3.1023 \(\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=40 \[ \frac{c (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

(c*(d + e*x)*Log[d + e*x])/(e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

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Rubi [A]  time = 0.0773892, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ \frac{c (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)*Log[d + e*x])/(e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

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Rubi in Sympy [A]  time = 19.3953, size = 37, normalized size = 0.92 \[ \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}} \log{\left (d + e x \right )}}{e \left (d + e x\right )} \]

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)**(1/2)/(e*x+d)**2,x)

[Out]

sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)*log(d + e*x)/(e*(d + e*x))

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Mathematica [A]  time = 0.014389, size = 29, normalized size = 0.72 \[ \frac{c (d+e x) \log (d+e x)}{e \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)*Log[d + e*x])/(e*Sqrt[c*(d + e*x)^2])

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Maple [A]  time = 0.005, size = 40, normalized size = 1. \[{\frac{\ln \left ( ex+d \right ) }{ \left ( ex+d \right ) e}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.226427, size = 54, normalized size = 1.35 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{e^{2} x + d e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*log(e*x + d)/(e^2*x + d*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c \left (d + e x\right )^{2}}}{\left (d + e x\right )^{2}}\, dx \]

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)**(1/2)/(e*x+d)**2,x)

[Out]

Integral(sqrt(c*(d + e*x)**2)/(d + e*x)**2, x)

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GIAC/XCAS [A]  time = 0.265782, size = 11, normalized size = 0.28 \[ 2 \, C_{0} \sqrt{c} e^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*C_0*sqrt(c)*e^3