Optimal. Leaf size=98 \[ -\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+\frac{1}{2} b x^2 (2 a C+A b)+a \log (x) (a C+2 A b)+\frac{1}{3} b x^3 (2 a D+b B)+a x (a D+2 b B)+\frac{1}{4} b^2 C x^4+\frac{1}{5} b^2 D x^5 \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.186014, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ -\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+\frac{1}{2} b x^2 (2 a C+A b)+a \log (x) (a C+2 A b)+\frac{1}{3} b x^3 (2 a D+b B)+a x (a D+2 b B)+\frac{1}{4} b^2 C x^4+\frac{1}{5} b^2 D x^5 \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{2}}{2 x^{2}} - \frac{B a^{2}}{x} + \frac{C b^{2} x^{4}}{4} + \frac{D b^{2} x^{5}}{5} + a \left (2 A b + C a\right ) \log{\left (x \right )} + \frac{b x^{3} \left (B b + 2 D a\right )}{3} + b \left (A b + 2 C a\right ) \int x\, dx + \frac{a \left (2 B b + D a\right ) \int D\, dx}{D} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(D*x**3+C*x**2+B*x+A)/x**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0850998, size = 87, normalized size = 0.89 \[ -\frac{a^2 \left (A+2 B x-2 D x^3\right )}{2 x^2}+a \log (x) (a C+2 A b)+\frac{1}{3} a b x (6 B+x (3 C+2 D x))+\frac{1}{60} b^2 x^2 (30 A+x (20 B+3 x (5 C+4 D x))) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^3,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 97, normalized size = 1. \[{\frac{{b}^{2}D{x}^{5}}{5}}+{\frac{{b}^{2}C{x}^{4}}{4}}+{\frac{B{b}^{2}{x}^{3}}{3}}+{\frac{2\,D{x}^{3}ab}{3}}+{\frac{A{x}^{2}{b}^{2}}{2}}+C{x}^{2}ab+2\,Bxab+Dx{a}^{2}+2\,A\ln \left ( x \right ) ab+C\ln \left ( x \right ){a}^{2}-{\frac{A{a}^{2}}{2\,{x}^{2}}}-{\frac{{a}^{2}B}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/x^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.35124, size = 130, normalized size = 1.33 \[ \frac{1}{5} \, D b^{2} x^{5} + \frac{1}{4} \, C b^{2} x^{4} + \frac{1}{3} \,{\left (2 \, D a b + B b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, C a b + A b^{2}\right )} x^{2} +{\left (D a^{2} + 2 \, B a b\right )} x +{\left (C a^{2} + 2 \, A a b\right )} \log \left (x\right ) - \frac{2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^2/x^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.219041, size = 139, normalized size = 1.42 \[ \frac{12 \, D b^{2} x^{7} + 15 \, C b^{2} x^{6} + 20 \,{\left (2 \, D a b + B b^{2}\right )} x^{5} + 30 \,{\left (2 \, C a b + A b^{2}\right )} x^{4} - 60 \, B a^{2} x + 60 \,{\left (D a^{2} + 2 \, B a b\right )} x^{3} + 60 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2} \log \left (x\right ) - 30 \, A a^{2}}{60 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^2/x^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.02945, size = 99, normalized size = 1.01 \[ \frac{C b^{2} x^{4}}{4} + \frac{D b^{2} x^{5}}{5} + a \left (2 A b + C a\right ) \log{\left (x \right )} + x^{3} \left (\frac{B b^{2}}{3} + \frac{2 D a b}{3}\right ) + x^{2} \left (\frac{A b^{2}}{2} + C a b\right ) + x \left (2 B a b + D a^{2}\right ) - \frac{A a^{2} + 2 B a^{2} x}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(D*x**3+C*x**2+B*x+A)/x**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.223473, size = 131, normalized size = 1.34 \[ \frac{1}{5} \, D b^{2} x^{5} + \frac{1}{4} \, C b^{2} x^{4} + \frac{2}{3} \, D a b x^{3} + \frac{1}{3} \, B b^{2} x^{3} + C a b x^{2} + \frac{1}{2} \, A b^{2} x^{2} + D a^{2} x + 2 \, B a b x +{\left (C a^{2} + 2 \, A a b\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^2/x^3,x, algorithm="giac")
[Out]