\(\int (1-x)^{7/3} (1+x)^n \, dx\) [1888]

Optimal result

Integrand size = 15, antiderivative size = 37 \[ \int (1-x)^{7/3} (1+x)^n \, dx=-\frac {3}{5} 2^{-1+n} (1-x)^{10/3} \operatorname {Hypergeometric2F1}\left (\frac {10}{3},-n,\frac {13}{3},\frac {1-x}{2}\right ) \]

[Out]

-3/5*2^(-1+n)*(1-x)^(10/3)*hypergeom([10/3, -n],[13/3],1/2-1/2*x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {71} \[ \int (1-x)^{7/3} (1+x)^n \, dx=-\frac {3}{5} 2^{n-1} (1-x)^{10/3} \operatorname {Hypergeometric2F1}\left (\frac {10}{3},-n,\frac {13}{3},\frac {1-x}{2}\right ) \]

[In]

Int[(1 - x)^(7/3)*(1 + x)^n,x]

[Out]

(-3*2^(-1 + n)*(1 - x)^(10/3)*Hypergeometric2F1[10/3, -n, 13/3, (1 - x)/2])/5

Rule 71

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] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rubi steps \begin{align*} \text {integral}& = -\frac {3}{5} 2^{-1+n} (1-x)^{10/3} \, _2F_1\left (\frac {10}{3},-n;\frac {13}{3};\frac {1-x}{2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int (1-x)^{7/3} (1+x)^n \, dx=-\frac {3}{5} 2^{-1+n} (1-x)^{10/3} \operatorname {Hypergeometric2F1}\left (\frac {10}{3},-n,\frac {13}{3},\frac {1-x}{2}\right ) \]

[In]

Integrate[(1 - x)^(7/3)*(1 + x)^n,x]

[Out]

(-3*2^(-1 + n)*(1 - x)^(10/3)*Hypergeometric2F1[10/3, -n, 13/3, (1 - x)/2])/5

Maple [F]

\[\int \left (1-x \right )^{\frac {7}{3}} \left (1+x \right )^{n}d x\]

[In]

int((1-x)^(7/3)*(1+x)^n,x)

[Out]

int((1-x)^(7/3)*(1+x)^n,x)

Fricas [F]

\[ \int (1-x)^{7/3} (1+x)^n \, dx=\int { {\left (x + 1\right )}^{n} {\left (-x + 1\right )}^{\frac {7}{3}} \,d x } \]

[In]

integrate((1-x)^(7/3)*(1+x)^n,x, algorithm="fricas")

[Out]

integral((x^2 - 2*x + 1)*(x + 1)^n*(-x + 1)^(1/3), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 46.61 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int (1-x)^{7/3} (1+x)^n \, dx=\frac {\sqrt [3]{-1} \cdot 2^{n} \left (x - 1\right )^{\frac {10}{3}} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {10}{3}, - n \\ \frac {13}{3} \end {matrix}\middle | {\frac {\left (x - 1\right ) e^{i \pi }}{2}} \right )}}{\Gamma \left (\frac {13}{3}\right )} \]

[In]

integrate((1-x)**(7/3)*(1+x)**n,x)

[Out]

(-1)**(1/3)*2**n*(x - 1)**(10/3)*gamma(10/3)*hyper((10/3, -n), (13/3,), (x - 1)*exp_polar(I*pi)/2)/gamma(13/3)

Maxima [F]

\[ \int (1-x)^{7/3} (1+x)^n \, dx=\int { {\left (x + 1\right )}^{n} {\left (-x + 1\right )}^{\frac {7}{3}} \,d x } \]

[In]

integrate((1-x)^(7/3)*(1+x)^n,x, algorithm="maxima")

[Out]

integrate((x + 1)^n*(-x + 1)^(7/3), x)

Giac [F]

\[ \int (1-x)^{7/3} (1+x)^n \, dx=\int { {\left (x + 1\right )}^{n} {\left (-x + 1\right )}^{\frac {7}{3}} \,d x } \]

[In]

integrate((1-x)^(7/3)*(1+x)^n,x, algorithm="giac")

[Out]

integrate((x + 1)^n*(-x + 1)^(7/3), x)

Mupad [F(-1)]

Timed out. \[ \int (1-x)^{7/3} (1+x)^n \, dx=\int {\left (1-x\right )}^{7/3}\,{\left (x+1\right )}^n \,d x \]

[In]

int((1 - x)^(7/3)*(x + 1)^n,x)

[Out]

int((1 - x)^(7/3)*(x + 1)^n, x)