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