3.3141 \(\int \frac{(-3-4 x)^n}{\sqrt{1-x} \sqrt{1+x}} \, dx\)

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 = 0.0550025, 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 \[ \sqrt{2} \sqrt{x+1} F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};4 (x+1),\frac{x+1}{2}\right ) \]

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]

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Rubi in Sympy [A]  time = 5.34981, size = 29, normalized size = 0.76 \[ \sqrt{2} \sqrt{x + 1} \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{2},- n,\frac{3}{2},\frac{x}{2} + \frac{1}{2},4 x + 4 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3-4*x)**n/(1-x)**(1/2)/(1+x)**(1/2),x)

[Out]

sqrt(2)*sqrt(x + 1)*appellf1(1/2, 1/2, -n, 3/2, x/2 + 1/2, 4*x + 4)

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Mathematica [A]  time = 0.049471, 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 = 0.049, size = 0, normalized size = 0. \[ \int{ \left ( -3-4\,x \right ) ^{n}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}\, dx \]

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. \[ \int \frac{{\left (-4 \, x - 3\right )}^{n}}{\sqrt{x + 1} \sqrt{-x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x - 3)^n/(sqrt(x + 1)*sqrt(-x + 1)),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. \[{\rm integral}\left (\frac{{\left (-4 \, x - 3\right )}^{n}}{\sqrt{x + 1} \sqrt{-x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x - 3)^n/(sqrt(x + 1)*sqrt(-x + 1)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x - 3)^n/(sqrt(x + 1)*sqrt(-x + 1)),x, algorithm="giac")

[Out]

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