3.983 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=13 \[ \frac{c^2 \log (d+e x)}{e} \]

[Out]

(c^2*Log[d + e*x])/e

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Rubi [A]  time = 0.0153995, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{c^2 \log (d+e x)}{e} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^5,x]

[Out]

(c^2*Log[d + e*x])/e

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Rubi in Sympy [A]  time = 18.2218, size = 10, normalized size = 0.77 \[ \frac{c^{2} \log{\left (d + e x \right )}}{e} \]

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)**2/(e*x+d)**5,x)

[Out]

c**2*log(d + e*x)/e

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Mathematica [A]  time = 0.00222452, size = 13, normalized size = 1. \[ \frac{c^2 \log (d+e x)}{e} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^5,x]

[Out]

(c^2*Log[d + e*x])/e

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Maple [A]  time = 0.003, size = 14, normalized size = 1.1 \[{\frac{{c}^{2}\ln \left ( ex+d \right ) }{e}} \]

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)^2/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 0.698079, size = 18, normalized size = 1.38 \[ \frac{c^{2} \log \left (e x + d\right )}{e} \]

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)^2/(e*x + d)^5,x, algorithm="maxima")

[Out]

c^2*log(e*x + d)/e

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Fricas [A]  time = 0.212696, size = 18, normalized size = 1.38 \[ \frac{c^{2} \log \left (e x + d\right )}{e} \]

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)^2/(e*x + d)^5,x, algorithm="fricas")

[Out]

c^2*log(e*x + d)/e

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Sympy [A]  time = 0.203589, size = 10, normalized size = 0.77 \[ \frac{c^{2} \log{\left (d + e x \right )}}{e} \]

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)**2/(e*x+d)**5,x)

[Out]

c**2*log(d + e*x)/e

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GIAC/XCAS [A]  time = 0.215014, size = 35, normalized size = 2.69 \[ -c^{2} e^{\left (-1\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) \]

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)^2/(e*x + d)^5,x, algorithm="giac")

[Out]

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