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

Optimal. Leaf size=38 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 c e (d+e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0503122, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 c e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.33267, size = 32, normalized size = 0.84 \[ - \frac{2 \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{3 c e \left (d + e x\right )^{\frac{3}{2}}} \]

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)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.032708, size = 40, normalized size = 1.05 \[ -\frac{2 (d-e x) \sqrt{c \left (d^2-e^2 x^2\right )}}{3 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 36, normalized size = 1. \[ -{\frac{-2\,ex+2\,d}{3\,e}\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}{\frac{1}{\sqrt{ex+d}}}} \]

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)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.733651, size = 35, normalized size = 0.92 \[ \frac{2 \,{\left (\sqrt{c} e x - \sqrt{c} d\right )} \sqrt{-e x + d}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.223597, size = 82, normalized size = 2.16 \[ -\frac{2 \,{\left (c e^{3} x^{3} - c d e^{2} x^{2} - c d^{2} e x + c d^{3}\right )}}{3 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(c*e^3*x^3 - c*d*e^2*x^2 - c*d^2*e*x + c*d^3)/(sqrt(-c*e^2*x^2 + c*d^2)*sqr
t(e*x + d)*e)

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

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)**(1/2),x)

[Out]

Integral(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. \[ \int \frac{\sqrt{-c e^{2} x^{2} + c d^{2}}}{\sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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