3.358 \(\int \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n \, dx\)

Optimal. Leaf size=327 \[ -\frac{\left (d^2-a f^2\right )^2 \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-2}}{4 e f (2-n) \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}}-\frac{\left (d^2-a f^2\right ) \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^n}{2 e f n \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}}+\frac{\sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+2}}{4 e f (n+2) \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}} \]

[Out]

-((d^2 - a*f^2)^2*Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2]*(d + e*x + f*Sqr
t[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-2 + n))/(4*e*f*(2 - n)*Sqrt[a + (2*d*e*x
)/f^2 + (e^2*x^2)/f^2]) - ((d^2 - a*f^2)*Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2
)/f^2]*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n)/(2*e*f*n*Sqrt[a
+ (2*d*e*x)/f^2 + (e^2*x^2)/f^2]) + (Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^
2]*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(2 + n))/(4*e*f*(2 + n)
*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])

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Rubi [A]  time = 1.05213, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 62, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{\left (d^2-a f^2\right )^2 \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-2}}{4 e f (2-n) \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}}-\frac{\left (d^2-a f^2\right ) \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^n}{2 e f n \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}}+\frac{\sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+2}}{4 e f (n+2) \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2]*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]

[Out]

-((d^2 - a*f^2)^2*Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2]*(d + e*x + f*Sqr
t[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-2 + n))/(4*e*f*(2 - n)*Sqrt[a + (2*d*e*x
)/f^2 + (e^2*x^2)/f^2]) - ((d^2 - a*f^2)*Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2
)/f^2]*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n)/(2*e*f*n*Sqrt[a
+ (2*d*e*x)/f^2 + (e^2*x^2)/f^2]) + (Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^
2]*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(2 + n))/(4*e*f*(2 + n)
*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*g+2*d*e*g*x/f**2+e**2*g*x**2/f**2)**(1/2)*(d+e*x+f*(a+2*d*e*x/f**2+e**2*x**2/f**2)**(1/2))**n,x)

[Out]

Timed out

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Mathematica [A]  time = 0.123497, size = 0, normalized size = 0. \[ \int \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2]*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]

[Out]

Integrate[Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2]*(d + e*x + f*Sqrt[a + (2
*d*e*x)/f^2 + (e^2*x^2)/f^2])^n, x]

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Maple [F]  time = 0.114, size = 0, normalized size = 0. \[ \int \sqrt{ag+2\,{\frac{degx}{{f}^{2}}}+{\frac{{e}^{2}g{x}^{2}}{{f}^{2}}}} \left ( d+ex+f\sqrt{a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^n,x)

[Out]

int((a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)
^(1/2))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{e^{2} g x^{2}}{f^{2}} + a g + \frac{2 \, d e g x}{f^{2}}}{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e^2*g*x^2/f^2 + a*g + 2*d*e*g*x/f^2)*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n,x, algorithm="maxima")

[Out]

integrate(sqrt(e^2*g*x^2/f^2 + a*g + 2*d*e*g*x/f^2)*(e*x + sqrt(e^2*x^2/f^2 + a
+ 2*d*e*x/f^2)*f + d)^n, x)

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Fricas [A]  time = 0.323563, size = 312, normalized size = 0.95 \[ -\frac{{\left (2 \, e^{3} n x^{3} + 6 \, d e^{2} n x^{2} + 2 \, a d f^{2} n + 2 \,{\left (a e f^{2} + 2 \, d^{2} e\right )} n x -{\left (e^{2} f n^{2} x^{2} + a f^{3} n^{2} + 2 \, d e f n^{2} x - 2 \, a f^{3} + 2 \, d^{2} f\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}\right )}{\left (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n} \sqrt{\frac{e^{2} g x^{2} + a f^{2} g + 2 \, d e g x}{f^{2}}}}{a e f^{2} n^{3} - 4 \, a e f^{2} n +{\left (e^{3} n^{3} - 4 \, e^{3} n\right )} x^{2} + 2 \,{\left (d e^{2} n^{3} - 4 \, d e^{2} n\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e^2*g*x^2/f^2 + a*g + 2*d*e*g*x/f^2)*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n,x, algorithm="fricas")

[Out]

-(2*e^3*n*x^3 + 6*d*e^2*n*x^2 + 2*a*d*f^2*n + 2*(a*e*f^2 + 2*d^2*e)*n*x - (e^2*f
*n^2*x^2 + a*f^3*n^2 + 2*d*e*f*n^2*x - 2*a*f^3 + 2*d^2*f)*sqrt((e^2*x^2 + a*f^2
+ 2*d*e*x)/f^2))*(e*x + f*sqrt((e^2*x^2 + a*f^2 + 2*d*e*x)/f^2) + d)^n*sqrt((e^2
*g*x^2 + a*f^2*g + 2*d*e*g*x)/f^2)/(a*e*f^2*n^3 - 4*a*e*f^2*n + (e^3*n^3 - 4*e^3
*n)*x^2 + 2*(d*e^2*n^3 - 4*d*e^2*n)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{g \left (a + \frac{2 d e x}{f^{2}} + \frac{e^{2} x^{2}}{f^{2}}\right )} \left (d + e x + f \sqrt{a + \frac{2 d e x}{f^{2}} + \frac{e^{2} x^{2}}{f^{2}}}\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*g+2*d*e*g*x/f**2+e**2*g*x**2/f**2)**(1/2)*(d+e*x+f*(a+2*d*e*x/f**2+e**2*x**2/f**2)**(1/2))**n,x)

[Out]

Integral(sqrt(g*(a + 2*d*e*x/f**2 + e**2*x**2/f**2))*(d + e*x + f*sqrt(a + 2*d*e
*x/f**2 + e**2*x**2/f**2))**n, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{e^{2} g x^{2}}{f^{2}} + a g + \frac{2 \, d e g x}{f^{2}}}{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e^2*g*x^2/f^2 + a*g + 2*d*e*g*x/f^2)*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n,x, algorithm="giac")

[Out]

integrate(sqrt(e^2*g*x^2/f^2 + a*g + 2*d*e*g*x/f^2)*(e*x + sqrt(e^2*x^2/f^2 + a
+ 2*d*e*x/f^2)*f + d)^n, x)