3.1397 \(\int \frac{(c e+d e x)^{5/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\)

Optimal. Leaf size=111 \[ -\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{3/2}}{5 d}-\frac{6 e^{5/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d}+\frac{6 e^{5/2} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d} \]

[Out]

(-2*e*(c*e + d*e*x)^(3/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(5*d) + (6*e^(5/2)*
EllipticE[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(5*d) - (6*e^(5/2)*EllipticF[A
rcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(5*d)

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Rubi [A]  time = 0.281521, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ -\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{3/2}}{5 d}-\frac{6 e^{5/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d}+\frac{6 e^{5/2} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*(c*e + d*e*x)^(3/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(5*d) + (6*e^(5/2)*
EllipticE[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(5*d) - (6*e^(5/2)*EllipticF[A
rcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(5*d)

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Rubi in Sympy [A]  time = 59.1936, size = 104, normalized size = 0.94 \[ \frac{6 e^{\frac{5}{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{5 d} - \frac{6 e^{\frac{5}{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{5 d} - \frac{2 e \left (c e + d e x\right )^{\frac{3}{2}} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{5 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*e**(5/2)*elliptic_e(asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/(5*d) - 6*e**(5/2)*el
liptic_f(asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/(5*d) - 2*e*(c*e + d*e*x)**(3/2)*s
qrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(5*d)

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Mathematica [A]  time = 0.197324, size = 90, normalized size = 0.81 \[ -\frac{2 (e (c+d x))^{5/2} \left (\sqrt{-c^2-2 c d x-d^2 x^2+1} (c+d x)^{3/2}+3 F\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )-3 E\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )\right )}{5 d (c+d x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 234, normalized size = 2.1 \[ -{\frac{{e}^{2}}{5\,d \left ({x}^{3}{d}^{3}+3\,c{d}^{2}{x}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) }\sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1} \left ( 2\,{d}^{4}{x}^{4}+8\,{x}^{3}c{d}^{3}+12\,{x}^{2}{c}^{2}{d}^{2}+8\,{c}^{3}dx+2\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) -5\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) -2\,{d}^{2}{x}^{2}+2\,{c}^{4}-4\,cdx-2\,{c}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(e*(d*x+c))^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*e^2*(2*d^4*x^4+8*x^3*c*d^3
+12*x^2*c^2*d^2+8*c^3*d*x+2*(-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))-5*(-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))-2*d^2*x^2+2*
c^4-4*c*d*x-2*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. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{5}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(5/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^(5/2)/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. \[{\rm integral}\left (\frac{{\left (d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}\right )} \sqrt{d e x + c e}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(5/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((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. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{5}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(5/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="giac")

[Out]

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