### 3.746 $$\int \frac{(3-4 x)^n}{\sqrt{1-x^2}} \, dx$$

Optimal. Leaf size=43 $\sqrt{2} 7^n \sqrt{x+1} F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{4 (x+1)}{7},\frac{x+1}{2}\right )$

[Out]

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

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Rubi [A]  time = 0.0209597, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.105, Rules used = {756, 138} $\sqrt{2} 7^n \sqrt{x+1} F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{4 (x+1)}{7},\frac{x+1}{2}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(3 - 4*x)^n/Sqrt[1 - x^2],x]

[Out]

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

Rule 756

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(d + e*x)^m*(Rt[a, 2] + Rt[-c, 2]*x)
^p*(Rt[a, 2] - Rt[-c, 2]*x)^p, x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&
GtQ[a, 0] && LtQ[c, 0]

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^2}} \, dx &=\int \frac{(3-4 x)^n}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\sqrt{2} 7^n \sqrt{1+x} F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{4 (1+x)}{7},\frac{1+x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0320823, size = 48, normalized size = 1.12 $-\frac{(3-4 x)^{n+1} F_1\left (n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{1}{7} (3-4 x),4 x-3\right )}{\sqrt{7} (n+1)}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.415, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 3-4\,x \right ) ^{n}{\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((3-4*x)^n/(-x^2+1)^(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^{2} + 1}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{2} + 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/(-x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-x^2 + 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 (3 - 4 x\right )^{n}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((3 - 4*x)**n/sqrt(-(x - 1)*(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^{2} + 1}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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