Optimal. Leaf size=87 \[ \frac{14 \sqrt{x+1}}{3 \sqrt{1-x}}-\frac{5 \sqrt{x+1}}{3 \sqrt{1-x} x}+\frac{2 \sqrt{x+1}}{3 (1-x)^{3/2} x}-3 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
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Rubi [A] time = 0.0182755, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {99, 151, 152, 12, 92, 206} \[ \frac{14 \sqrt{x+1}}{3 \sqrt{1-x}}-\frac{5 \sqrt{x+1}}{3 \sqrt{1-x} x}+\frac{2 \sqrt{x+1}}{3 (1-x)^{3/2} x}-3 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 99
Rule 151
Rule 152
Rule 12
Rule 92
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1+x}}{(1-x)^{5/2} x^2} \, dx &=\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x}-\frac{2}{3} \int \frac{-\frac{5}{2}-2 x}{(1-x)^{3/2} x^2 \sqrt{1+x}} \, dx\\ &=\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x}-\frac{5 \sqrt{1+x}}{3 \sqrt{1-x} x}+\frac{2}{3} \int \frac{\frac{9}{2}+\frac{5 x}{2}}{(1-x)^{3/2} x \sqrt{1+x}} \, dx\\ &=\frac{14 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x}-\frac{5 \sqrt{1+x}}{3 \sqrt{1-x} x}-\frac{2}{3} \int -\frac{9}{2 \sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=\frac{14 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x}-\frac{5 \sqrt{1+x}}{3 \sqrt{1-x} x}+3 \int \frac{1}{\sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=\frac{14 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x}-\frac{5 \sqrt{1+x}}{3 \sqrt{1-x} x}-3 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x} \sqrt{1+x}\right )\\ &=\frac{14 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x}-\frac{5 \sqrt{1+x}}{3 \sqrt{1-x} x}-3 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.02435, size = 67, normalized size = 0.77 \[ \frac{14 x^3-5 x^2-9 (x-1) \sqrt{1-x^2} x \tanh ^{-1}\left (\sqrt{1-x^2}\right )-16 x+3}{3 (x-1) x \sqrt{1-x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.01, size = 113, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,x \left ( -1+x \right ) ^{2}} \left ( 9\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{3}-18\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{2}+14\,{x}^{2}\sqrt{-{x}^{2}+1}+9\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) x-19\,x\sqrt{-{x}^{2}+1}+3\,\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04439, size = 116, normalized size = 1.33 \begin{align*} \frac{14 \, x}{3 \, \sqrt{-x^{2} + 1}} + \frac{3}{\sqrt{-x^{2} + 1}} + \frac{7 \, x}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{4}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{1}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x} - 3 \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56637, size = 209, normalized size = 2.4 \begin{align*} \frac{13 \, x^{3} - 26 \, x^{2} -{\left (14 \, x^{2} - 19 \, x + 3\right )} \sqrt{x + 1} \sqrt{-x + 1} + 9 \,{\left (x^{3} - 2 \, x^{2} + x\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 13 \, x}{3 \,{\left (x^{3} - 2 \, x^{2} + x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x + 1}}{x^{2} \left (1 - x\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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