3.3166 \(\int \frac {(3+4 x)^n}{\sqrt {1-x} \sqrt {1+x}} \, dx\)

Optimal. Leaf size=50 \[ -\sqrt {2} 7^n \sqrt {1-x} F_1\left (\frac {1}{2};\frac {1}{2},-n;\frac {3}{2};\frac {1-x}{2},\frac {4 (1-x)}{7}\right ) \]

[Out]

-7^n*AppellF1(1/2,1/2,-n,3/2,1/2-1/2*x,4/7-4/7*x)*2^(1/2)*(1-x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, 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} \[ -\sqrt {2} 7^n \sqrt {1-x} F_1\left (\frac {1}{2};\frac {1}{2},-n;\frac {3}{2};\frac {1-x}{2},\frac {4 (1-x)}{7}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 47, normalized size = 0.94 \[ \frac {(4 x+3)^{n+1} F_1\left (n+1;\frac {1}{2},\frac {1}{2};n+2;-4 x-3,\frac {1}{7} (4 x+3)\right )}{\sqrt {7} (n+1)} \]

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))

________________________________________________________________________________________

fricas [F]  time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (4 \, x + 3\right )}^{n} \sqrt {x + 1} \sqrt {-x + 1}}{x^{2} - 1}, x\right ) \]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (4 \, x + 3\right )}^{n}}{\sqrt {x + 1} \sqrt {-x + 1}}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\left (4 x +3\right )^{n}}{\sqrt {-x +1}\, \sqrt {x +1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (4 \, x + 3\right )}^{n}}{\sqrt {x + 1} \sqrt {-x + 1}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (4\,x+3\right )}^n}{\sqrt {1-x}\,\sqrt {x+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (4 x + 3\right )^{n}}{\sqrt {1 - x} \sqrt {x + 1}}\, dx \]

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)

________________________________________________________________________________________