Optimal. Leaf size=59 \[ -\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{e \log \left (a+c x^2\right )}{2 a}-\frac{d}{a x}+\frac{e \log (x)}{a} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.106766, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{e \log \left (a+c x^2\right )}{2 a}-\frac{d}{a x}+\frac{e \log (x)}{a} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(x^2*(a + c*x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 15.1534, size = 49, normalized size = 0.83 \[ - \frac{d}{a x} + \frac{e \log{\left (x \right )}}{a} - \frac{e \log{\left (a + c x^{2} \right )}}{2 a} - \frac{\sqrt{c} d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/x**2/(c*x**2+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0559503, size = 59, normalized size = 1. \[ -\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{e \log \left (a+c x^2\right )}{2 a}-\frac{d}{a x}+\frac{e \log (x)}{a} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(x^2*(a + c*x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 53, normalized size = 0.9 \[ -{\frac{d}{ax}}+{\frac{e\ln \left ( x \right ) }{a}}-{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,a}}-{\frac{cd}{a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/x^2/(c*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((e*x + d)/((c*x^2 + a)*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.286209, size = 1, normalized size = 0.02 \[ \left [\frac{d x \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} - 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) - e x \log \left (c x^{2} + a\right ) + 2 \, e x \log \left (x\right ) - 2 \, d}{2 \, a x}, -\frac{2 \, d x \sqrt{\frac{c}{a}} \arctan \left (\frac{c x}{a \sqrt{\frac{c}{a}}}\right ) + e x \log \left (c x^{2} + a\right ) - 2 \, e x \log \left (x\right ) + 2 \, d}{2 \, a x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c*x^2 + a)*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.81003, size = 326, normalized size = 5.53 \[ \left (- \frac{e}{2 a} - \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) \log{\left (x + \frac{12 a^{4} e \left (- \frac{e}{2 a} - \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right )^{2} - 6 a^{3} e^{2} \left (- \frac{e}{2 a} - \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) - 2 a^{2} c d^{2} \left (- \frac{e}{2 a} - \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) - 6 a^{2} e^{3} + 2 a c d^{2} e}{9 a c d e^{2} + c^{2} d^{3}} \right )} + \left (- \frac{e}{2 a} + \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) \log{\left (x + \frac{12 a^{4} e \left (- \frac{e}{2 a} + \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right )^{2} - 6 a^{3} e^{2} \left (- \frac{e}{2 a} + \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) - 2 a^{2} c d^{2} \left (- \frac{e}{2 a} + \frac{d \sqrt{- a^{3} c}}{2 a^{3}}\right ) - 6 a^{2} e^{3} + 2 a c d^{2} e}{9 a c d e^{2} + c^{2} d^{3}} \right )} - \frac{d}{a x} + \frac{e \log{\left (x \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/x**2/(c*x**2+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.269637, size = 74, normalized size = 1.25 \[ -\frac{c d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} a} - \frac{e{\rm ln}\left (c x^{2} + a\right )}{2 \, a} + \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a} - \frac{d}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c*x^2 + a)*x^2),x, algorithm="giac")
[Out]