3.6.11 \(\int \frac {(d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}})^n}{(a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2})^2} \, dx\) [511]

3.6.11.1 Optimal result

Integrand size = 56, antiderivative size = 122 \[ \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2} \, dx=-\frac {8 f^4 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{3+n} \operatorname {Hypergeometric2F1}\left (3,\frac {3+n}{2},\frac {5+n}{2},\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right )^3 (3+n)} \]

output

-8*f^4*hypergeom([3, 3/2+1/2*n],[5/2+1/2*n],(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^ 
2/f^2)^(1/2))^2/(-a*f^2+d^2))*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^ 
(3+n)/e/(-a*f^2+d^2)^3/(3+n)
 

3.6.11.2 Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2} \, dx=-\frac {8 f^4 \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^{3+n} \operatorname {Hypergeometric2F1}\left (3,\frac {3+n}{2},\frac {5+n}{2},\frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right )^3 (3+n)} \]

input

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)^2,x]
 

output

(-8*f^4*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(3 + n)*Hypergeometr 
ic2F1[3, (3 + n)/2, (5 + n)/2, (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2 
])^2/(d^2 - a*f^2)])/(e*(d^2 - a*f^2)^3*(3 + n))
 

3.6.11.3 Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2546, 27, 278}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2546

\(\displaystyle 2 f^4 \int -\frac {4 \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n+2}}{e \left (d^2-a f^2-\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2\right )^3}d\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {8 f^4 \int \frac {\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n+2}}{\left (d^2-a f^2-\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2\right )^3}d\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )}{e}\)

\(\Big \downarrow \) 278

\(\displaystyle -\frac {8 f^4 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+3} \operatorname {Hypergeometric2F1}\left (3,\frac {n+3}{2},\frac {n+5}{2},\frac {\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2}{d^2-a f^2}\right )}{e (n+3) \left (d^2-a f^2\right )^3}\)

input

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)^2,x]
 

output

(-8*f^4*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(3 + n)*Hype 
rgeometric2F1[3, (3 + n)/2, (5 + 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)^3*(3 + n))
 

3.6.11.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 2546

Int[((g_.) + (h_.)*(x_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*S 
qrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Simp[(2/f^(2*m) 
)*(i/c)^m   Subst[Int[x^n*((d^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x + e 
*x^2)^(2*m + 1)/(-2*d*e + b*f^2 + 2*e*x)^(2*(m + 1))), x], x, d + e*x + f*S 
qrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n}, x] && Eq 
Q[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && EqQ[c*h - b*i, 0] && IntegerQ[2*m 
] && (IntegerQ[m] || GtQ[i/c, 0])
 

3.6.11.4 Maple [F]

\[\int \frac {{\left (d +e x +f \sqrt {a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}}\right )}^{n}}{\left (a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}\right )^{2}}d x\]

input

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)^2,x)
 

output

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)^2,x)
 

3.6.11.5 Fricas [F]

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

input

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)^2,x, algorithm="fricas")
 

output

integral((e*x + f*sqrt((e^2*x^2 + a*f^2 + 2*d*e*x)/f^2) + d)^n*f^4/(e^4*x^ 
4 + 4*d*e^3*x^3 + a^2*f^4 + 4*a*d*e*f^2*x + 2*(a*e^2*f^2 + 2*d^2*e^2)*x^2) 
, x)
 

3.6.11.6 Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

input

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)**2,x)
 

output

Exception raised: HeuristicGCDFailed >> no luck
 

3.6.11.7 Maxima [F]

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

input

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)^2,x, algorithm="maxima")
 

output

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)^2, x)
 

3.6.11.8 Giac [F]

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

input

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)^2,x, algorithm="giac")
 

output

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)^2, x)
 

3.6.11.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2} \, dx=\int \frac {{\left (d+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}}+e\,x\right )}^n}{{\left (a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}\right )}^2} \,d x \]

input

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

output

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