Optimal. Leaf size=30 \[ 2^{1+n} \sqrt {1+x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {1+x}{2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {71}
\begin {gather*} 2^{n+1} \sqrt {x+1} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {x+1}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 71
Rubi steps
\begin {align*} \int \frac {(1-x)^n}{\sqrt {1+x}} \, dx &=2^{1+n} \sqrt {1+x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {1+x}{2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 30, normalized size = 1.00 \begin {gather*} 2^{1+n} \sqrt {1+x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {1+x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (1-x \right )^{n}}{\sqrt {1+x}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 1.17, size = 29, normalized size = 0.97 \begin {gather*} 2 \cdot 2^{n} \sqrt {x + 1} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - n \\ \frac {3}{2} \end {matrix}\middle | {\frac {\left (x + 1\right ) e^{2 i \pi }}{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (1-x\right )}^n}{\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________