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

Optimal. Leaf size=122 \[ -\frac{2 f^2 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\frac{\left (d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+\frac{2 d e x}{f^2}+a}\right )^2}{d^2-a f^2}\right )}{e (n+1) \left (d^2-a f^2\right )} \]

[Out]

(-2*f^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(1 + n)*Hypergeome
tric2F1[1, (1 + n)/2, (3 + n)/2, (d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)
/f^2])^2/(d^2 - a*f^2)])/(e*(d^2 - a*f^2)*(1 + n))

_______________________________________________________________________________________

Rubi [A]  time = 0.485895, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ -\frac{2 f^2 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\frac{\left (d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+\frac{2 d e x}{f^2}+a}\right )^2}{d^2-a f^2}\right )}{e (n+1) \left (d^2-a f^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*f^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(1 + n)*Hypergeome
tric2F1[1, (1 + n)/2, (3 + n)/2, (d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)
/f^2])^2/(d^2 - a*f^2)])/(e*(d^2 - a*f^2)*(1 + n))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 111.889, size = 110, normalized size = 0.9 \[ - \frac{2 f^{2} \left (d + e x + f \sqrt{a + \frac{2 d e x}{f^{2}} + \frac{e^{2} x^{2}}{f^{2}}}\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n}{2} + \frac{1}{2} \\ \frac{n}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{\left (d + e x + f \sqrt{a + \frac{2 d e x}{f^{2}} + \frac{e^{2} x^{2}}{f^{2}}}\right )^{2}}{a f^{2} - d^{2}}} \right )}}{e \left (n + 1\right ) \left (- a f^{2} + d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*f**2*(d + e*x + f*sqrt(a + 2*d*e*x/f**2 + e**2*x**2/f**2))**(n + 1)*hyper((1,
n/2 + 1/2), (n/2 + 3/2,), -(d + e*x + f*sqrt(a + 2*d*e*x/f**2 + e**2*x**2/f**2)
)**2/(a*f**2 - d**2))/(e*(n + 1)*(-a*f**2 + d**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.152238, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n}{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.133, size = 0, normalized size = 0. \[ \int{1 \left ( d+ex+f\sqrt{a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n} \left ( a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}} \right ) ^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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