43 \[ \frac{B x^{2}}{2 \, b^{3}} - \frac{{\left (3 \, B a - A b\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac{9 \, B a b^{2} x^{4} - 3 \, A b^{3} x^{4} + 12 \, B a^{2} b x^{2} - 2 \, A a b^{2} x^{2} + 4 \, B a^{3}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^5/(b*x^2 + a)^3,x, algorithm="giac")
[Out]