Optimal. Leaf size=65 \[ \frac {15}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 65, 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 = {49, 52, 41, 222}
\begin {gather*} \frac {2 (x+1)^{5/2}}{\sqrt {1-x}}+\frac {5}{2} \sqrt {1-x} (x+1)^{3/2}+\frac {15}{2} \sqrt {1-x} \sqrt {x+1}-\frac {15}{2} \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 49
Rule 52
Rule 222
Rubi steps
\begin {align*} \int \frac {(1+x)^{5/2}}{(1-x)^{3/2}} \, dx &=\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-5 \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=\frac {15}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {15}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {15}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \sin ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 49, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {1+x} \left (-24+7 x+x^2\right )}{2 \sqrt {1-x}}+15 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 7.25, size = 110, normalized size = 1.69 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (\left (1+x\right )^{\frac {5}{2}}-30 \sqrt {1+x}+5 \left (1+x\right )^{\frac {3}{2}}+30 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {-1+x}\right )}{2 \sqrt {-1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},-15 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-\frac {5 \left (1+x\right )^{\frac {3}{2}}}{2 \sqrt {1-x}}-\frac {\left (1+x\right )^{\frac {5}{2}}}{2 \sqrt {1-x}}+\frac {15 \sqrt {1+x}}{\sqrt {1-x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.18, size = 77, normalized size = 1.18
method | result | size |
risch | \(-\frac {\left (x^{3}+8 x^{2}-17 x -24\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}-\frac {15 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.38, size = 56, normalized size = 0.86 \begin {gather*} -\frac {x^{3}}{2 \, \sqrt {-x^{2} + 1}} - \frac {4 \, x^{2}}{\sqrt {-x^{2} + 1}} + \frac {17 \, x}{2 \, \sqrt {-x^{2} + 1}} + \frac {12}{\sqrt {-x^{2} + 1}} - \frac {15}{2} \, \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.30, size = 58, normalized size = 0.89 \begin {gather*} \frac {{\left (x^{2} + 7 \, x - 24\right )} \sqrt {x + 1} \sqrt {-x + 1} + 30 \, {\left (x - 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 24 \, x - 24}{2 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 6.18, size = 138, normalized size = 2.12 \begin {gather*} \begin {cases} 15 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 {5 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} - \frac {15 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- 15 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} - \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} + \frac {15 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.01, size = 84, normalized size = 1.29 \begin {gather*} \frac {2 \left (\left (-\frac {5}{4}-\frac {1}{4} \sqrt {x+1} \sqrt {x+1}\right ) \sqrt {x+1} \sqrt {x+1}+\frac {15}{2}\right ) \sqrt {x+1} \sqrt {-x+1}}{-x+1}-15 \arcsin \left (\frac {\sqrt {x+1}}{\sqrt {2}}\right ) \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 (x+1\right )}^{5/2}}{{\left (1-x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________