3.989 \(\int \frac{(d+e x)^3}{c d^2+2 c d e x+c e^2 x^2} \, dx\)

Optimal. Leaf size=17 \[ \frac{(d+e x)^2}{2 c e} \]

[Out]

(d + e*x)^2/(2*c*e)

_______________________________________________________________________________________

Rubi [A]  time = 0.0138367, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{(d+e x)^2}{2 c e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]

[Out]

(d + e*x)^2/(2*c*e)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{e \int x\, dx}{c} + \frac{\int d\, dx}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2),x)

[Out]

e*Integral(x, x)/c + Integral(d, x)/c

_______________________________________________________________________________________

Mathematica [A]  time = 0.00114778, size = 16, normalized size = 0.94 \[ \frac{d x+\frac{e x^2}{2}}{c} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]

[Out]

(d*x + (e*x^2)/2)/c

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \[{\frac{1}{c} \left ({\frac{e{x}^{2}}{2}}+dx \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2),x)

[Out]

1/c*(1/2*e*x^2+d*x)

_______________________________________________________________________________________

Maxima [A]  time = 0.698572, size = 20, normalized size = 1.18 \[ \frac{e x^{2} + 2 \, d x}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="maxima")

[Out]

1/2*(e*x^2 + 2*d*x)/c

_______________________________________________________________________________________

Fricas [A]  time = 0.203636, size = 20, normalized size = 1.18 \[ \frac{e x^{2} + 2 \, d x}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="fricas")

[Out]

1/2*(e*x^2 + 2*d*x)/c

_______________________________________________________________________________________

Sympy [A]  time = 0.140857, size = 12, normalized size = 0.71 \[ \frac{d x}{c} + \frac{e x^{2}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2),x)

[Out]

d*x/c + e*x**2/(2*c)

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError