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

Optimal. Leaf size=11 \[ \frac{c \log (d+e x)}{e} \]

[Out]

(c*Log[d + e*x])/e

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Rubi [A]  time = 0.01888, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{c \log (d+e x)}{e} \]

Antiderivative was successfully verified.

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

[Out]

(c*Log[d + e*x])/e

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Rubi in Sympy [A]  time = 10.1403, size = 8, normalized size = 0.73 \[ \frac{c \log{\left (d + e x \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*log(d + e*x)/e

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Mathematica [A]  time = 0.00223444, size = 11, normalized size = 1. \[ \frac{c \log (d+e x)}{e} \]

Antiderivative was successfully verified.

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

[Out]

(c*Log[d + e*x])/e

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Maple [A]  time = 0.002, size = 12, normalized size = 1.1 \[{\frac{c\ln \left ( ex+d \right ) }{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*ln(e*x+d)/e

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Maxima [A]  time = 0.697294, size = 15, normalized size = 1.36 \[ \frac{c \log \left (e x + d\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*log(e*x + d)/e

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Fricas [A]  time = 0.247163, size = 15, normalized size = 1.36 \[ \frac{c \log \left (e x + d\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*log(e*x + d)/e

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Sympy [A]  time = 0.142877, size = 8, normalized size = 0.73 \[ \frac{c \log{\left (d + e x \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*log(d + e*x)/e

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GIAC/XCAS [A]  time = 0.210933, size = 16, normalized size = 1.45 \[ c e^{\left (-1\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*e^(-1)*ln(abs(x*e + d))