3.3140 \(\int \frac{(-3+4 x)^n}{\sqrt{1-x} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=45 \[ -\sqrt{2} \sqrt{1-x} F_1\left (\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1-x}{2},4 (1-x)\right ) \]

[Out]

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

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Rubi [A]  time = 0.0611376, antiderivative size = 45, 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 \[ -\sqrt{2} \sqrt{1-x} F_1\left (\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1-x}{2},4 (1-x)\right ) \]

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

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Rubi in Sympy [A]  time = 5.36534, size = 31, normalized size = 0.69 \[ - \sqrt{2} \sqrt{- x + 1} \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{2},- n,\frac{3}{2},- \frac{x}{2} + \frac{1}{2},- 4 x + 4 \right )} \]

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 = 0.0483507, size = 47, normalized size = 1.04 \[ \frac{(4 x-3)^{n+1} F_1\left (n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{1}{7} (3-4 x),4 x-3\right )}{\sqrt{7} (n+1)} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.081, 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)