3.6.13 \(\int \frac {(d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}})^n}{a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \, dx\) [513]

3.6.13.1 Optimal result

Integrand size = 56, antiderivative size = 122 \[ \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \, dx=-\frac {2 f^2 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right ) (1+n)} \]

output

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

3.6.13.2 Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \, dx=-\frac {2 f^2 \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},\frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right ) (1+n)} \]

input

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

output

(-2*f^2*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(1 + n)*Hypergeometr 
ic2F1[1, (1 + n)/2, (3 + 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)*(1 + n))
 

3.6.13.3 Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {2552, 2546, 25, 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.

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

\(\Big \downarrow \) 2552

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

\(\Big \downarrow \) 2546

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 278

\(\displaystyle -\frac {2 f^2 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\frac {\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2}{d^2-a f^2}\right )}{e (n+1) \left (d^2-a f^2\right )}\)

input

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

output

(-2*f^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(1 + n)*Hype 
rgeometric2F1[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))
 

3.6.13.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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])
 

rule 2552

Int[((u_) + (f_.)*((j_.) + (k_.)*Sqrt[v_]))^(n_.)*(w_)^(m_.), x_Symbol] :> 
Int[ExpandToSum[w, x]^m*(ExpandToSum[u + f*j, x] + f*k*Sqrt[ExpandToSum[v, 
x]])^n, x] /; FreeQ[{f, j, k, m, n}, x] && LinearQ[u, x] && QuadraticQ[{v, 
w}, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[{v, w}, x] && (EqQ[j, 0] 
 || EqQ[f, 1])) && EqQ[Coefficient[u, x, 1]^2 - Coefficient[v, x, 2]*f^2*k^ 
2, 0]
 

3.6.13.4 Maple [F]

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

input

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

output

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

3.6.13.5 Fricas [F]

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

input

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

3.6.13.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.6.13.7 Maxima [F]

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

input

integrate((d+e*x+f*((a*f^2+e*x*(e*x+2*d))/f^2)^(1/2))^n/(a+2*d*e*x/f^2+e^2 
*x^2/f^2),x, algorithm="maxima")
 

output

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

3.6.13.8 Giac [F]

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

input

integrate((d+e*x+f*((a*f^2+e*x*(e*x+2*d))/f^2)^(1/2))^n/(a+2*d*e*x/f^2+e^2 
*x^2/f^2),x, algorithm="giac")
 

output

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

3.6.13.9 Mupad [F(-1)]

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

input

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

output

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