3.873 \(\int \frac{1}{(d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}} \, dx\)

Optimal. Leaf size=109 \[ -\frac{\sqrt{c d^2-c e^2 x^2}}{2 c d e (d+e x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{2 \sqrt{2} \sqrt{c} d^{3/2} e} \]

[Out]

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

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Rubi [A]  time = 0.149939, antiderivative size = 109, 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 \[ -\frac{\sqrt{c d^2-c e^2 x^2}}{2 c d e (d+e x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{2 \sqrt{2} \sqrt{c} d^{3/2} e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.074, size = 95, normalized size = 0.87 \[ - \frac{\sqrt{c d^{2} - c e^{2} x^{2}}}{2 c d e \left (d + e x\right )^{\frac{3}{2}}} - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{4 \sqrt{c} d^{\frac{3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c*d**2 - c*e**2*x**2)/(2*c*d*e*(d + e*x)**(3/2)) - sqrt(2)*atanh(sqrt(2)*s
qrt(c*d**2 - c*e**2*x**2)/(2*sqrt(c)*sqrt(d)*sqrt(d + e*x)))/(4*sqrt(c)*d**(3/2)
*e)

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Mathematica [A]  time = 0.0921135, size = 122, normalized size = 1.12 \[ \frac{-\sqrt{2} \sqrt{d+e x} \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )-2 \sqrt{d} (d-e x)}{4 d^{3/2} e \sqrt{d+e x} \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d]*(d - e*x) - Sqrt[2]*Sqrt[d + e*x]*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d
^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(4*d^(3/2)*e*Sqrt[d + e*x]*Sqrt[
c*(d^2 - e^2*x^2)])

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Maple [A]  time = 0.023, size = 133, normalized size = 1.2 \[ -{\frac{1}{4\,ced}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( \sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{- \left ( ex-d \right ) c}{\frac{1}{\sqrt{cd}}}} \right ) xce+cd\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{- \left ( ex-d \right ) c}{\frac{1}{\sqrt{cd}}}} \right ) +2\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd} \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/(e*x+d)^(3/2)*(-c*(e^2*x^2-d^2))^(1/2)/c*(2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^
(1/2)*2^(1/2)/(c*d)^(1/2))*x*c*e+c*d*2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1
/2)/(c*d)^(1/2))+2*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2))/(-(e*x-d)*c)^(1/2)/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(1/(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 = 0.222747, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d} -{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \log \left (-\frac{\sqrt{2}{\left (e^{2} x^{2} - 2 \, d e x - 3 \, d^{2}\right )} \sqrt{c d} + 4 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )\right )}}{8 \,{\left (c d e^{3} x^{2} + 2 \, c d^{2} e^{2} x + c d^{3} e\right )} \sqrt{c d}}, -\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d} -{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d}}{c e^{2} x^{2} - c d^{2}}\right )\right )}}{4 \,{\left (c d e^{3} x^{2} + 2 \, c d^{2} e^{2} x + c d^{3} e\right )} \sqrt{-c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-c*e^2*x^2 + c*d^2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[-1/8*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 + 2*c*d*e*x + 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*d*
e^3*x^2 + 2*c*d^2*e^2*x + c*d^3*e)*sqrt(c*d)), -1/4*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 + 2*c*d*e*x + c*d^2)*arctan(
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)))/
((c*d*e^3*x^2 + 2*c*d^2*e^2*x + c*d^3*e)*sqrt(-c*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(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. \[ \int \frac{1}{\sqrt{-c e^{2} x^{2} + c d^{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-c*e^2*x^2 + c*d^2)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-c*e^2*x^2 + c*d^2)*(e*x + d)^(3/2)), x)