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]
________________________________________________________________________________________
Rubi [A] time = 0.0155432, 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, Rules used = {138} \[ -\sqrt{2} 7^n \sqrt{1-x} F_1\left (\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1-x}{2},\frac{4 (1-x)}{7}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 138
Rubi steps
\begin{align*} \int \frac{(3+4 x)^n}{\sqrt{1-x} \sqrt{1+x}} \, dx &=-\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 )\\ \end{align*}
Mathematica [A] time = 0.0350669, 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]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 4\,x+3 \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (4 \, x + 3\right )}^{n} \sqrt{x + 1} \sqrt{-x + 1}}{x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]