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

Optimal. Leaf size=365 \[ \frac{\left (d^2-a f^2\right )^5 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-5}}{32 e f^4 (5-n)}-\frac{5 \left (d^2-a f^2\right )^4 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{32 e f^4 (3-n)}+\frac{5 \left (d^2-a f^2\right )^3 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{16 e f^4 (1-n)}+\frac{5 \left (d^2-a f^2\right )^2 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{16 e f^4 (n+1)}-\frac{5 \left (d^2-a f^2\right ) \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+3}}{32 e f^4 (n+3)}+\frac{\left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+5}}{32 e f^4 (n+5)} \]

[Out]

((d^2 - a*f^2)^5*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-5 + n))
/(32*e*f^4*(5 - n)) - (5*(d^2 - a*f^2)^4*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (
e^2*x^2)/f^2])^(-3 + n))/(32*e*f^4*(3 - n)) + (5*(d^2 - a*f^2)^3*(d + e*x + f*Sq
rt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-1 + n))/(16*e*f^4*(1 - n)) + (5*(d^2 -
a*f^2)^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(1 + n))/(16*e*f^
4*(1 + n)) - (5*(d^2 - a*f^2)*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^
2])^(3 + n))/(32*e*f^4*(3 + n)) + (d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2
)/f^2])^(5 + n)/(32*e*f^4*(5 + n))

_______________________________________________________________________________________

Rubi [A]  time = 0.863627, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{\left (d^2-a f^2\right )^5 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-5}}{32 e f^4 (5-n)}-\frac{5 \left (d^2-a f^2\right )^4 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{32 e f^4 (3-n)}+\frac{5 \left (d^2-a f^2\right )^3 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{16 e f^4 (1-n)}+\frac{5 \left (d^2-a f^2\right )^2 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{16 e f^4 (n+1)}-\frac{5 \left (d^2-a f^2\right ) \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+3}}{32 e f^4 (n+3)}+\frac{\left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+5}}{32 e f^4 (n+5)} \]

Antiderivative was successfully verified.

[In]  Int[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)^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)^5*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-5 + n))
/(32*e*f^4*(5 - n)) - (5*(d^2 - a*f^2)^4*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (
e^2*x^2)/f^2])^(-3 + n))/(32*e*f^4*(3 - n)) + (5*(d^2 - a*f^2)^3*(d + e*x + f*Sq
rt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-1 + n))/(16*e*f^4*(1 - n)) + (5*(d^2 -
a*f^2)^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(1 + n))/(16*e*f^
4*(1 + n)) - (5*(d^2 - a*f^2)*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^
2])^(3 + n))/(32*e*f^4*(3 + n)) + (d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2
)/f^2])^(5 + n)/(32*e*f^4*(5 + n))

_______________________________________________________________________________________

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+2*d*e*x/f**2+e**2*x**2/f**2)**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

_______________________________________________________________________________________

Mathematica [A]  time = 0.646068, size = 0, normalized size = 0. \[ \int \left (a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}\right )^2 \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.124, size = 0, normalized size = 0. \[ \int \left ( a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}} \right ) ^{2} \left ( d+ex+f\sqrt{a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

