Optimal. Leaf size=59 \[ \frac {(x+1)^{3/2}}{3 (1-x)^{3/2}}+\frac {2 \sqrt {x+1}}{\sqrt {1-x}}-\tanh ^{-1}\left (\sqrt {1-x} \sqrt {x+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {96, 94, 92, 206} \[ \frac {(x+1)^{3/2}}{3 (1-x)^{3/2}}+\frac {2 \sqrt {x+1}}{\sqrt {1-x}}-\tanh ^{-1}\left (\sqrt {1-x} \sqrt {x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 92
Rule 94
Rule 96
Rule 206
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x} \, dx &=\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}+\int \frac {\sqrt {1+x}}{(1-x)^{3/2} x} \, dx\\ &=\frac {2 \sqrt {1+x}}{\sqrt {1-x}}+\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}+\int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx\\ &=\frac {2 \sqrt {1+x}}{\sqrt {1-x}}+\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x} \sqrt {1+x}\right )\\ &=\frac {2 \sqrt {1+x}}{\sqrt {1-x}}+\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}-\tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 0.98 \[ \frac {5 x^2-3 (x-1) \sqrt {1-x^2} \tanh ^{-1}\left (\sqrt {1-x^2}\right )-2 x-7}{3 (x-1) \sqrt {1-x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.10, size = 71, normalized size = 1.20 \[ \frac {7 \, x^{2} - {\left (5 \, x - 7\right )} \sqrt {x + 1} \sqrt {-x + 1} + 3 \, {\left (x^{2} - 2 \, x + 1\right )} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 14 \, x + 7}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 93, normalized size = 1.58 \[ -\frac {\left (3 x^{2} \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )-6 x \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )+5 \sqrt {-x^{2}+1}\, x +3 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )-7 \sqrt {-x^{2}+1}\right ) \sqrt {-x +1}\, \sqrt {x +1}}{3 \left (x -1\right )^{2} \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 70, normalized size = 1.19 \[ \frac {5 \, x}{3 \, \sqrt {-x^{2} + 1}} + \frac {1}{\sqrt {-x^{2} + 1}} + \frac {4 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {4}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
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 \[ \int \frac {\sqrt {x+1}}{x\,{\left (1-x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x + 1}}{x \left (1 - x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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