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

Optimal. Leaf size=89 \[ -\frac{2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.87695, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 \sqrt{d + e x} \int ^{\frac{1}{\sqrt{x}}} \frac{x^{2}}{\sqrt{1 - \frac{x^{4}}{c^{2}}} \sqrt{d x^{2} + e}}\, dx}{\sqrt{x} \sqrt{\frac{d}{x} + e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(d + e*x)*Integral(x**2/(sqrt(1 - x**4/c**2)*sqrt(d*x**2 + e)), (x, 1/sqr
t(x)))/(sqrt(x)*sqrt(d/x + e))

_______________________________________________________________________________________

Mathematica [C]  time = 0.5741, size = 188, normalized size = 2.11 \[ -\frac{2 i (d+e x) \sqrt{\frac{e (c x-1)}{c (d+e x)}} \sqrt{\frac{c e x+e}{c d+c e x}} \left (F\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{c d+e}{c}}}{\sqrt{d+e x}}\right )|\frac{c d-e}{c d+e}\right )-\Pi \left (\frac{c d}{c d+e};i \sinh ^{-1}\left (\frac{\sqrt{-\frac{c d+e}{c}}}{\sqrt{d+e x}}\right )|\frac{c d-e}{c d+e}\right )\right )}{d x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c d+e}{c}}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*I)*Sqrt[(e*(-1 + c*x))/(c*(d + e*x))]*(d + e*x)*Sqrt[(e + c*e*x)/(c*d + c*e
*x)]*(EllipticF[I*ArcSinh[Sqrt[-((c*d + e)/c)]/Sqrt[d + e*x]], (c*d - e)/(c*d +
e)] - EllipticPi[(c*d)/(c*d + e), I*ArcSinh[Sqrt[-((c*d + e)/c)]/Sqrt[d + e*x]],
(c*d - e)/(c*d + e)]))/(d*Sqrt[-((c*d + e)/c)]*Sqrt[1 - 1/(c^2*x^2)]*x)

_______________________________________________________________________________________

Maple [A]  time = 0.165, size = 148, normalized size = 1.7 \[ -2\,{\frac{cd-e}{x\sqrt{ex+d}cd}{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( ex+d \right ) c}{cd-e}}},{\frac{cd-e}{cd}},\sqrt{{\frac{cd-e}{cd+e}}} \right ) \sqrt{-{\frac{ \left ( cx+1 \right ) e}{cd-e}}}\sqrt{-{\frac{ \left ( cx-1 \right ) e}{cd+e}}}\sqrt{{\frac{ \left ( ex+d \right ) c}{cd-e}}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{e x + d} x^{2} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*sqrt(-(-1 + 1/(c*x))*(1 + 1/(c*x)))*sqrt(d + e*x)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{e x + d} x^{2} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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