Optimal. Leaf size=41 \[ \frac{(x+1)^{3/2}}{3 (1-x)^{3/2}}-\frac{2 \sqrt{x+1}}{\sqrt{1-x}}+\sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.005062, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 47, 41, 216} \[ \frac{(x+1)^{3/2}}{3 (1-x)^{3/2}}-\frac{2 \sqrt{x+1}}{\sqrt{1-x}}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 47
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{x \sqrt{1+x}}{(1-x)^{5/2}} \, dx &=\frac{(1+x)^{3/2}}{3 (1-x)^{3/2}}-\int \frac{\sqrt{1+x}}{(1-x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1+x}}{\sqrt{1-x}}+\frac{(1+x)^{3/2}}{3 (1-x)^{3/2}}+\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{2 \sqrt{1+x}}{\sqrt{1-x}}+\frac{(1+x)^{3/2}}{3 (1-x)^{3/2}}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{2 \sqrt{1+x}}{\sqrt{1-x}}+\frac{(1+x)^{3/2}}{3 (1-x)^{3/2}}+\sin ^{-1}(x)\\ \end{align*}
Mathematica [C] time = 0.0313656, size = 47, normalized size = 1.15 \[ -\frac{(x+1)^{3/2}-4 \sqrt{2} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{1-x}{2}\right )}{3 (1-x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.01, size = 69, normalized size = 1.7 \begin{align*}{\frac{1}{3\, \left ( -1+x \right ) ^{2}} \left ( 3\,\arcsin \left ( x \right ){x}^{2}-6\,\arcsin \left ( x \right ) x+7\,x\sqrt{-{x}^{2}+1}+3\,\arcsin \left ( x \right ) -5\,\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \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{\sqrt{x + 1} x}{{\left (-x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.80709, size = 188, normalized size = 4.59 \begin{align*} -\frac{5 \, x^{2} -{\left (7 \, x - 5\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \,{\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 10 \, x + 5}{3 \,{\left (x^{2} - 2 \, x + 1\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{x \sqrt{x + 1}}{\left (1 - x\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 3.40459, size = 51, normalized size = 1.24 \begin{align*} \frac{{\left (7 \, x - 5\right )} \sqrt{x + 1} \sqrt{-x + 1}}{3 \,{\left (x - 1\right )}^{2}} + 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]