Optimal. Leaf size=38 \[ \sqrt{2} \sqrt{x+1} F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};4 (x+1),\frac{x+1}{2}\right ) \]
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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]
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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)
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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]
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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]
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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]
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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]
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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]
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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]