\(\int \frac {(e+f x)^n}{(4-d^2 x^2)^{3/2}} \, dx\) [118]

Optimal result

Integrand size = 22, antiderivative size = 84 \[ \int \frac {(e+f x)^n}{\left (4-d^2 x^2\right )^{3/2}} \, dx=\frac {(e+f x)^n \left (\frac {d (e+f x)}{d e+2 f}\right )^{-n} \operatorname {AppellF1}\left (-\frac {1}{2},-n,\frac {3}{2},\frac {1}{2},\frac {f (2-d x)}{d e+2 f},\frac {1}{4} (2-d x)\right )}{4 d \sqrt {2-d x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.80 \[ \int \frac {(e+f x)^n}{\left (4-d^2 x^2\right )^{3/2}} \, dx=\frac {\left (\frac {f \left (2 \sqrt {\frac {1}{d^2}}-x\right )}{e+2 \sqrt {\frac {1}{d^2}} f}\right )^{3/2} \left (\frac {f \left (2 \sqrt {\frac {1}{d^2}}+x\right )}{-e+2 \sqrt {\frac {1}{d^2}} f}\right )^{3/2} (e+f x)^{1+n} \operatorname {AppellF1}\left (1+n,\frac {3}{2},\frac {3}{2},2+n,\frac {e+f x}{e-2 \sqrt {\frac {1}{d^2}} f},\frac {e+f x}{e+2 \sqrt {\frac {1}{d^2}} f}\right )}{f (1+n) \left (4-d^2 x^2\right )^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {513, 27, 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.

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

\(\Big \downarrow \) 513

\(\displaystyle \frac {1}{8} \int \frac {8 (e+f x)^n}{(2-d x)^{3/2} (d x+2)^{3/2}}dx\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 156

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

\(\Big \downarrow \) 155

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

Input:

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

Output:

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

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 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 513

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
a^p   Int[(c + d*x)^n*(1 + Rt[-b/a, 2]*x)^p*(1 - Rt[-b/a, 2]*x)^p, x], x] / 
; FreeQ[{a, b, c, d, n, p}, x] && GtQ[a, 0] && NegQ[b/a]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(e+f x)^n}{\left (4-d^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{{\left (-d^{2} x^{2} + 4\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((f*x+e)^n/(-d^2*x^2+4)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-d^2*x^2 + 4)*(f*x + e)^n/(d^4*x^4 - 8*d^2*x^2 + 16), x)
 

Sympy [F]

\[ \int \frac {(e+f x)^n}{\left (4-d^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (e + f x\right )^{n}}{\left (- \left (d x - 2\right ) \left (d x + 2\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((f*x+e)**n/(-d**2*x**2+4)**(3/2),x)
 

Output:

Integral((e + f*x)**n/(-(d*x - 2)*(d*x + 2))**(3/2), x)
 

Maxima [F]

\[ \int \frac {(e+f x)^n}{\left (4-d^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{{\left (-d^{2} x^{2} + 4\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((f*x+e)^n/(-d^2*x^2+4)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {(e+f x)^n}{\left (4-d^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{{\left (-d^{2} x^{2} + 4\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((f*x+e)^n/(-d^2*x^2+4)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^n}{\left (4-d^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (e+f\,x\right )}^n}{{\left (4-d^2\,x^2\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(e+f x)^n}{\left (4-d^2 x^2\right )^{3/2}} \, dx=-\left (\int \frac {\left (f x +e \right )^{n}}{\sqrt {-d^{2} x^{2}+4}\, d^{2} x^{2}-4 \sqrt {-d^{2} x^{2}+4}}d x \right ) \] Input:

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

Output:

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