Optimal. Leaf size=42 \[ -\frac{(c d-b e)^2 \log (b+c x)}{b c^2}+\frac{d^2 \log (x)}{b}+\frac{e^2 x}{c} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0903946, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{(c d-b e)^2 \log (b+c x)}{b c^2}+\frac{d^2 \log (x)}{b}+\frac{e^2 x}{c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2/(b*x + c*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ e^{2} \int \frac{1}{c}\, dx + \frac{d^{2} \log{\left (x \right )}}{b} - \frac{\left (b e - c d\right )^{2} \log{\left (b + c x \right )}}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2/(c*x**2+b*x),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0302237, size = 42, normalized size = 1. \[ \frac{-(c d-b e)^2 \log (b+c x)+b c e^2 x+c^2 d^2 \log (x)}{b c^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2/(b*x + c*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 61, normalized size = 1.5 \[{\frac{{e}^{2}x}{c}}+{\frac{{d}^{2}\ln \left ( x \right ) }{b}}-{\frac{b\ln \left ( cx+b \right ){e}^{2}}{{c}^{2}}}+2\,{\frac{\ln \left ( cx+b \right ) de}{c}}-{\frac{\ln \left ( cx+b \right ){d}^{2}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2/(c*x^2+b*x),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.693834, size = 72, normalized size = 1.71 \[ \frac{e^{2} x}{c} + \frac{d^{2} \log \left (x\right )}{b} - \frac{{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \log \left (c x + b\right )}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(c*x^2 + b*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.226047, size = 72, normalized size = 1.71 \[ \frac{b c e^{2} x + c^{2} d^{2} \log \left (x\right ) -{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \log \left (c x + b\right )}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(c*x^2 + b*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.53863, size = 73, normalized size = 1.74 \[ \frac{e^{2} x}{c} + \frac{d^{2} \log{\left (x \right )}}{b} - \frac{\left (b e - c d\right )^{2} \log{\left (x + \frac{b c d^{2} + \frac{b \left (b e - c d\right )^{2}}{c}}{b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}} \right )}}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2/(c*x**2+b*x),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.205213, size = 73, normalized size = 1.74 \[ \frac{d^{2}{\rm ln}\left ({\left | x \right |}\right )}{b} + \frac{x e^{2}}{c} - \frac{{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(c*x^2 + b*x),x, algorithm="giac")
[Out]