Optimal. Leaf size=47 \[ -\frac {1}{2} \sqrt {1-x} (x+1)^{3/2}-\frac {3}{2} \sqrt {1-x} \sqrt {x+1}+\frac {3}{2} \sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {50, 41, 216} \[ -\frac {1}{2} \sqrt {1-x} (x+1)^{3/2}-\frac {3}{2} \sqrt {1-x} \sqrt {x+1}+\frac {3}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 50
Rule 216
Rubi steps
\begin {align*} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx &=-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-\frac {3}{2} \sqrt {1-x} \sqrt {1+x}-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {3}{2} \sqrt {1-x} \sqrt {1+x}-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {3}{2} \sqrt {1-x} \sqrt {1+x}-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \sin ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 37, normalized size = 0.79 \[ -\frac {1}{2} \sqrt {1-x^2} (x+4)-3 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 40, normalized size = 0.85 \[ -\frac {1}{2} \, {\left (x + 4\right )} \sqrt {x + 1} \sqrt {-x + 1} - 3 \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.10, size = 31, normalized size = 0.66 \[ -\frac {1}{2} \, {\left (x + 4\right )} \sqrt {x + 1} \sqrt {-x + 1} + 3 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 57, normalized size = 1.21 \[ \frac {3 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{2 \sqrt {x +1}\, \sqrt {-x +1}}-\frac {\sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{2}-\frac {3 \sqrt {-x +1}\, \sqrt {x +1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.79, size = 28, normalized size = 0.60 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 1} x - 2 \, \sqrt {-x^{2} + 1} + \frac {3}{2} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (x+1\right )}^{3/2}}{\sqrt {1-x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.37, size = 136, normalized size = 2.89 \[ \begin {cases} - 3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} + \frac {3 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} - \frac {3 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________