\(\int \frac {(e+f x)^n}{(2+d x) \sqrt {4-d^2 x^2}} \, dx\) [114]

Optimal result

Integrand size = 29, antiderivative size = 84 \[ \int \frac {(e+f x)^n}{(2+d x) \sqrt {4-d^2 x^2}} \, dx=-\frac {\sqrt {2-d x} (e+f x)^n \left (\frac {d (e+f x)}{d e+2 f}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{2},-n,\frac {3}{2},\frac {1}{4} (2-d x),\frac {f (2-d x)}{d e+2 f}\right )}{4 d} \] Output:

-1/4*(-d*x+2)^(1/2)*(f*x+e)^n*AppellF1(1/2,-n,3/2,3/2,f*(-d*x+2)/(d*e+2*f) 
,-1/4*d*x+1/2)/d/((d*(f*x+e)/(d*e+2*f))^n)
 

Mathematica [F]

\[ \int \frac {(e+f x)^n}{(2+d x) \sqrt {4-d^2 x^2}} \, dx=\int \frac {(e+f x)^n}{(2+d x) \sqrt {4-d^2 x^2}} \, dx \] Input:

Integrate[(e + f*x)^n/((2 + d*x)*Sqrt[4 - d^2*x^2]),x]
 

Output:

Integrate[(e + f*x)^n/((2 + d*x)*Sqrt[4 - d^2*x^2]), x]
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {717, 156, 155}

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.

\(\displaystyle \int \frac {(e+f x)^n}{(d x+2) \sqrt {4-d^2 x^2}} \, dx\)

\(\Big \downarrow \) 717

\(\displaystyle \int \frac {(e+f x)^n}{\sqrt {2-d x} (d x+2)^{3/2}}dx\)

\(\Big \downarrow \) 156

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

\(\Big \downarrow \) 155

\(\displaystyle -\frac {\sqrt {2-d x} (e+f x)^n \left (\frac {d (e+f x)}{d e+2 f}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{2},-n,\frac {3}{2},\frac {1}{4} (2-d x),\frac {f (2-d x)}{d e+2 f}\right )}{4 d}\)

Input:

Int[(e + f*x)^n/((2 + d*x)*Sqrt[4 - d^2*x^2]),x]
 

Output:

-1/4*(Sqrt[2 - d*x]*(e + f*x)^n*AppellF1[1/2, 3/2, -n, 3/2, (2 - d*x)/4, ( 
f*(2 - d*x))/(d*e + 2*f)])/(d*((d*(e + f*x))/(d*e + 2*f))^n)
 

Defintions of rubi rules used

rule 155

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* 
Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ 
(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, 
 m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[Sim 
plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] &&  !(GtQ[Simpl 
ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d 
*x, a + b*x]) &&  !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c 
- e*d)], 0] && SimplerQ[e + f*x, a + b*x])
 

rule 156

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p 
]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p])   Int[(a + b*x)^m*(c + d*x)^n*Si 
mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] & 
& GtQ[Simplify[b/(b*c - a*d)], 0] &&  !GtQ[Simplify[b/(b*e - a*f)], 0]
 

rule 717

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_) 
^2)^(p_), x_Symbol] :> Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, 
 x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a 
, 0] && GtQ[d, 0]
 

Maple [F]

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

Input:

int((f*x+e)^n/(d*x+2)/(-d^2*x^2+4)^(1/2),x)
 

Output:

int((f*x+e)^n/(d*x+2)/(-d^2*x^2+4)^(1/2),x)
 

Fricas [F]

\[ \int \frac {(e+f x)^n}{(2+d x) \sqrt {4-d^2 x^2}} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{\sqrt {-d^{2} x^{2} + 4} {\left (d x + 2\right )}} \,d x } \] Input:

integrate((f*x+e)^n/(d*x+2)/(-d^2*x^2+4)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-d^2*x^2 + 4)*(f*x + e)^n/(d^3*x^3 + 2*d^2*x^2 - 4*d*x - 8) 
, x)
 

Sympy [F]

\[ \int \frac {(e+f x)^n}{(2+d x) \sqrt {4-d^2 x^2}} \, dx=\int \frac {\left (e + f x\right )^{n}}{\sqrt {- \left (d x - 2\right ) \left (d x + 2\right )} \left (d x + 2\right )}\, dx \] Input:

integrate((f*x+e)**n/(d*x+2)/(-d**2*x**2+4)**(1/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {(e+f x)^n}{(2+d x) \sqrt {4-d^2 x^2}} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{\sqrt {-d^{2} x^{2} + 4} {\left (d x + 2\right )}} \,d x } \] Input:

integrate((f*x+e)^n/(d*x+2)/(-d^2*x^2+4)^(1/2),x, algorithm="maxima")
 

Output:

integrate((f*x + e)^n/(sqrt(-d^2*x^2 + 4)*(d*x + 2)), x)
 

Giac [F]

\[ \int \frac {(e+f x)^n}{(2+d x) \sqrt {4-d^2 x^2}} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{\sqrt {-d^{2} x^{2} + 4} {\left (d x + 2\right )}} \,d x } \] Input:

integrate((f*x+e)^n/(d*x+2)/(-d^2*x^2+4)^(1/2),x, algorithm="giac")
 

Output:

integrate((f*x + e)^n/(sqrt(-d^2*x^2 + 4)*(d*x + 2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^n}{(2+d x) \sqrt {4-d^2 x^2}} \, dx=\int \frac {{\left (e+f\,x\right )}^n}{\sqrt {4-d^2\,x^2}\,\left (d\,x+2\right )} \,d x \] Input:

int((e + f*x)^n/((4 - d^2*x^2)^(1/2)*(d*x + 2)),x)
 

Output:

int((e + f*x)^n/((4 - d^2*x^2)^(1/2)*(d*x + 2)), x)
 

Reduce [F]

\[ \int \frac {(e+f x)^n}{(2+d x) \sqrt {4-d^2 x^2}} \, dx=\int \frac {\left (f x +e \right )^{n}}{\sqrt {-d^{2} x^{2}+4}\, d x +2 \sqrt {-d^{2} x^{2}+4}}d x \] Input:

int((f*x+e)^n/(d*x+2)/(-d^2*x^2+4)^(1/2),x)
 

Output:

int((e + f*x)**n/(sqrt( - d**2*x**2 + 4)*d*x + 2*sqrt( - d**2*x**2 + 4)),x 
)