Optimal. Leaf size=96 \[ -\frac{3 \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{c d \log \left (a+c x^2\right )}{a^3}-\frac{2 c d \log (x)}{a^3}-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.195158, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{3 \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{c d \log \left (a+c x^2\right )}{a^3}-\frac{2 c d \log (x)}{a^3}-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(x^3*(a + c*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 33.69, size = 92, normalized size = 0.96 \[ \frac{d + e x}{2 a x^{2} \left (a + c x^{2}\right )} - \frac{d}{a^{2} x^{2}} - \frac{3 e}{2 a^{2} x} - \frac{2 c d \log{\left (x \right )}}{a^{3}} + \frac{c d \log{\left (a + c x^{2} \right )}}{a^{3}} - \frac{3 \sqrt{c} e \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/x**3/(c*x**2+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.202738, size = 82, normalized size = 0.85 \[ -\frac{\frac{a c (d+e x)}{a+c x^2}-2 c d \log \left (a+c x^2\right )+3 \sqrt{a} \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )+\frac{a d}{x^2}+\frac{2 a e}{x}+4 c d \log (x)}{2 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(x^3*(a + c*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.018, size = 97, normalized size = 1. \[ -{\frac{e}{{a}^{2}x}}-{\frac{d}{2\,{a}^{2}{x}^{2}}}-2\,{\frac{cd\ln \left ( x \right ) }{{a}^{3}}}-{\frac{xec}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{cd}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{cd\ln \left ( c{x}^{2}+a \right ) }{{a}^{3}}}-{\frac{3\,ce}{2\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/x^3/(c*x^2+a)^2,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)^2*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.30353, size = 1, normalized size = 0.01 \[ \left [-\frac{6 \, a c e x^{3} + 4 \, a c d x^{2} + 4 \, a^{2} e x + 2 \, a^{2} d - 3 \,{\left (a c e x^{4} + a^{2} e x^{2}\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} - 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) - 4 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (c x^{2} + a\right ) + 8 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{3} c x^{4} + a^{4} x^{2}\right )}}, -\frac{3 \, a c e x^{3} + 2 \, a c d x^{2} + 2 \, a^{2} e x + a^{2} d + 3 \,{\left (a c e x^{4} + a^{2} e x^{2}\right )} \sqrt{\frac{c}{a}} \arctan \left (\frac{c x}{a \sqrt{\frac{c}{a}}}\right ) - 2 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (c x^{2} + a\right ) + 4 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} c x^{4} + a^{4} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c*x^2 + a)^2*x^3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 6.4664, size = 396, normalized size = 4.12 \[ \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) \log{\left (x + \frac{- 64 a^{6} d \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right )^{2} - 12 a^{4} e^{2} \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 64 a^{3} c d^{2} \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 24 a c d e^{2} + 128 c^{2} d^{3}}{9 a c e^{3} + 144 c^{2} d^{2} e} \right )} + \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) \log{\left (x + \frac{- 64 a^{6} d \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right )^{2} - 12 a^{4} e^{2} \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 64 a^{3} c d^{2} \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 24 a c d e^{2} + 128 c^{2} d^{3}}{9 a c e^{3} + 144 c^{2} d^{2} e} \right )} - \frac{a d + 2 a e x + 2 c d x^{2} + 3 c e x^{3}}{2 a^{3} x^{2} + 2 a^{2} c x^{4}} - \frac{2 c d \log{\left (x \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/x**3/(c*x**2+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.274071, size = 128, normalized size = 1.33 \[ -\frac{3 \, c \arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{2 \, \sqrt{a c} a^{2}} + \frac{c d{\rm ln}\left (c x^{2} + a\right )}{a^{3}} - \frac{2 \, c d{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{3 \, a c x^{3} e + 2 \, a c d x^{2} + 2 \, a^{2} x e + a^{2} d}{2 \,{\left (c x^{2} + a\right )} a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((c*x^2 + a)^2*x^3),x, algorithm="giac")
[Out]