3.856 \(\int \frac{\sqrt{c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx\)

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]

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

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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]

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

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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]

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

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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]

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

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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]

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))*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)^(3/2)/(-(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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

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]

Integral(sqrt(-c*(-d + e*x)*(d + e*x))/(d + e*x)**(5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-c e^{2} x^{2} + c d^{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

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]

integrate(sqrt(-c*e^2*x^2 + c*d^2)/(e*x + d)^(5/2), x)