3.1394 \(\int \frac{1}{(c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{10 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{21 d e^3 (c e+d e x)^{3/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}+\frac{10 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{21 d e^{9/2}} \]

[Out]

(-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(7*d*e*(c*e + d*e*x)^(7/2)) - (10*Sqrt[1
- c^2 - 2*c*d*x - d^2*x^2])/(21*d*e^3*(c*e + d*e*x)^(3/2)) + (10*EllipticF[ArcSi
n[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(21*d*e^(9/2))

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Rubi [A]  time = 0.265131, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ -\frac{10 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{21 d e^3 (c e+d e x)^{3/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}+\frac{10 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{21 d e^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*e + d*e*x)^(9/2)*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])/(7*d*e*(c*e + d*e*x)^(7/2)) - (10*Sqrt[1
- c^2 - 2*c*d*x - d^2*x^2])/(21*d*e^3*(c*e + d*e*x)^(3/2)) + (10*EllipticF[ArcSi
n[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(21*d*e^(9/2))

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Rubi in Sympy [A]  time = 63.2216, size = 114, normalized size = 0.9 \[ - \frac{2 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{7 d e \left (c e + d e x\right )^{\frac{7}{2}}} - \frac{10 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{21 d e^{3} \left (c e + d e x\right )^{\frac{3}{2}}} + \frac{10 F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{21 d e^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*e*x+c*e)**(9/2)/(-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)/(7*d*e*(c*e + d*e*x)**(7/2)) - 10*sqrt(
-c**2 - 2*c*d*x - d**2*x**2 + 1)/(21*d*e**3*(c*e + d*e*x)**(3/2)) + 10*elliptic_
f(asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/(21*d*e**(9/2))

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Mathematica [A]  time = 0.230343, size = 110, normalized size = 0.87 \[ \frac{(c+d x)^{9/2} \left (-\frac{2 \left (1-(c+d x)^2\right ) \left (5 (c+d x)^2+3\right )}{(c+d x)^{7/2}}-10 (c+d x) \sqrt{1-\frac{1}{(c+d x)^2}} F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{c+d x}}\right )\right |-1\right )\right )}{21 d \sqrt{1-(c+d x)^2} (e (c+d x))^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*e + d*e*x)^(9/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]

[Out]

((c + d*x)^(9/2)*((-2*(1 - (c + d*x)^2)*(3 + 5*(c + d*x)^2))/(c + d*x)^(7/2) - 1
0*(c + d*x)*Sqrt[1 - (c + d*x)^(-2)]*EllipticF[ArcSin[1/Sqrt[c + d*x]], -1]))/(2
1*d*(e*(c + d*x))^(9/2)*Sqrt[1 - (c + d*x)^2])

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Maple [B]  time = 0.072, size = 564, normalized size = 4.5 \[ -{\frac{1}{21\,{e}^{5} \left ( dx+c \right ) ^{4} \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2}-1 \right ) d} \left ( 19\,\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ){x}^{3}{d}^{3}+14\,\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ){x}^{3}{d}^{3}+57\,\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ){x}^{2}c{d}^{2}+42\,\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ){x}^{2}c{d}^{2}+57\,\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) x{c}^{2}d+42\,\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) x{c}^{2}d+10\,{d}^{4}{x}^{4}+19\,\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ){c}^{3}+14\,\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ){c}^{3}+40\,{x}^{3}c{d}^{3}+60\,{x}^{2}{c}^{2}{d}^{2}+40\,{c}^{3}dx-4\,{d}^{2}{x}^{2}+10\,{c}^{4}-8\,cdx-4\,{c}^{2}-6 \right ) \sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1}\sqrt{e \left ( dx+c \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*e*x+c*e)^(9/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)

[Out]

-1/21*(19*(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^3*d^3+14*(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^3*d^3+57*(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^2*c*d^2+42*(2*d*x+2*c+2)^(1/2)*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*Ell
ipticF(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*x^2*c*d^2+57*(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*c^2*d
+42*(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*c^2*d+10*d^4*x^4+19*(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+14*(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^3+40*x^3*c*d^3+60*x^2*c^2*d^2+40*c^3*d*x-4*d^2*x^2+10*c^4-8*c*d*x-4*
c^2-6)/e^5*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*(e*(d*x+c))^(1/2)/(d*x+c)^4/(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. \[ \int \frac{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(9/2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(9/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt{d e x + c e}}, x\right ) \]

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)^(9/2)),x, algorithm="fricas")

[Out]

integral(1/((d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x +
c^4*e^4)*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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*e*x+c*e)**(9/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(9/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(9/2)), x)