Optimal. Leaf size=72 \[ -\frac{(A b-a C) \log \left (a+b x^2\right )}{2 a b}+\frac{A \log (x)}{a}+\frac{(b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{D x}{b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.209655, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{(A b-a C) \log \left (a+b x^2\right )}{2 a b}+\frac{A \log (x)}{a}+\frac{(b B-a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{D x}{b} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x + C*x^2 + D*x^3)/(x*(a + b*x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{A \log{\left (x \right )}}{a} + \frac{\int D\, dx}{b} - \frac{\left (A b - C a\right ) \log{\left (a + b x^{2} \right )}}{2 a b} + \frac{\left (B b - D a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((D*x**3+C*x**2+B*x+A)/x/(b*x**2+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0994818, size = 73, normalized size = 1.01 \[ \frac{(a C-A b) \log \left (a+b x^2\right )}{2 a b}+\frac{A \log (x)}{a}-\frac{(a D-b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{D x}{b} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x + C*x^2 + D*x^3)/(x*(a + b*x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 80, normalized size = 1.1 \[{\frac{Dx}{b}}+{\frac{A\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{2}+a \right ) A}{2\,a}}+{\frac{\ln \left ( b{x}^{2}+a \right ) C}{2\,b}}+{B\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{aD}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((D*x^3+C*x^2+B*x+A)/x/(b*x^2+a),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
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),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.270926, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (D a^{2} - B a b\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) -{\left (2 \, D a x + 2 \, A b \log \left (x\right ) +{\left (C a - A b\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{-a b}}{2 \, \sqrt{-a b} a b}, -\frac{2 \,{\left (D a^{2} - B a b\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (2 \, D a x + 2 \, A b \log \left (x\right ) +{\left (C a - A b\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{a b}}{2 \, \sqrt{a b} a b}\right ] \]
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),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 39.1185, size = 1268, normalized size = 17.61 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x**3+C*x**2+B*x+A)/x/(b*x**2+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.224664, size = 89, normalized size = 1.24 \[ \frac{D x}{b} + \frac{A{\rm ln}\left ({\left | x \right |}\right )}{a} - \frac{{\left (D a - B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{{\left (C a - A b\right )}{\rm ln}\left (b x^{2} + a\right )}{2 \, a b} \]
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),x, algorithm="giac")
[Out]