Optimal. Leaf size=45 \[ \frac{d (c d-b e) \log (d+e x)}{e^3}-\frac{x (c d-b e)}{e^2}+\frac{c x^2}{2 e} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0962125, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{d (c d-b e) \log (d+e x)}{e^3}-\frac{x (c d-b e)}{e^2}+\frac{c x^2}{2 e} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)/(d + e*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{c \int x\, dx}{e} - \frac{d \left (b e - c d\right ) \log{\left (d + e x \right )}}{e^{3}} + \left (b e - c d\right ) \int \frac{1}{e^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)/(e*x+d),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0247568, size = 41, normalized size = 0.91 \[ \frac{e x (2 b e-2 c d+c e x)+2 d (c d-b e) \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)/(d + e*x),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.004, size = 52, normalized size = 1.2 \[{\frac{c{x}^{2}}{2\,e}}+{\frac{bx}{e}}-{\frac{cdx}{{e}^{2}}}-{\frac{d\ln \left ( ex+d \right ) b}{{e}^{2}}}+{\frac{{d}^{2}\ln \left ( ex+d \right ) c}{{e}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)/(e*x+d),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.774945, size = 61, normalized size = 1.36 \[ \frac{c e x^{2} - 2 \,{\left (c d - b e\right )} x}{2 \, e^{2}} + \frac{{\left (c d^{2} - b d e\right )} \log \left (e x + d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.218798, size = 63, normalized size = 1.4 \[ \frac{c e^{2} x^{2} - 2 \,{\left (c d e - b e^{2}\right )} x + 2 \,{\left (c d^{2} - b d e\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.32684, size = 37, normalized size = 0.82 \[ \frac{c x^{2}}{2 e} - \frac{d \left (b e - c d\right ) \log{\left (d + e x \right )}}{e^{3}} + \frac{x \left (b e - c d\right )}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)/(e*x+d),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.20736, size = 63, normalized size = 1.4 \[{\left (c d^{2} - b d e\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (c x^{2} e - 2 \, c d x + 2 \, b x e\right )} e^{\left (-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d),x, algorithm="giac")
[Out]