3.1399 \(\int \frac{1}{(c e+d e x)^{3/2} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\)

Optimal. Leaf size=107 \[ -\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{d e \sqrt{c e+d e x}}+\frac{2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d e^{3/2}}-\frac{2 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d e^{3/2}} \]

[Out]

(-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(d*e*Sqrt[c*e + d*e*x]) - (2*EllipticE[Ar
cSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(d*e^(3/2)) + (2*EllipticF[ArcSin[Sqrt[c*e
+ d*e*x]/Sqrt[e]], -1])/(d*e^(3/2))

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Rubi [A]  time = 0.277698, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ -\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{d e \sqrt{c e+d e x}}+\frac{2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d e^{3/2}}-\frac{2 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d e^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(d*e*Sqrt[c*e + d*e*x]) - (2*EllipticE[Ar
cSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(d*e^(3/2)) + (2*EllipticF[ArcSin[Sqrt[c*e
+ d*e*x]/Sqrt[e]], -1])/(d*e^(3/2))

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Rubi in Sympy [A]  time = 59.1368, size = 99, normalized size = 0.93 \[ - \frac{2 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d e \sqrt{c e + d e x}} - \frac{2 E\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{d e^{\frac{3}{2}}} + \frac{2 F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{d e^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(d*e*sqrt(c*e + d*e*x)) - 2*elliptic_e(
asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/(d*e**(3/2)) + 2*elliptic_f(asin(sqrt(c*e +
d*e*x)/sqrt(e)), -1)/(d*e**(3/2))

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Mathematica [A]  time = 0.0804994, size = 80, normalized size = 0.75 \[ -\frac{2 \left (\sqrt{1-(c+d x)^2}-\sqrt{c+d x} F\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )+\sqrt{c+d x} E\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )\right )}{d e \sqrt{e (c+d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.069, size = 193, normalized size = 1.8 \[{\frac{1}{d{e}^{2} \left ({x}^{3}{d}^{3}+3\,c{d}^{2}{x}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) } \left ( -2\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) +\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticE} \left ({\frac{1}{2}\sqrt{2\,dx+2\,c+2}},\sqrt{2} \right ) -2\,{d}^{2}{x}^{2}-4\,cdx-2\,{c}^{2}+2 \right ) \sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticE(1/2*(-2*d*x
-2*c+2)^(1/2),2^(1/2))+(-2*d*x-2*c+2)^(1/2)*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/2)*E
llipticE(1/2*(2*d*x+2*c+2)^(1/2),2^(1/2))-2*d^2*x^2-4*c*d*x-2*c^2+2)*(e*(d*x+c))
^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d/e^2/(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3-d
*x-c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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