3.3169 \(\int \frac{(-3-4 x)^n}{\sqrt{1-x} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=38 \[ \sqrt{2} \sqrt{x+1} F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};4 (x+1),\frac{x+1}{2}\right ) \]

[Out]

Sqrt[2]*Sqrt[1 + x]*AppellF1[1/2, -n, 1/2, 3/2, 4*(1 + x), (1 + x)/2]

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Rubi [A]  time = 0.0120558, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {138} \[ \sqrt{2} \sqrt{x+1} F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};4 (x+1),\frac{x+1}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-3 - 4*x)^n/(Sqrt[1 - x]*Sqrt[1 + x]),x]

[Out]

Sqrt[2]*Sqrt[1 + x]*AppellF1[1/2, -n, 1/2, 3/2, 4*(1 + x), (1 + x)/2]

Rule 138

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b*(m + 1
)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rubi steps

\begin{align*} \int \frac{(-3-4 x)^n}{\sqrt{1-x} \sqrt{1+x}} \, dx &=\sqrt{2} \sqrt{1+x} F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};4 (1+x),\frac{1+x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0303905, size = 48, normalized size = 1.26 \[ -\frac{(-4 x-3)^{n+1} F_1\left (n+1;\frac{1}{2},\frac{1}{2};n+2;-4 x-3,\frac{1}{7} (4 x+3)\right )}{\sqrt{7} (n+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(-3 - 4*x)^n/(Sqrt[1 - x]*Sqrt[1 + x]),x]

[Out]

-(((-3 - 4*x)^(1 + n)*AppellF1[1 + n, 1/2, 1/2, 2 + n, -3 - 4*x, (3 + 4*x)/7])/(Sqrt[7]*(1 + n)))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -3-4\,x \right ) ^{n}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3-4*x)^n/(1-x)^(1/2)/(1+x)^(1/2),x)

[Out]

int((-3-4*x)^n/(1-x)^(1/2)/(1+x)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-4 \, x - 3\right )}^{n}}{\sqrt{x + 1} \sqrt{-x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3-4*x)^n/(1-x)^(1/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

integrate((-4*x - 3)^n/(sqrt(x + 1)*sqrt(-x + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x + 1} \sqrt{-x + 1}{\left (-4 \, x - 3\right )}^{n}}{x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3-4*x)^n/(1-x)^(1/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(x + 1)*sqrt(-x + 1)*(-4*x - 3)^n/(x^2 - 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- 4 x - 3\right )^{n}}{\sqrt{1 - x} \sqrt{x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3-4*x)**n/(1-x)**(1/2)/(1+x)**(1/2),x)

[Out]

Integral((-4*x - 3)**n/(sqrt(1 - x)*sqrt(x + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-4 \, x - 3\right )}^{n}}{\sqrt{x + 1} \sqrt{-x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3-4*x)^n/(1-x)^(1/2)/(1+x)^(1/2),x, algorithm="giac")

[Out]

integrate((-4*x - 3)^n/(sqrt(x + 1)*sqrt(-x + 1)), x)