Optimal. Leaf size=170 \[ -\frac{30 e^5 \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{77 d}-\frac{18 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{77 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{9/2}}{11 d}+\frac{30 e^{11/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{77 d} \]
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Rubi [A] time = 0.398824, antiderivative size = 170, 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 \[ -\frac{30 e^5 \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{77 d}-\frac{18 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{77 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{9/2}}{11 d}+\frac{30 e^{11/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{77 d} \]
Antiderivative was successfully verified.
[In] Int[(c*e + d*e*x)^(11/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]
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Rubi in Sympy [A] time = 80.5499, size = 156, normalized size = 0.92 \[ \frac{30 e^{\frac{11}{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{77 d} - \frac{30 e^{5} \sqrt{c e + d e x} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{77 d} - \frac{18 e^{3} \left (c e + d e x\right )^{\frac{5}{2}} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{77 d} - \frac{2 e \left (c e + d e x\right )^{\frac{9}{2}} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{11 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*e*x+c*e)**(11/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)
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Mathematica [A] time = 0.430987, size = 120, normalized size = 0.71 \[ \frac{e (e (c+d x))^{9/2} \left (2 \sqrt{c+d x} \left (1-(c+d x)^2\right ) \left (-7 (c+d x)^4-9 (c+d x)^2-15\right )-30 (c+d x) \sqrt{1-\frac{1}{(c+d x)^2}} F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{c+d x}}\right )\right |-1\right )\right )}{77 d (c+d x)^{9/2} \sqrt{1-(c+d x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*e + d*e*x)^(11/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]
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Maple [B] time = 0.215, size = 647, normalized size = 3.8 \[{\frac{{e}^{5}}{1155\,d \left ({x}^{3}{d}^{3}+3\,c{d}^{2}{x}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) }\sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1} \left ( 450\,c+462\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) c-1470\,{x}^{6}c{d}^{6}-180\,{c}^{3}+450\,dx-210\,{c}^{7}+1386\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ){c}^{3}-4410\,{x}^{5}{c}^{2}{d}^{5}-7350\,{x}^{4}{c}^{3}{d}^{4}-7350\,{x}^{3}{c}^{4}{d}^{3}-300\,{x}^{4}c{d}^{4}-4410\,{x}^{2}{c}^{5}{d}^{2}-600\,{x}^{3}{c}^{2}{d}^{3}-1470\,x{c}^{6}d-600\,{x}^{2}{c}^{3}{d}^{2}-300\,x{c}^{4}d-540\,x{c}^{2}d-540\,c{d}^{2}{x}^{2}-462\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) c-160\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) -385\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) -180\,{x}^{3}{d}^{3}-210\,{x}^{7}{d}^{7}-60\,{x}^{5}{d}^{5}-60\,{c}^{5}-660\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ){c}^{2}-1386\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ){c}^{3}-660\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ){c}^{2} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*e*x+c*e)^(11/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{11}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*e*x + c*e)^(11/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d^{5} e^{5} x^{5} + 5 \, c d^{4} e^{5} x^{4} + 10 \, c^{2} d^{3} e^{5} x^{3} + 10 \, c^{3} d^{2} e^{5} x^{2} + 5 \, c^{4} d e^{5} x + c^{5} e^{5}\right )} \sqrt{d e x + c e}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*e*x + c*e)^(11/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="fricas")
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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((d*e*x+c*e)**(11/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{11}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*e*x + c*e)^(11/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="giac")
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