3.1396 \(\int \frac{(c e+d e x)^{9/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\)

Optimal. Leaf size=157 \[ -\frac{14 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{3/2}}{45 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{7/2}}{9 d}-\frac{14 e^{9/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{15 d}+\frac{14 e^{9/2} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{15 d} \]

[Out]

(-14*e^3*(c*e + d*e*x)^(3/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(45*d) - (2*e*(c
*e + d*e*x)^(7/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(9*d) + (14*e^(9/2)*Ellipti
cE[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(15*d) - (14*e^(9/2)*EllipticF[ArcSin
[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(15*d)

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Rubi [A]  time = 0.399287, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ -\frac{14 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{3/2}}{45 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{7/2}}{9 d}-\frac{14 e^{9/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{15 d}+\frac{14 e^{9/2} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{15 d} \]

Antiderivative was successfully verified.

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

[Out]

(-14*e^3*(c*e + d*e*x)^(3/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(45*d) - (2*e*(c
*e + d*e*x)^(7/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(9*d) + (14*e^(9/2)*Ellipti
cE[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(15*d) - (14*e^(9/2)*EllipticF[ArcSin
[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(15*d)

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Rubi in Sympy [A]  time = 79.6109, size = 146, normalized size = 0.93 \[ \frac{14 e^{\frac{9}{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{15 d} - \frac{14 e^{\frac{9}{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{15 d} - \frac{14 e^{3} \left (c e + d e x\right )^{\frac{3}{2}} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{45 d} - \frac{2 e \left (c e + d e x\right )^{\frac{7}{2}} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{9 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14*e**(9/2)*elliptic_e(asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/(15*d) - 14*e**(9/2)
*elliptic_f(asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/(15*d) - 14*e**3*(c*e + d*e*x)*
*(3/2)*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(45*d) - 2*e*(c*e + d*e*x)**(7/2)*s
qrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(9*d)

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Mathematica [A]  time = 0.391372, size = 96, normalized size = 0.61 \[ \frac{(e (c+d x))^{9/2} \left (\frac{2}{3} \left (-5 (c+d x)^2-7\right ) \sqrt{1-(c+d x)^2} (c+d x)^{3/2}+14 \left (E\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )-F\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )\right )\right )}{15 d (c+d x)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

((e*(c + d*x))^(9/2)*((2*(c + d*x)^(3/2)*(-7 - 5*(c + d*x)^2)*Sqrt[1 - (c + d*x)
^2])/3 + 14*(EllipticE[ArcSin[Sqrt[c + d*x]], -1] - EllipticF[ArcSin[Sqrt[c + d*
x]], -1])))/(15*d*(c + d*x)^(9/2))

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Maple [B]  time = 0.03, size = 502, normalized size = 3.2 \[{\frac{{e}^{4}}{315\,d \left ({x}^{3}{d}^{3}+3\,c{d}^{2}{x}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) }\sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1} \left ( -70\,{d}^{6}{x}^{6}-420\,{x}^{5}c{d}^{5}-1050\,{x}^{4}{c}^{2}{d}^{4}-1400\,{x}^{3}{c}^{3}{d}^{3}-28\,{d}^{4}{x}^{4}-1050\,{x}^{2}{c}^{4}{d}^{2}-112\,{x}^{3}c{d}^{3}-420\,x{c}^{5}d+126\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ){c}^{2}-126\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ){c}^{2}-168\,{x}^{2}{c}^{2}{d}^{2}-70\,{c}^{6}-60\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) c-60\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) c-112\,{c}^{3}dx-126\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) +273\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) +98\,{d}^{2}{x}^{2}-28\,{c}^{4}+196\,cdx+98\,{c}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*(e*(d*x+c))^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*e^4*(-70*d^6*x^6-420*x^5*
c*d^5-1050*x^4*c^2*d^4-1400*x^3*c^3*d^3-28*d^4*x^4-1050*x^2*c^4*d^2-112*x^3*c*d^
3-420*x*c^5*d+126*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*Ellipti
cE(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*c^2-126*(-2*d*x-2*c+2)^(1/2)*(2*d*x+2*c+2)^
(1/2)*(-d*x-c)^(1/2)*EllipticE(1/2*(2*d*x+2*c+2)^(1/2),2^(1/2))*c^2-168*x^2*c^2*
d^2-70*c^6-60*(-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-60*(-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-112*c^3*d*x-126*(-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))+273*(-2*d*x-2*c+2)^(1/2)*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/2)*Ellipt
icE(1/2*(2*d*x+2*c+2)^(1/2),2^(1/2))+98*d^2*x^2-28*c^4+196*c*d*x+98*c^2)/d/(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. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{9}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(9/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(9/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="fricas")

[Out]

integral((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*e*x + c*e)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1), 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((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. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{9}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(9/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="giac")

[Out]

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