3.1888 \(\int (1-x)^{7/3} (1+x)^n \, dx\)

Optimal. Leaf size=37 \[ -\frac{3}{5} 2^{n-1} (1-x)^{10/3} \, _2F_1\left (\frac{10}{3},-n;\frac{13}{3};\frac{1-x}{2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0052561, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {69} \[ -\frac{3}{5} 2^{n-1} (1-x)^{10/3} \, _2F_1\left (\frac{10}{3},-n;\frac{13}{3};\frac{1-x}{2}\right ) \]

Antiderivative was successfully verified.

[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 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), 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*} \int (1-x)^{7/3} (1+x)^n \, dx &=-\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]  time = 0.0119005, size = 37, normalized size = 1. \[ -\frac{3}{5} 2^{n-1} (1-x)^{10/3} \, _2F_1\left (\frac{10}{3},-n;\frac{13}{3};\frac{1-x}{2}\right ) \]

Antiderivative was successfully verified.

[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

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int \left ( 1-x \right ) ^{{\frac{7}{3}}} \left ( 1+x \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x + 1\right )}^{n}{\left (-x + 1\right )}^{\frac{7}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (x^{2} - 2 \, x + 1\right )}{\left (x + 1\right )}^{n}{\left (-x + 1\right )}^{\frac{1}{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x + 1\right )}^{n}{\left (-x + 1\right )}^{\frac{7}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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