\(\int (1-x)^n x^3 (1+x)^{-n} \, dx\) [326]

Optimal result

Integrand size = 18, antiderivative size = 130 \[ \int (1-x)^n x^3 (1+x)^{-n} \, dx=-\frac {1}{6} \left (3-n+n^2\right ) (1-x)^{1+n} (1+x)^{1-n}+\frac {1}{6} (3-n) (1-x)^{2+n} (1+x)^{1-n}-\frac {1}{4} (1-x)^{3+n} (1+x)^{1-n}-\frac {2^n n \left (2+n^2\right ) (1+x)^{1-n} \operatorname {Hypergeometric2F1}\left (1-n,-n,2-n,\frac {1+x}{2}\right )}{3 (1-n)} \] Output:

-1/6*(n^2-n+3)*(1-x)^(1+n)*(1+x)^(1-n)+1/6*(3-n)*(1-x)^(2+n)*(1+x)^(1-n)-1 
/4*(1-x)^(3+n)*(1+x)^(1-n)-2^n*n*(n^2+2)*(1+x)^(1-n)*hypergeom([-n, 1-n],[ 
2-n],1/2+1/2*x)/(3-3*n)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int (1-x)^n x^3 (1+x)^{-n} \, dx=\frac {2^{-2-n} (1-x)^n (-1+x) (1+x)^{-n} \left (-8 n (1+x)^n \operatorname {Hypergeometric2F1}\left (-2+n,1+n,2+n,\frac {1-x}{2}\right )+4 (1+2 n) (1+x)^n \operatorname {Hypergeometric2F1}\left (-1+n,1+n,2+n,\frac {1-x}{2}\right )+(1+n) \left (2^n x^2 (1+x)-2 (1+x)^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {1-x}{2}\right )\right )\right )}{1+n} \] Input:

Integrate[((1 - x)^n*x^3)/(1 + x)^n,x]
 

Output:

(2^(-2 - n)*(1 - x)^n*(-1 + x)*(-8*n*(1 + x)^n*Hypergeometric2F1[-2 + n, 1 
 + n, 2 + n, (1 - x)/2] + 4*(1 + 2*n)*(1 + x)^n*Hypergeometric2F1[-1 + n, 
1 + n, 2 + n, (1 - x)/2] + (1 + n)*(2^n*x^2*(1 + x) - 2*(1 + x)^n*Hypergeo 
metric2F1[n, 1 + n, 2 + n, (1 - x)/2])))/((1 + n)*(1 + x)^n)
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {111, 27, 164, 79}

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.

\(\displaystyle \int x^3 (1-x)^n (x+1)^{-n} \, dx\)

\(\Big \downarrow \) 111

\(\displaystyle -\frac {1}{4} \int -2 (1-x)^n x (x+1)^{-n} (1-n x)dx-\frac {1}{4} x^2 (1-x)^{n+1} (x+1)^{1-n}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int (1-x)^n x (x+1)^{-n} (1-n x)dx-\frac {1}{4} x^2 (1-x)^{n+1} (x+1)^{1-n}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} n \left (n^2+2\right ) \int (1-x)^n (x+1)^{-n}dx-\frac {1}{6} (1-x)^{n+1} \left (2 n^2-2 n x+3\right ) (x+1)^{1-n}\right )-\frac {1}{4} x^2 (1-x)^{n+1} (x+1)^{1-n}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {1}{2} \left (\frac {2^{1-n} n \left (n^2+2\right ) (1-x)^{n+1} \operatorname {Hypergeometric2F1}\left (n,n+1,n+2,\frac {1-x}{2}\right )}{3 (n+1)}-\frac {1}{6} (1-x)^{n+1} (x+1)^{1-n} \left (2 n^2-2 n x+3\right )\right )-\frac {1}{4} x^2 (1-x)^{n+1} (x+1)^{1-n}\)

Input:

Int[((1 - x)^n*x^3)/(1 + x)^n,x]
 

Output:

-1/4*((1 - x)^(1 + n)*x^2*(1 + x)^(1 - n)) + (-1/6*((1 - x)^(1 + n)*(1 + x 
)^(1 - n)*(3 + 2*n^2 - 2*n*x)) + (2^(1 - n)*n*(2 + n^2)*(1 - x)^(1 + n)*Hy 
pergeometric2F1[n, 1 + n, 2 + n, (1 - x)/2])/(3*(1 + n)))/2
 

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 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 111

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

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

Maple [F]

\[\int \left (1-x \right )^{n} x^{3} \left (1+x \right )^{-n}d x\]

Input:

int((1-x)^n*x^3/((1+x)^n),x)
 

Output:

int((1-x)^n*x^3/((1+x)^n),x)
 

Fricas [F]

\[ \int (1-x)^n x^3 (1+x)^{-n} \, dx=\int { \frac {x^{3} {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}} \,d x } \] Input:

integrate((1-x)^n*x^3/((1+x)^n),x, algorithm="fricas")
 

Output:

integral(x^3*(-x + 1)^n/(x + 1)^n, x)
 

Sympy [F]

\[ \int (1-x)^n x^3 (1+x)^{-n} \, dx=\int x^{3} \left (1 - x\right )^{n} \left (x + 1\right )^{- n}\, dx \] Input:

integrate((1-x)**n*x**3/((1+x)**n),x)
 

Output:

Integral(x**3*(1 - x)**n/(x + 1)**n, x)
 

Maxima [F]

\[ \int (1-x)^n x^3 (1+x)^{-n} \, dx=\int { \frac {x^{3} {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}} \,d x } \] Input:

integrate((1-x)^n*x^3/((1+x)^n),x, algorithm="maxima")
 

Output:

integrate(x^3*(-x + 1)^n/(x + 1)^n, x)
 

Giac [F]

\[ \int (1-x)^n x^3 (1+x)^{-n} \, dx=\int { \frac {x^{3} {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}} \,d x } \] Input:

integrate((1-x)^n*x^3/((1+x)^n),x, algorithm="giac")
 

Output:

integrate(x^3*(-x + 1)^n/(x + 1)^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int (1-x)^n x^3 (1+x)^{-n} \, dx=\int \frac {x^3\,{\left (1-x\right )}^n}{{\left (x+1\right )}^n} \,d x \] Input:

int((x^3*(1 - x)^n)/(x + 1)^n,x)
 

Output:

int((x^3*(1 - x)^n)/(x + 1)^n, x)
 

Reduce [F]

\[ \int (1-x)^n x^3 (1+x)^{-n} \, dx=\int \frac {\left (1-x \right )^{n} x^{3}}{\left (x +1\right )^{n}}d x \] Input:

int((1-x)^n*x^3/((1+x)^n),x)
 

Output:

int((( - x + 1)**n*x**3)/(x + 1)**n,x)