Optimal. Leaf size=54 \[ \log (x) (a C+A b)-\frac{a A}{2 x^2}+x (a D+b B)-\frac{a B}{x}+\frac{1}{2} b C x^2+\frac{1}{3} b D x^3 \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0990235, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \log (x) (a C+A b)-\frac{a A}{2 x^2}+x (a D+b B)-\frac{a B}{x}+\frac{1}{2} b C x^2+\frac{1}{3} b D x^3 \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^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. \[ - \frac{A a}{2 x^{2}} - \frac{B a}{x} + C b \int x\, dx + \frac{D b x^{3}}{3} + \left (A b + C a\right ) \log{\left (x \right )} + \frac{\left (B b + D a\right ) \int B\, dx}{B} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)*(D*x**3+C*x**2+B*x+A)/x**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0524996, size = 51, normalized size = 0.94 \[ \log (x) (a C+A b)-\frac{a \left (A+2 B x-2 D x^3\right )}{2 x^2}+\frac{1}{6} b x \left (6 B+3 C x+2 D x^2\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)*(A + B*x + C*x^2 + D*x^3))/x^3,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 48, normalized size = 0.9 \[{\frac{bD{x}^{3}}{3}}+{\frac{bC{x}^{2}}{2}}+bBx+Dxa+A\ln \left ( x \right ) b+C\ln \left ( x \right ) a-{\frac{Aa}{2\,{x}^{2}}}-{\frac{Ba}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)*(D*x^3+C*x^2+B*x+A)/x^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.34612, size = 65, normalized size = 1.2 \[ \frac{1}{3} \, D b x^{3} + \frac{1}{2} \, C b x^{2} +{\left (D a + B b\right )} x +{\left (C a + A b\right )} \log \left (x\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \]
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)/x^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.263849, size = 74, normalized size = 1.37 \[ \frac{2 \, D b x^{5} + 3 \, C b x^{4} + 6 \,{\left (D a + B b\right )} x^{3} + 6 \,{\left (C a + A b\right )} x^{2} \log \left (x\right ) - 6 \, B a x - 3 \, A a}{6 \, x^{2}} \]
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)/x^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.850776, size = 49, normalized size = 0.91 \[ \frac{C b x^{2}}{2} + \frac{D b x^{3}}{3} + x \left (B b + D a\right ) + \left (A b + C a\right ) \log{\left (x \right )} - \frac{A a + 2 B a x}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)*(D*x**3+C*x**2+B*x+A)/x**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.214842, size = 65, normalized size = 1.2 \[ \frac{1}{3} \, D b x^{3} + \frac{1}{2} \, C b x^{2} + D a x + B b x +{\left (C a + A b\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \]
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)/x^3,x, algorithm="giac")
[Out]