Optimal. Leaf size=33 \[ \frac{c d \log (d+e x)}{e^2}-\frac{a-\frac{c d^2}{e^2}}{d+e x} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.072832, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{c d \log (d+e x)}{e^2}-\frac{a-\frac{c d^2}{e^2}}{d+e x} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 15.7369, size = 31, normalized size = 0.94 \[ \frac{c d \log{\left (d + e x \right )}}{e^{2}} - \frac{a e^{2} - c d^{2}}{e^{2} \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0244201, size = 36, normalized size = 1.09 \[ \frac{c d^2-a e^2}{e^2 (d+e x)}+\frac{c d \log (d+e x)}{e^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^3,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 39, normalized size = 1.2 \[{\frac{cd\ln \left ( ex+d \right ) }{{e}^{2}}}-{\frac{a}{ex+d}}+{\frac{c{d}^{2}}{{e}^{2} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.725454, size = 53, normalized size = 1.61 \[ \frac{c d \log \left (e x + d\right )}{e^{2}} + \frac{c d^{2} - a e^{2}}{e^{3} x + d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.206861, size = 59, normalized size = 1.79 \[ \frac{c d^{2} - a e^{2} +{\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{e^{3} x + d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.51148, size = 32, normalized size = 0.97 \[ \frac{c d \log{\left (d + e x \right )}}{e^{2}} - \frac{a e^{2} - c d^{2}}{d e^{2} + e^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.211926, size = 70, normalized size = 2.12 \[ c d e^{\left (-2\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^3,x, algorithm="giac")
[Out]