Optimal. Leaf size=63 \[ -\frac {2 (1-x)^{5/2}}{3 (x+1)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {x+1}}+5 \sqrt {x+1} \sqrt {1-x}+5 \sin ^{-1}(x) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {47, 50, 41, 216} \begin {gather*} -\frac {2 (1-x)^{5/2}}{3 (x+1)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {x+1}}+5 \sqrt {x+1} \sqrt {1-x}+5 \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 47
Rule 50
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-x)^{5/2}}{(1+x)^{5/2}} \, dx &=-\frac {2 (1-x)^{5/2}}{3 (1+x)^{3/2}}-\frac {5}{3} \int \frac {(1-x)^{3/2}}{(1+x)^{3/2}} \, dx\\ &=-\frac {2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {1+x}}+5 \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {1+x}}+5 \sqrt {1-x} \sqrt {1+x}+5 \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {1+x}}+5 \sqrt {1-x} \sqrt {1+x}+5 \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {1+x}}+5 \sqrt {1-x} \sqrt {1+x}+5 \sin ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 37, normalized size = 0.59 \begin {gather*} -\frac {(1-x)^{7/2} \, _2F_1\left (\frac {5}{2},\frac {7}{2};\frac {9}{2};\frac {1-x}{2}\right )}{14 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [C] time = 0.17, size = 61, normalized size = 0.97 \begin {gather*} \frac {\sqrt {1-x} \left (3 (x+1)^2+28 (x+1)-8\right )}{3 (x+1)^{3/2}}+10 i \log \left (\sqrt {1-x}-i \sqrt {x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.26, size = 75, normalized size = 1.19 \begin {gather*} \frac {23 \, x^{2} + {\left (3 \, x^{2} + 34 \, x + 23\right )} \sqrt {x + 1} \sqrt {-x + 1} - 30 \, {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 46 \, x + 23}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.75, size = 115, normalized size = 1.83 \begin {gather*} \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{6 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \sqrt {x + 1} \sqrt {-x + 1} - \frac {9 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{2 \, \sqrt {x + 1}} + \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {27 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{6 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} + 10 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 79, normalized size = 1.25 \begin {gather*} \frac {5 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}}-\frac {\left (3 x^{3}+31 x^{2}-11 x -23\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{3 \left (x +1\right )^{\frac {3}{2}} \sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 2.97, size = 98, normalized size = 1.56 \begin {gather*} \frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} - \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} - \frac {10 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {35 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x + 1\right )}} + 5 \, \arcsin \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (1-x\right )}^{5/2}}{{\left (x+1\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 6.46, size = 160, normalized size = 2.54 \begin {gather*} \begin {cases} \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) + \frac {28 \sqrt {-1 + \frac {2}{x + 1}}}{3} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}}}{3 \left (x + 1\right )} + 5 i \log {\left (\frac {1}{x + 1} \right )} + 5 i \log {\left (x + 1 \right )} + 10 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + \frac {28 i \sqrt {1 - \frac {2}{x + 1}}}{3} - \frac {8 i \sqrt {1 - \frac {2}{x + 1}}}{3 \left (x + 1\right )} + 5 i \log {\left (\frac {1}{x + 1} \right )} - 10 i \log {\left (\sqrt {1 - \frac {2}{x + 1}} + 1 \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________