Optimal. Leaf size=63 \[ \frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\frac {2 (1+x)^{3/2}}{3 (1-x)^{3/2}}+\frac {2 (1+x)^{5/2}}{5 (1-x)^{5/2}}-\sin ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 5, 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)^{5/2}}{5 (1-x)^{5/2}}-\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)^{5/2}}{(1-x)^{7/2}} \, dx &=\frac {2 (1+x)^{5/2}}{5 (1-x)^{5/2}}-\int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx\\ &=-\frac {2 (1+x)^{3/2}}{3 (1-x)^{3/2}}+\frac {2 (1+x)^{5/2}}{5 (1-x)^{5/2}}+\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}}+\frac {2 (1+x)^{5/2}}{5 (1-x)^{5/2}}-\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}}+\frac {2 (1+x)^{5/2}}{5 (1-x)^{5/2}}-\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}}+\frac {2 (1+x)^{5/2}}{5 (1-x)^{5/2}}-\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 51, normalized size = 0.81 \begin {gather*} \frac {2 \sqrt {1+x} \left (13-24 x+23 x^2\right )}{15 (1-x)^{5/2}}+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 = 18.35, size = 746, normalized size = 11.84 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {15 \text {Pi}+48 I x \sqrt {-1+x} \sqrt {1+x}+90 I x \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-90 I x^2 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-46 I x^2 \sqrt {-1+x} \sqrt {1+x}+30 I x^3 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-45 \text {Pi} x+45 \text {Pi} x^2-15 \text {Pi} x^3-30 I \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-26 I \sqrt {-1+x} \sqrt {1+x}}{15 \left (-1+3 x-3 x^2+x^3\right )},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-360 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \left (1+x\right )^{\frac {31}{2}} \sqrt {1-x}}{-120 \left (1+x\right )^{\frac {29}{2}} \sqrt {1-x}-90 \left (1+x\right )^{\frac {33}{2}} \sqrt {1-x}+15 \left (1+x\right )^{\frac {35}{2}} \sqrt {1-x}+180 \left (1+x\right )^{\frac {31}{2}} \sqrt {1-x}}-\frac {240 \left (1+x\right )^{15}}{-120 \left (1+x\right )^{\frac {29}{2}} \sqrt {1-x}-90 \left (1+x\right )^{\frac {33}{2}} \sqrt {1-x}+15 \left (1+x\right )^{\frac {35}{2}} \sqrt {1-x}+180 \left (1+x\right )^{\frac {31}{2}} \sqrt {1-x}}-\frac {232 \left (1+x\right )^{17}}{-120 \left (1+x\right )^{\frac {29}{2}} \sqrt {1-x}-90 \left (1+x\right )^{\frac {33}{2}} \sqrt {1-x}+15 \left (1+x\right )^{\frac {35}{2}} \sqrt {1-x}+180 \left (1+x\right )^{\frac {31}{2}} \sqrt {1-x}}-\frac {30 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \left (1+x\right )^{\frac {35}{2}} \sqrt {1-x}}{-120 \left (1+x\right )^{\frac {29}{2}} \sqrt {1-x}-90 \left (1+x\right )^{\frac {33}{2}} \sqrt {1-x}+15 \left (1+x\right )^{\frac {35}{2}} \sqrt {1-x}+180 \left (1+x\right )^{\frac {31}{2}} \sqrt {1-x}}+\frac {46 \left (1+x\right )^{18}}{-120 \left (1+x\right )^{\frac {29}{2}} \sqrt {1-x}-90 \left (1+x\right )^{\frac {33}{2}} \sqrt {1-x}+15 \left (1+x\right )^{\frac {35}{2}} \sqrt {1-x}+180 \left (1+x\right )^{\frac {31}{2}} \sqrt {1-x}}+\frac {180 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \left (1+x\right )^{\frac {33}{2}} \sqrt {1-x}}{-120 \left (1+x\right )^{\frac {29}{2}} \sqrt {1-x}-90 \left (1+x\right )^{\frac {33}{2}} \sqrt {1-x}+15 \left (1+x\right )^{\frac {35}{2}} \sqrt {1-x}+180 \left (1+x\right )^{\frac {31}{2}} \sqrt {1-x}}+\frac {240 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \left (1+x\right )^{\frac {29}{2}} \sqrt {1-x}}{-120 \left (1+x\right )^{\frac {29}{2}} \sqrt {1-x}-90 \left (1+x\right )^{\frac {33}{2}} \sqrt {1-x}+15 \left (1+x\right )^{\frac {35}{2}} \sqrt {1-x}+180 \left (1+x\right )^{\frac {31}{2}} \sqrt {1-x}}+\frac {400 \left (1+x\right )^{16}}{-120 \left (1+x\right )^{\frac {29}{2}} \sqrt {1-x}-90 \left (1+x\right )^{\frac {33}{2}} \sqrt {1-x}+15 \left (1+x\right )^{\frac {35}{2}} \sqrt {1-x}+180 \left (1+x\right )^{\frac {31}{2}} \sqrt {1-x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 84, normalized size = 1.33
method | result | size |
risch | \(\frac {2 \left (23 x^{3}-x^{2}-11 x +13\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{15 \left (-1+x \right )^{2} \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(84\) |
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. 160 vs.
\(2 (47) = 94\).
time = 0.36, size = 160, normalized size = 2.54 \begin {gather*} -\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{5 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1} + \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {6 \, \sqrt {-x^{2} + 1}}{5 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {7 \, \sqrt {-x^{2} + 1}}{15 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {38 \, \sqrt {-x^{2} + 1}}{15 \, {\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 [A]
time = 0.29, size = 91, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (13 \, x^{3} - 39 \, x^{2} - {\left (23 \, x^{2} - 24 \, x + 13\right )} \sqrt {x + 1} \sqrt {-x + 1} + 15 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 39 \, x - 13\right )}}{15 \, {\left (x^{3} - 3 \, x^{2} + 3 \, 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 = 8.54, size = 1606, normalized size = 25.49
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 83, normalized size = 1.32 \begin {gather*} \frac {2 \left (\left (\frac {23}{15} \sqrt {x+1} \sqrt {x+1}-\frac {14}{3}\right ) \sqrt {x+1} \sqrt {x+1}+4\right ) \sqrt {x+1} \sqrt {-x+1}}{\left (-x+1\right )^{3}}-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 )}^{5/2}}{{\left (1-x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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