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 = 0.03, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 12
Rule 92
Rule 99
Rule 151
Rule 152
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 1.16, size = 95, normalized size = 0.85 \[ \frac {38 \, x^{4} - 76 \, x^{3} + 38 \, x^{2} - {\left (52 \, x^{3} - 71 \, x^{2} + 12 \, x + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 33 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right )}{6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )}} \]
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 [A] time = 0.02, size = 129, normalized size = 1.15 \[ -\frac {\left (33 x^{4} \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )-66 x^{3} \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )+52 \sqrt {-x^{2}+1}\, x^{3}+33 x^{2} \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )-71 \sqrt {-x^{2}+1}\, x^{2}+12 \sqrt {-x^{2}+1}\, x +3 \sqrt {-x^{2}+1}\right ) \sqrt {-x +1}\, \sqrt {x +1}}{6 \left (x -1\right )^{2} \sqrt {-x^{2}+1}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 100, normalized size = 0.89 \[ \frac {26 \, x}{3 \, \sqrt {-x^{2} + 1}} + \frac {11}{2 \, \sqrt {-x^{2} + 1}} + \frac {13 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {11}{6 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {3}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} x} - \frac {1}{2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} - \frac {11}{2} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {x+1}}{x^3\,{\left (1-x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\sqrt {x + 1}}{x^{3} \left (1 - x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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