Optimal. Leaf size=99 \[ \frac{2 \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{2 \sqrt{2} \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e} \]
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Rubi [A] time = 0.160846, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{2 \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{2 \sqrt{2} \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c*d^2 - c*e^2*x^2]/(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 15.7176, size = 90, normalized size = 0.91 \[ - \frac{2 \sqrt{2} \sqrt{c} \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{e} + \frac{2 \sqrt{c d^{2} - c e^{2} x^{2}}}{e \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-c*e**2*x**2+c*d**2)**(1/2)/(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.119321, size = 98, normalized size = 0.99 \[ \frac{2 \sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{1}{\sqrt{d+e x}}-\frac{\sqrt{2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c*d^2 - c*e^2*x^2]/(d + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.033, size = 97, normalized size = 1. \[ -2\,{\frac{\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) }}{\sqrt{ex+d}\sqrt{- \left ( ex-d \right ) c}e\sqrt{cd}} \left ( cd\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) -\sqrt{- \left ( ex-d \right ) c}\sqrt{cd} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-c*e^2*x^2+c*d^2)^(1/2)/(e*x+d)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 + c*d^2)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.242035, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, c e^{2} x^{2} - 2 \, c d^{2} - \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d} \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{\sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e}, -\frac{2 \,{\left (c e^{2} x^{2} - c d^{2} - \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d} \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-c d}}\right )\right )}}{\sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 + c*d^2)/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*e**2*x**2+c*d**2)**(1/2)/(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-c e^{2} x^{2} + c d^{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 + c*d^2)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]