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

Optimal. Leaf size=99 \[ \frac{2 \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{2 \sqrt{2} \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e} \]

[Out]

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

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

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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]

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

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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]

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

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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]

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

[Out]

Exception raised: ValueError

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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]

[-(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(-(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)*s
qrt(c*d)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)))/(sqrt(-c*e^2*x^2 + c*d^2)*sq
rt(e*x + d)*e), -2*(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(sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*d/((e^2*
x^2 - d^2)*sqrt(-c*d))))/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*e)]

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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]

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

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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]

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