\(\int (1+d x) (e+f x)^n (1-d^2 x^2)^p \, dx\) [141]

Optimal result

Integrand size = 25, antiderivative size = 90 \[ \int (1+d x) (e+f x)^n \left (1-d^2 x^2\right )^p \, dx=-\frac {2^{1+p} (1-d x)^{1+p} (e+f x)^n \left (\frac {d (e+f x)}{d e+f}\right )^{-n} \operatorname {AppellF1}\left (1+p,-1-p,-n,2+p,\frac {1}{2} (1-d x),\frac {f (1-d x)}{d e+f}\right )}{d (1+p)} \] Output:

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

Mathematica [F]

\[ \int (1+d x) (e+f x)^n \left (1-d^2 x^2\right )^p \, dx=\int (1+d x) (e+f x)^n \left (1-d^2 x^2\right )^p \, dx \] Input:

Integrate[(1 + d*x)*(e + f*x)^n*(1 - d^2*x^2)^p,x]
 

Output:

Integrate[(1 + d*x)*(e + f*x)^n*(1 - d^2*x^2)^p, x]
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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.

\(\displaystyle \int (d x+1) \left (1-d^2 x^2\right )^p (e+f x)^n \, dx\)

\(\Big \downarrow \) 717

\(\displaystyle \int (1-d x)^p (d x+1)^{p+1} (e+f x)^ndx\)

\(\Big \downarrow \) 156

\(\displaystyle (e+f x)^n \left (\frac {d (e+f x)}{d e+f}\right )^{-n} \int (1-d x)^p (d x+1)^{p+1} \left (\frac {d e}{d e+f}+\frac {d f x}{d e+f}\right )^ndx\)

\(\Big \downarrow \) 155

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

Input:

Int[(1 + d*x)*(e + f*x)^n*(1 - d^2*x^2)^p,x]
 

Output:

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

\[\int \left (d x +1\right ) \left (f x +e \right )^{n} \left (-d^{2} x^{2}+1\right )^{p}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int (1+d x) (e+f x)^n \left (1-d^2 x^2\right )^p \, dx=\int { {\left (d x + 1\right )} {\left (-d^{2} x^{2} + 1\right )}^{p} {\left (f x + e\right )}^{n} \,d x } \] Input:

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

Output:

integral((d*x + 1)*(-d^2*x^2 + 1)^p*(f*x + e)^n, x)
 

Sympy [F]

\[ \int (1+d x) (e+f x)^n \left (1-d^2 x^2\right )^p \, dx=\int \left (- \left (d x - 1\right ) \left (d x + 1\right )\right )^{p} \left (e + f x\right )^{n} \left (d x + 1\right )\, dx \] Input:

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

Output:

Integral((-(d*x - 1)*(d*x + 1))**p*(e + f*x)**n*(d*x + 1), x)
 

Maxima [F]

\[ \int (1+d x) (e+f x)^n \left (1-d^2 x^2\right )^p \, dx=\int { {\left (d x + 1\right )} {\left (-d^{2} x^{2} + 1\right )}^{p} {\left (f x + e\right )}^{n} \,d x } \] Input:

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

Output:

integrate((d*x + 1)*(-d^2*x^2 + 1)^p*(f*x + e)^n, x)
 

Giac [F]

\[ \int (1+d x) (e+f x)^n \left (1-d^2 x^2\right )^p \, dx=\int { {\left (d x + 1\right )} {\left (-d^{2} x^{2} + 1\right )}^{p} {\left (f x + e\right )}^{n} \,d x } \] Input:

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

Output:

integrate((d*x + 1)*(-d^2*x^2 + 1)^p*(f*x + e)^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int (1+d x) (e+f x)^n \left (1-d^2 x^2\right )^p \, dx=\int {\left (e+f\,x\right )}^n\,{\left (1-d^2\,x^2\right )}^p\,\left (d\,x+1\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (1+d x) (e+f x)^n \left (1-d^2 x^2\right )^p \, dx=\text {too large to display} \] Input:

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

Output:

((e + f*x)**n*( - d**2*x**2 + 1)**p*d**3*e**2*n*x + (e + f*x)**n*( - d**2* 
x**2 + 1)**p*d**3*e*f*n*x**2 + 2*(e + f*x)**n*( - d**2*x**2 + 1)**p*d**3*e 
*f*p*x**2 + (e + f*x)**n*( - d**2*x**2 + 1)**p*d**3*e*f*x**2 + (e + f*x)** 
n*( - d**2*x**2 + 1)**p*d**2*e*f*n*x + 2*(e + f*x)**n*( - d**2*x**2 + 1)** 
p*d**2*e*f*p*x + 2*(e + f*x)**n*( - d**2*x**2 + 1)**p*d**2*e*f*x - 2*(e + 
f*x)**n*( - d**2*x**2 + 1)**p*d*e*f*n - 2*(e + f*x)**n*( - d**2*x**2 + 1)* 
*p*d*e*f*p - (e + f*x)**n*( - d**2*x**2 + 1)**p*d*e*f - (e + f*x)**n*( - d 
**2*x**2 + 1)**p*f**2*n - 2*(e + f*x)**n*( - d**2*x**2 + 1)**p*f**2*p - 2* 
(e + f*x)**n*( - d**2*x**2 + 1)**p*f**2 - 2*int(((e + f*x)**n*( - d**2*x** 
2 + 1)**p*x**2)/(d**2*e*n**2*x**2 + 4*d**2*e*n*p*x**2 + 3*d**2*e*n*x**2 + 
4*d**2*e*p**2*x**2 + 6*d**2*e*p*x**2 + 2*d**2*e*x**2 + d**2*f*n**2*x**3 + 
4*d**2*f*n*p*x**3 + 3*d**2*f*n*x**3 + 4*d**2*f*p**2*x**3 + 6*d**2*f*p*x**3 
 + 2*d**2*f*x**3 - e*n**2 - 4*e*n*p - 3*e*n - 4*e*p**2 - 6*e*p - 2*e - f*n 
**2*x - 4*f*n*p*x - 3*f*n*x - 4*f*p**2*x - 6*f*p*x - 2*f*x),x)*d**5*e**3*n 
**3*p - int(((e + f*x)**n*( - d**2*x**2 + 1)**p*x**2)/(d**2*e*n**2*x**2 + 
4*d**2*e*n*p*x**2 + 3*d**2*e*n*x**2 + 4*d**2*e*p**2*x**2 + 6*d**2*e*p*x**2 
 + 2*d**2*e*x**2 + d**2*f*n**2*x**3 + 4*d**2*f*n*p*x**3 + 3*d**2*f*n*x**3 
+ 4*d**2*f*p**2*x**3 + 6*d**2*f*p*x**3 + 2*d**2*f*x**3 - e*n**2 - 4*e*n*p 
- 3*e*n - 4*e*p**2 - 6*e*p - 2*e - f*n**2*x - 4*f*n*p*x - 3*f*n*x - 4*f*p* 
*2*x - 6*f*p*x - 2*f*x),x)*d**5*e**3*n**3 - 8*int(((e + f*x)**n*( - d**...