Fricas [A]  time = 0.332434, size = 883, normalized size = 2.42 \[ -\frac{{\left (5 \, a^{2} d f^{4} n^{4} + 225 \, a^{2} d f^{4} - 300 \, a d^{3} f^{2} + 5 \,{\left (e^{5} n^{4} - 10 \, e^{5} n^{2} + 9 \, e^{5}\right )} x^{5} + 120 \, d^{5} + 25 \,{\left (d e^{4} n^{4} - 10 \, d e^{4} n^{2} + 9 \, d e^{4}\right )} x^{4} + 10 \,{\left (15 \, a e^{3} f^{2} + 30 \, d^{2} e^{3} +{\left (a e^{3} f^{2} + 4 \, d^{2} e^{3}\right )} n^{4} - 2 \,{\left (8 \, a e^{3} f^{2} + 17 \, d^{2} e^{3}\right )} n^{2}\right )} x^{3} - 10 \,{\left (11 \, a^{2} d f^{4} - 6 \, a d^{3} f^{2}\right )} n^{2} + 10 \,{\left (45 \, a d e^{2} f^{2} +{\left (3 \, a d e^{2} f^{2} + 2 \, d^{3} e^{2}\right )} n^{4} - 2 \,{\left (24 \, a d e^{2} f^{2} + d^{3} e^{2}\right )} n^{2}\right )} x^{2} + 5 \,{\left (45 \, a^{2} e f^{4} +{\left (a^{2} e f^{4} + 4 \, a d^{2} e f^{2}\right )} n^{4} - 2 \,{\left (11 \, a^{2} e f^{4} + 26 \, a d^{2} e f^{2} - 12 \, d^{4} e\right )} n^{2}\right )} x -{\left (a^{2} f^{5} n^{5} +{\left (e^{4} f n^{5} - 10 \, e^{4} f n^{3} + 9 \, e^{4} f n\right )} x^{4} - 10 \,{\left (3 \, a^{2} f^{5} - 2 \, a d^{2} f^{3}\right )} n^{3} + 4 \,{\left (d e^{3} f n^{5} - 10 \, d e^{3} f n^{3} + 9 \, d e^{3} f n\right )} x^{3} + 2 \,{\left ({\left (a e^{2} f^{3} + 2 \, d^{2} e^{2} f\right )} n^{5} - 10 \,{\left (2 \, a e^{2} f^{3} + d^{2} e^{2} f\right )} n^{3} +{\left (19 \, a e^{2} f^{3} + 8 \, d^{2} e^{2} f\right )} n\right )} x^{2} +{\left (149 \, a^{2} f^{5} - 260 \, a d^{2} f^{3} + 120 \, d^{4} f\right )} n + 4 \,{\left (a d e f^{3} n^{5} - 10 \,{\left (2 \, a d e f^{3} - d^{3} e f\right )} n^{3} +{\left (19 \, a d e f^{3} - 10 \, d^{3} e f\right )} n\right )} x\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}\right )}{\left (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n}}{e f^{4} n^{6} - 35 \, e f^{4} n^{4} + 259 \, e f^{4} n^{2} - 225 \, e f^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(5*a^2*d*f^4*n^4 + 225*a^2*d*f^4 - 300*a*d^3*f^2 + 5*(e^5*n^4 - 10*e^5*n^2 + 9*
e^5)*x^5 + 120*d^5 + 25*(d*e^4*n^4 - 10*d*e^4*n^2 + 9*d*e^4)*x^4 + 10*(15*a*e^3*
f^2 + 30*d^2*e^3 + (a*e^3*f^2 + 4*d^2*e^3)*n^4 - 2*(8*a*e^3*f^2 + 17*d^2*e^3)*n^
2)*x^3 - 10*(11*a^2*d*f^4 - 6*a*d^3*f^2)*n^2 + 10*(45*a*d*e^2*f^2 + (3*a*d*e^2*f
^2 + 2*d^3*e^2)*n^4 - 2*(24*a*d*e^2*f^2 + d^3*e^2)*n^2)*x^2 + 5*(45*a^2*e*f^4 +
(a^2*e*f^4 + 4*a*d^2*e*f^2)*n^4 - 2*(11*a^2*e*f^4 + 26*a*d^2*e*f^2 - 12*d^4*e)*n
^2)*x - (a^2*f^5*n^5 + (e^4*f*n^5 - 10*e^4*f*n^3 + 9*e^4*f*n)*x^4 - 10*(3*a^2*f^
5 - 2*a*d^2*f^3)*n^3 + 4*(d*e^3*f*n^5 - 10*d*e^3*f*n^3 + 9*d*e^3*f*n)*x^3 + 2*((
a*e^2*f^3 + 2*d^2*e^2*f)*n^5 - 10*(2*a*e^2*f^3 + d^2*e^2*f)*n^3 + (19*a*e^2*f^3
+ 8*d^2*e^2*f)*n)*x^2 + (149*a^2*f^5 - 260*a*d^2*f^3 + 120*d^4*f)*n + 4*(a*d*e*f
^3*n^5 - 10*(2*a*d*e*f^3 - d^3*e*f)*n^3 + (19*a*d*e*f^3 - 10*d^3*e*f)*n)*x)*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/(e*f^4*n^6 - 35*e*f^4*n^4 + 259*e*f^4*n^2 - 225*e*f^4)

_______________________________________________________________________________________

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((a+2*d*e*x/f**2+e**2*x**2/f**2)**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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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