Optimal. Leaf size=112 \[ -\frac{7 \sqrt{x+1}}{6 \sqrt{1-x} x^2}+\frac{2 \sqrt{x+1}}{3 (1-x)^{3/2} x^2}+\frac{26 \sqrt{x+1}}{3 \sqrt{1-x}}-\frac{19 \sqrt{x+1}}{6 \sqrt{1-x} x}-\frac{11}{2} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
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Rubi [A] time = 0.025062, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{7 \sqrt{x+1}}{6 \sqrt{1-x} x^2}+\frac{2 \sqrt{x+1}}{3 (1-x)^{3/2} x^2}+\frac{26 \sqrt{x+1}}{3 \sqrt{1-x}}-\frac{19 \sqrt{x+1}}{6 \sqrt{1-x} x}-\frac{11}{2} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
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^3} \, dx &=\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x^2}-\frac{2}{3} \int \frac{-\frac{7}{2}-3 x}{(1-x)^{3/2} x^3 \sqrt{1+x}} \, dx\\ &=\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x^2}-\frac{7 \sqrt{1+x}}{6 \sqrt{1-x} x^2}+\frac{1}{3} \int \frac{\frac{19}{2}+7 x}{(1-x)^{3/2} x^2 \sqrt{1+x}} \, dx\\ &=\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x^2}-\frac{7 \sqrt{1+x}}{6 \sqrt{1-x} x^2}-\frac{19 \sqrt{1+x}}{6 \sqrt{1-x} x}-\frac{1}{3} \int \frac{-\frac{33}{2}-\frac{19 x}{2}}{(1-x)^{3/2} x \sqrt{1+x}} \, dx\\ &=\frac{26 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x^2}-\frac{7 \sqrt{1+x}}{6 \sqrt{1-x} x^2}-\frac{19 \sqrt{1+x}}{6 \sqrt{1-x} x}+\frac{1}{3} \int \frac{33}{2 \sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=\frac{26 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x^2}-\frac{7 \sqrt{1+x}}{6 \sqrt{1-x} x^2}-\frac{19 \sqrt{1+x}}{6 \sqrt{1-x} x}+\frac{11}{2} \int \frac{1}{\sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=\frac{26 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x^2}-\frac{7 \sqrt{1+x}}{6 \sqrt{1-x} x^2}-\frac{19 \sqrt{1+x}}{6 \sqrt{1-x} x}-\frac{11}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x} \sqrt{1+x}\right )\\ &=\frac{26 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 \sqrt{1+x}}{3 (1-x)^{3/2} x^2}-\frac{7 \sqrt{1+x}}{6 \sqrt{1-x} x^2}-\frac{19 \sqrt{1+x}}{6 \sqrt{1-x} x}-\frac{11}{2} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0272796, size = 74, normalized size = 0.66 \[ \frac{52 x^4-19 x^3-59 x^2-33 (x-1) \sqrt{1-x^2} x^2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+15 x+3}{6 (x-1) x^2 \sqrt{1-x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.012, size = 129, normalized size = 1.2 \begin{align*} -{\frac{1}{6\,{x}^{2} \left ( -1+x \right ) ^{2}} \left ( 33\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{4}-66\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{3}+52\,{x}^{3}\sqrt{-{x}^{2}+1}+33\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{2}-71\,{x}^{2}\sqrt{-{x}^{2}+1}+12\,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 = 1.02477, size = 135, normalized size = 1.21 \begin{align*} \frac{26 \, x}{3 \, \sqrt{-x^{2} + 1}} + \frac{11}{2 \, \sqrt{-x^{2} + 1}} + \frac{13 \, x}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{11}{6 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{3}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x} - \frac{1}{2 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{2}} - \frac{11}{2} \, \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 = 1.58483, size = 231, normalized size = 2.06 \begin{align*} \frac{38 \, x^{4} - 76 \, x^{3} + 38 \, x^{2} -{\left (52 \, x^{3} - 71 \, x^{2} + 12 \, x + 3\right )} \sqrt{x + 1} \sqrt{-x + 1} + 33 \,{\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right )}{6 \,{\left (x^{4} - 2 \, x^{3} + x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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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