Optimal. Leaf size=30 \[ -\frac{c}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
[Out]
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Rubi [A] time = 0.065545, antiderivative size = 30, 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 \[ -\frac{c}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 18.2955, size = 29, normalized size = 0.97 \[ - \frac{c}{e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2)/(e*x+d)**3,x)
[Out]
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Mathematica [A] time = 0.0142786, size = 19, normalized size = 0.63 \[ -\frac{c}{e \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^3,x]
[Out]
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Maple [A] time = 0.004, size = 35, normalized size = 1.2 \[ -{\frac{1}{ \left ( ex+d \right ) ^{2}e}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]
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)^3,x)
[Out]
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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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224338, size = 62, normalized size = 2.07 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e} \]
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)^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c \left (d + e x\right )^{2}}}{\left (d + e x\right )^{3}}\, dx \]
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)**3,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(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)^3,x, algorithm="giac")
[Out]