3.1393 \(\int \frac{1}{(c e+d e x)^{5/2} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\)

Optimal. Leaf size=80 \[ \frac{2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{3 d e^{5/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}} \]

[Out]

(-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(3*d*e*(c*e + d*e*x)^(3/2)) + (2*Elliptic
F[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(3*d*e^(5/2))

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Rubi [A]  time = 0.174724, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ \frac{2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{3 d e^{5/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*e + d*e*x)^(5/2)*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])/(3*d*e*(c*e + d*e*x)^(3/2)) + (2*Elliptic
F[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(3*d*e^(5/2))

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Rubi in Sympy [A]  time = 43.225, size = 71, normalized size = 0.89 \[ - \frac{2 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{3 d e \left (c e + d e x\right )^{\frac{3}{2}}} + \frac{2 F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{3 d e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*e*x+c*e)**(5/2)/(-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)/(3*d*e*(c*e + d*e*x)**(3/2)) + 2*ellipt
ic_f(asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/(3*d*e**(5/2))

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Mathematica [A]  time = 0.237762, size = 105, normalized size = 1.31 \[ \frac{2 \left (-\sqrt{\frac{c^2+2 c d x+d^2 x^2-1}{(c+d x)^2}} (c+d x)^{5/2} F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{c+d x}}\right )\right |-1\right )+c^2+2 c d x+d^2 x^2-1\right )}{3 d e \sqrt{1-(c+d x)^2} (e (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*e + d*e*x)^(5/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]

[Out]

(2*(-1 + c^2 + 2*c*d*x + d^2*x^2 - (c + d*x)^(5/2)*Sqrt[(-1 + c^2 + 2*c*d*x + d^
2*x^2)/(c + d*x)^2]*EllipticF[ArcSin[1/Sqrt[c + d*x]], -1]))/(3*d*e*(e*(c + d*x)
)^(3/2)*Sqrt[1 - (c + d*x)^2])

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Maple [B]  time = 0.076, size = 290, normalized size = 3.6 \[ -{\frac{1}{3\,{e}^{3} \left ( dx+c \right ) ^{2} \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2}-1 \right ) d} \left ( 2\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) xd+3\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) xd+2\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) c+3\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) c+2\,{d}^{2}{x}^{2}+4\,cdx+2\,{c}^{2}-2 \right ) \sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1}\sqrt{e \left ( dx+c \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*e*x+c*e)^(5/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{5}{2}}}\,{d x} \]

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)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(5/2)), x)

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

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)^(5/2)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (e \left (c + d x\right )\right )^{\frac{5}{2}} \sqrt{- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*e*x+c*e)**(5/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

Integral(1/((e*(c + d*x))**(5/2)*sqrt(-(c + d*x - 1)*(c + d*x + 1))), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{5}{2}}}\,{d x} \]

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)^(5/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(5/2)), x)