3.872 \(\int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{c} \sqrt{d} e} \]

[Out]

-((Sqrt[2]*ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d + e*x
])])/(Sqrt[c]*Sqrt[d]*e))

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Rubi [A]  time = 0.0867285, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{c} \sqrt{d} e} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2]),x]

[Out]

-((Sqrt[2]*ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d + e*x
])])/(Sqrt[c]*Sqrt[d]*e))

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Rubi in Sympy [A]  time = 9.61222, size = 61, normalized size = 0.94 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{\sqrt{c} \sqrt{d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(1/2)/(-c*e**2*x**2+c*d**2)**(1/2),x)

[Out]

-sqrt(2)*atanh(sqrt(2)*sqrt(c*d**2 - c*e**2*x**2)/(2*sqrt(c)*sqrt(d)*sqrt(d + e*
x)))/(sqrt(c)*sqrt(d)*e)

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Mathematica [A]  time = 0.0488022, size = 86, normalized size = 1.32 \[ -\frac{\sqrt{2} \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} e \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2]),x]

[Out]

-((Sqrt[2]*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt
[d + e*x])])/(Sqrt[d]*e*Sqrt[c*(d^2 - e^2*x^2)]))

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Maple [A]  time = 0.018, size = 74, normalized size = 1.1 \[ -{\frac{\sqrt{2}}{e}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) }{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{- \left ( ex-d \right ) c}{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(1/2),x)

[Out]

-1/(e*x+d)^(1/2)*(-c*(e^2*x^2-d^2))^(1/2)/(-(e*x-d)*c)^(1/2)/e*2^(1/2)/(c*d)^(1/
2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c*d)^(1/2))

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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(1/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226284, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{2} \sqrt{\frac{1}{c d}} \log \left (-\frac{e^{2} x^{2} - 2 \, d e x + 2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d \sqrt{\frac{1}{c d}} - 3 \, d^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, e}, \frac{\sqrt{2} \sqrt{-\frac{1}{c d}} \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{{\left (c e^{2} x^{2} - c d^{2}\right )} \sqrt{-\frac{1}{c d}}}\right )}{e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[1/2*sqrt(2)*sqrt(1/(c*d))*log(-(e^2*x^2 - 2*d*e*x + 2*sqrt(2)*sqrt(-c*e^2*x^2 +
c*d^2)*sqrt(e*x + d)*d*sqrt(1/(c*d)) - 3*d^2)/(e^2*x^2 + 2*d*e*x + d^2))/e, sqr
t(2)*sqrt(-1/(c*d))*arctan(sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/((c*e^
2*x^2 - c*d^2)*sqrt(-1/(c*d))))/e]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(1/2)/(-c*e**2*x**2+c*d**2)**(1/2),x)

[Out]

Integral(1/(sqrt(-c*(-d + e*x)*(d + e*x))*sqrt(d + e*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)), x)