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

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 = {71} \begin {gather*} -\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 ) \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.08, size = 37, normalized size = 1.00 \begin {gather*} -\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 {gather*}

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 = 0.05, size = 0, normalized size = 0.00 \[\int \left (1-x \right )^{\frac {7}{3}} \left (1+x \right )^{n}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 53.26, size = 42, normalized size = 1.14 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (1-x\right )}^{7/3}\,{\left (x+1\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(7/3)*(x + 1)^n,x)

[Out]

int((1 - x)^(7/3)*(x + 1)^n, x)

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