Optimal. Leaf size=98 \[ \frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{2} \sqrt{d} e}-\frac{\sqrt{c d^2-c e^2 x^2}}{e (d+e x)^{3/2}} \]
[Out]
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Rubi [A] time = 0.146002, antiderivative size = 98, 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 \[ \frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{2} \sqrt{d} e}-\frac{\sqrt{c d^2-c e^2 x^2}}{e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c*d^2 - c*e^2*x^2]/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 15.4399, size = 88, normalized size = 0.9 \[ \frac{\sqrt{2} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{2 \sqrt{d} e} - \frac{\sqrt{c d^{2} - c e^{2} x^{2}}}{e \left (d + e x\right )^{\frac{3}{2}}} \]
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)**(5/2),x)
[Out]
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Mathematica [A] time = 0.144369, size = 101, normalized size = 1.03 \[ \frac{\sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} \sqrt{d^2-e^2 x^2}}-\frac{2}{(d+e x)^{3/2}}\right )}{2 e} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c*d^2 - c*e^2*x^2]/(d + e*x)^(5/2),x]
[Out]
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Maple [A] time = 0.026, size = 127, normalized size = 1.3 \[{\frac{1}{2\,e}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( \sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{- \left ( ex-d \right ) c}{\frac{1}{\sqrt{cd}}}} \right ) xce+cd\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{- \left ( ex-d \right ) c}{\frac{1}{\sqrt{cd}}}} \right ) -2\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd} \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]
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)^(5/2),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)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234465, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{1}{2}}{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{\frac{c}{d}} \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} - 4 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d \sqrt{\frac{c}{d}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}}, -\frac{\sqrt{\frac{1}{2}}{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{-\frac{c}{d}} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-\frac{c}{d}}}\right ) + \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 + c*d^2)/(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]
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)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-c e^{2} x^{2} + c d^{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 + c*d^2)/(e*x + d)^(5/2),x, algorithm="giac")
[Out]