3.3138 \(\int \frac{(3+4 x)^n}{\sqrt{1-x} \sqrt{1+x}} \, dx\)

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]

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

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Rubi [A]  time = 0.0583057, antiderivative size = 50, 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 \[ -\sqrt{2} 7^n \sqrt{1-x} F_1\left (\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1-x}{2},\frac{4 (1-x)}{7}\right ) \]

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])

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Rubi in Sympy [A]  time = 5.41709, size = 37, normalized size = 0.74 \[ - \sqrt{2} \cdot 7^{n} \sqrt{- x + 1} \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{2},- n,\frac{3}{2},- \frac{x}{2} + \frac{1}{2},- \frac{4 x}{7} + \frac{4}{7} \right )} \]

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)*7**n*sqrt(-x + 1)*appellf1(1/2, 1/2, -n, 3/2, -x/2 + 1/2, -4*x/7 + 4/7)

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Mathematica [A]  time = 0.0499337, 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])/(Sqr
t[7]*(1 + n))

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Maple [F]  time = 0.06, 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)