Optimal. Leaf size=172 \[ -\frac{30 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{77 d e^5 (c e+d e x)^{3/2}}-\frac{18 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{77 d e^3 (c e+d e x)^{7/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{11 d e (c e+d e x)^{11/2}}+\frac{30 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{77 d e^{13/2}} \]
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Rubi [A] time = 0.357219, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ -\frac{30 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{77 d e^5 (c e+d e x)^{3/2}}-\frac{18 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{77 d e^3 (c e+d e x)^{7/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{11 d e (c e+d e x)^{11/2}}+\frac{30 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{77 d e^{13/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*e + d*e*x)^(13/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 83.5431, size = 156, normalized size = 0.91 \[ - \frac{2 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{11 d e \left (c e + d e x\right )^{\frac{11}{2}}} - \frac{18 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{77 d e^{3} \left (c e + d e x\right )^{\frac{7}{2}}} - \frac{30 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{77 d e^{5} \left (c e + d e x\right )^{\frac{3}{2}}} + \frac{30 F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{77 d e^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d*e*x+c*e)**(13/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.284468, size = 119, normalized size = 0.69 \[ \frac{(c+d x)^{13/2} \left (-\frac{2 \left (1-(c+d x)^2\right ) \left (15 (c+d x)^4+9 (c+d x)^2+7\right )}{(c+d x)^{11/2}}-30 (c+d x) \sqrt{1-\frac{1}{(c+d x)^2}} F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{c+d x}}\right )\right |-1\right )\right )}{77 d \sqrt{1-(c+d x)^2} (e (c+d x))^{13/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c*e + d*e*x)^(13/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]
[Out]
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Maple [B] time = 0.08, size = 860, normalized size = 5. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(d*e*x+c*e)^(13/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{13}{2}}}\,{d x} \]
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)^(13/2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (d^{6} e^{6} x^{6} + 6 \, c d^{5} e^{6} x^{5} + 15 \, c^{2} d^{4} e^{6} x^{4} + 20 \, c^{3} d^{3} e^{6} x^{3} + 15 \, c^{4} d^{2} e^{6} x^{2} + 6 \, c^{5} d e^{6} x + c^{6} e^{6}\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt{d e x + c e}}, x\right ) \]
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)^(13/2)),x, algorithm="fricas")
[Out]
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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)**(13/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{13}{2}}}\,{d x} \]
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)^(13/2)),x, algorithm="giac")
[Out]