3.879 \(\int \frac{\sqrt{d+e x}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=104 \[ \frac{\sqrt{d+e x}}{c d e \sqrt{c d^2-c e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{2} c^{3/2} d^{3/2} e} \]

[Out]

Sqrt[d + e*x]/(c*d*e*Sqrt[c*d^2 - c*e^2*x^2]) - ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/
(Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])]/(Sqrt[2]*c^(3/2)*d^(3/2)*e)

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Rubi [A]  time = 0.150352, antiderivative size = 104, 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 \[ \frac{\sqrt{d+e x}}{c d e \sqrt{c d^2-c e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{2} c^{3/2} d^{3/2} e} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

Sqrt[d + e*x]/(c*d*e*Sqrt[c*d^2 - c*e^2*x^2]) - ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/
(Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])]/(Sqrt[2]*c^(3/2)*d^(3/2)*e)

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Rubi in Sympy [A]  time = 16.8193, size = 92, normalized size = 0.88 \[ \frac{\sqrt{d + e x}}{c d e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{2 c^{\frac{3}{2}} d^{\frac{3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

sqrt(d + e*x)/(c*d*e*sqrt(c*d**2 - c*e**2*x**2)) - sqrt(2)*atanh(sqrt(2)*sqrt(c*
d**2 - c*e**2*x**2)/(2*sqrt(c)*sqrt(d)*sqrt(d + e*x)))/(2*c**(3/2)*d**(3/2)*e)

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Mathematica [A]  time = 0.0662998, size = 110, normalized size = 1.06 \[ \frac{2 \sqrt{d} \sqrt{d+e x}-\sqrt{2} \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{2 c d^{3/2} e \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(2*Sqrt[d]*Sqrt[d + e*x] - 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])])/(2*c*d^(3/2)*e*Sqrt[c*(d^2 - e^2*x^2)])

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Maple [A]  time = 0.026, size = 98, normalized size = 0.9 \[{\frac{1}{2\,{c}^{2} \left ( ex-d \right ) ed}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( \sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{- \left ( ex-d \right ) c}{\frac{1}{\sqrt{cd}}}} \right ) \sqrt{- \left ( ex-d \right ) c}-2\,\sqrt{cd} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

1/2/(e*x+d)^(1/2)*(-c*(e^2*x^2-d^2))^(1/2)*(2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/
2)*2^(1/2)/(c*d)^(1/2))*(-(e*x-d)*c)^(1/2)-2*(c*d)^(1/2))/c^2/(e*x-d)/e/d/(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(sqrt(e*x + d)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228437, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d} -{\left (c e^{2} x^{2} - c d^{2}\right )} \log \left (-\frac{\sqrt{2}{\left (e^{2} x^{2} - 2 \, d e x - 3 \, d^{2}\right )} \sqrt{c d} + 4 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )\right )}}{4 \,{\left (c^{2} d e^{3} x^{2} - c^{2} d^{3} e\right )} \sqrt{c d}}, -\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d} -{\left (c e^{2} x^{2} - c d^{2}\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d}}{c e^{2} x^{2} - c d^{2}}\right )\right )}}{2 \,{\left (c^{2} d e^{3} x^{2} - c^{2} d^{3} e\right )} \sqrt{-c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/4*sqrt(2)*(2*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(c*d)*sqrt(e*x + d) - (c*e
^2*x^2 - c*d^2)*log(-(sqrt(2)*(e^2*x^2 - 2*d*e*x - 3*d^2)*sqrt(c*d) + 4*sqrt(-c*
e^2*x^2 + c*d^2)*sqrt(e*x + d)*d)/(e^2*x^2 + 2*d*e*x + d^2)))/((c^2*d*e^3*x^2 -
c^2*d^3*e)*sqrt(c*d)), -1/2*sqrt(2)*(sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(-c*d)
*sqrt(e*x + d) - (c*e^2*x^2 - c*d^2)*arctan(sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqr
t(-c*d)*sqrt(e*x + d)/(c*e^2*x^2 - c*d^2)))/((c^2*d*e^3*x^2 - c^2*d^3*e)*sqrt(-c
*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(1/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Integral(sqrt(d + e*x)/(-c*(-d + e*x)*(d + e*x))**(3/2), x)

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GIAC/XCAS [A]  time = 0.619948, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x