Optimal. Leaf size=29 \[ -\frac{1}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.0672767, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{1}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 18.8979, size = 29, normalized size = 1. \[ - \frac{1}{e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
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Mathematica [A] time = 0.0202188, size = 18, normalized size = 0.62 \[ -\frac{1}{e \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]
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Maple [A] time = 0.004, size = 28, normalized size = 1. \[ -{\frac{1}{e}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.68584, size = 26, normalized size = 0.9 \[ -\frac{1}{\sqrt{c} e^{2} x + \sqrt{c} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.214857, size = 66, normalized size = 2.28 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c e^{3} x^{2} + 2 \, c d e^{2} x + c d^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.12497, size = 41, normalized size = 1.41 \[ \begin{cases} - \frac{1}{e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x}{d \sqrt{c d^{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
[Out]
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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(1/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)),x, algorithm="giac")
[Out]