Optimal. Leaf size=32 \[ -\frac{1}{c e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
[Out]
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Rubi [A] time = 0.0208159, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ -\frac{1}{c e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 9.43641, size = 31, normalized size = 0.97 \[ - \frac{1}{c e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0193484, size = 21, normalized size = 0.66 \[ -\frac{1}{c e \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.005, size = 35, normalized size = 1.1 \[ -{\frac{ \left ( ex+d \right ) ^{2}}{e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x)
[Out]
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Maxima [A] time = 0.678, size = 41, normalized size = 1.28 \[ -\frac{1}{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219641, size = 74, normalized size = 2.31 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c^{2} e^{3} x^{2} + 2 \, c^{2} d e^{2} x + c^{2} d^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.3923, size = 42, normalized size = 1.31 \[ \begin{cases} - \frac{1}{c e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{d x}{\left (c d^{2}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.255445, size = 55, normalized size = 1.72 \[ \frac{2 \, C_{0} d e^{\left (-1\right )} + 2 \, C_{0} x - \frac{e^{\left (-1\right )}}{c}}{\sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="giac")
[Out]