3.1390 \(\int \frac{(c e+d e x)^{7/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{10 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{21 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{7 d}+\frac{10 e^{7/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{21 d} \]

[Out]

(-10*e^3*Sqrt[c*e + d*e*x]*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(21*d) - (2*e*(c*e
+ d*e*x)^(5/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(7*d) + (10*e^(7/2)*EllipticF
[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(21*d)

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Rubi [A]  time = 0.272325, antiderivative size = 124, 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 \[ -\frac{10 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{21 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{7 d}+\frac{10 e^{7/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{21 d} \]

Antiderivative was successfully verified.

[In]  Int[(c*e + d*e*x)^(7/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]

[Out]

(-10*e^3*Sqrt[c*e + d*e*x]*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(21*d) - (2*e*(c*e
+ d*e*x)^(5/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(7*d) + (10*e^(7/2)*EllipticF
[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(21*d)

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Rubi in Sympy [A]  time = 61.48, size = 114, normalized size = 0.92 \[ \frac{10 e^{\frac{7}{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{21 d} - \frac{10 e^{3} \sqrt{c e + d e x} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{21 d} - \frac{2 e \left (c e + d e x\right )^{\frac{5}{2}} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{7 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*e*x+c*e)**(7/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

10*e**(7/2)*elliptic_f(asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/(21*d) - 10*e**3*sqr
t(c*e + d*e*x)*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(21*d) - 2*e*(c*e + d*e*x)*
*(5/2)*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(7*d)

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Mathematica [A]  time = 0.151956, size = 136, normalized size = 1.1 \[ \frac{\sqrt{1-(c+d x)^2} \left (-\frac{2}{7} (c+d x)^{5/2}-\frac{10}{21} \sqrt{c+d x}\right ) (e (c+d x))^{7/2}}{d (c+d x)^{7/2}}-\frac{10 \sqrt{1-\frac{1}{(c+d x)^2}} (e (c+d x))^{7/2} F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{c+d x}}\right )\right |-1\right )}{21 d (c+d x)^{5/2} \sqrt{1-(c+d x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*e + d*e*x)^(7/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]

[Out]

((e*(c + d*x))^(7/2)*Sqrt[1 - (c + d*x)^2]*((-10*Sqrt[c + d*x])/21 - (2*(c + d*x
)^(5/2))/7))/(d*(c + d*x)^(7/2)) - (10*(e*(c + d*x))^(7/2)*Sqrt[1 - (c + d*x)^(-
2)]*EllipticF[ArcSin[1/Sqrt[c + d*x]], -1])/(21*d*(c + d*x)^(5/2)*Sqrt[1 - (c +
d*x)^2])

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Maple [B]  time = 0.028, size = 263, normalized size = 2.1 \[ -{\frac{{e}^{3}}{21\,d \left ({x}^{3}{d}^{3}+3\,c{d}^{2}{x}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) }\sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1} \left ( 6\,{x}^{5}{d}^{5}+30\,{x}^{4}c{d}^{4}+60\,{x}^{3}{c}^{2}{d}^{3}+60\,{x}^{2}{c}^{3}{d}^{2}+4\,{x}^{3}{d}^{3}+30\,x{c}^{4}d+12\,c{d}^{2}{x}^{2}+6\,{c}^{5}+2\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) +7\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) +12\,x{c}^{2}d+4\,{c}^{3}-10\,dx-10\,c \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*e*x+c*e)^(7/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)

[Out]

-1/21*(e*(d*x+c))^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*e^3*(6*x^5*d^5+30*x^4*c*d
^4+60*x^3*c^2*d^3+60*x^2*c^3*d^2+4*x^3*d^3+30*x*c^4*d+12*c*d^2*x^2+6*c^5+2*(-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))+7*(-2*d*x-2*c+2)^(1/2)*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/2)*Elliptic
F(1/2*(2*d*x+2*c+2)^(1/2),2^(1/2))+12*x*c^2*d+4*c^3-10*d*x-10*c)/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. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{7}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(7/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^(7/2)/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. \[{\rm integral}\left (\frac{{\left (d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}\right )} \sqrt{d e x + c e}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(7/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="fricas")

[Out]

integral((d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3)*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)**(7/2)/(-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. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{7}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(7/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(7/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1), x)