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 = 0.0120558, 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, Rules used = {138} \[ \sqrt{2} \sqrt{x+1} F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};4 (x+1),\frac{x+1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 138
Rubi steps
\begin{align*} \int \frac{(-3-4 x)^n}{\sqrt{1-x} \sqrt{1+x}} \, dx &=\sqrt{2} \sqrt{1+x} F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};4 (1+x),\frac{1+x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0303905, 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.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -3-4\,x \right ) ^{n}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-4 \, x - 3\right )}^{n}}{\sqrt{x + 1} \sqrt{-x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x + 1} \sqrt{-x + 1}{\left (-4 \, x - 3\right )}^{n}}{x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- 4 x - 3\right )^{n}}{\sqrt{1 - x} \sqrt{x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-4 \, x - 3\right )}^{n}}{\sqrt{x + 1} \sqrt{-x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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