Optimal. Leaf size=41 \[ -\frac {2 \sqrt {1+x}}{\sqrt {1-x}}+\frac {2 (1+x)^{3/2}}{3 (1-x)^{3/2}}+\sin ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 41, 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 = {49, 41, 222}
\begin {gather*} \frac {2 (x+1)^{3/2}}{3 (1-x)^{3/2}}-\frac {2 \sqrt {x+1}}{\sqrt {1-x}}+\sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 49
Rule 222
Rubi steps
\begin {align*} \int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx &=\frac {2 (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 {2 (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 {2 (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 {2 (1+x)^{3/2}}{3 (1-x)^{3/2}}+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 46, normalized size = 1.12 \begin {gather*} \frac {4 \sqrt {1+x} (-1+2 x)}{3 (1-x)^{3/2}}-2 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 6.39, size = 266, normalized size = 6.49 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {4 I \left (-1+x\right ) \left (-1+2 x\right ) \sqrt {1+x}+3 \left (-2 \text {Pi}+\text {Pi} \left (1+x\right )-2 I \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \left (1+x\right )+4 I \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]\right ) \left (-1+x\right )^{\frac {3}{2}}}{3 \left (-1+x\right )^{\frac {5}{2}}},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-12 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \left (1+x\right )^{\frac {13}{2}} \sqrt {1-x}}{-6 \left (1+x\right )^{\frac {13}{2}} \sqrt {1-x}+3 \left (1+x\right )^{\frac {15}{2}} \sqrt {1-x}}-\frac {8 \left (1+x\right )^8}{-6 \left (1+x\right )^{\frac {13}{2}} \sqrt {1-x}+3 \left (1+x\right )^{\frac {15}{2}} \sqrt {1-x}}+\frac {6 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \left (1+x\right )^{\frac {15}{2}} \sqrt {1-x}}{-6 \left (1+x\right )^{\frac {13}{2}} \sqrt {1-x}+3 \left (1+x\right )^{\frac {15}{2}} \sqrt {1-x}}+\frac {12 \left (1+x\right )^7}{-6 \left (1+x\right )^{\frac {13}{2}} \sqrt {1-x}+3 \left (1+x\right )^{\frac {15}{2}} \sqrt {1-x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs.
\(2(31)=62\).
time = 0.16, size = 76, normalized size = 1.85
method | result | size |
risch | \(-\frac {4 \left (2 x^{2}+x -1\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 \left (-1+x \right ) \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (31) = 62\).
time = 0.36, size = 66, normalized size = 1.61 \begin {gather*} -\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {7 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} + \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (31) = 62\).
time = 0.30, size = 71, normalized size = 1.73 \begin {gather*} -\frac {2 \, {\left (2 \, x^{2} - 2 \, {\left (2 \, x - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} + 3 \, {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 4 \, x + 2\right )}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.18, size = 498, normalized size = 12.15 \begin {gather*} \begin {cases} - \frac {6 i \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} + \frac {3 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} + \frac {12 i \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} - \frac {6 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} + \frac {8 i \left (x + 1\right )^{8}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} - \frac {12 i \left (x + 1\right )^{7}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {6 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} - \frac {12 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} - \frac {8 \left (x + 1\right )^{8}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} + \frac {12 \left (x + 1\right )^{7}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 62, normalized size = 1.51 \begin {gather*} \frac {2 \left (\frac {4}{3} \sqrt {x+1} \sqrt {x+1}-2\right ) \sqrt {x+1} \sqrt {-x+1}}{\left (-x+1\right )^{2}}+2 \arcsin \left (\frac {\sqrt {x+1}}{\sqrt {2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (x+1\right )}^{3/2}}{{\left (1-x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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