Optimal. Leaf size=90 \[ -\left ((1-x) \sqrt {\frac {2 x}{x^2+1}+1} (x+1)\right )-\frac {x \left (x^2+1\right ) \sqrt {\frac {2 x}{x^2+1}+1}}{x+1}+\frac {3 \sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1} \sinh ^{-1}(x)}{x+1} \]
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Rubi [A] time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6723, 970, 739, 517, 388, 215} \[ -(1-x) \sqrt {\frac {2 x}{x^2+1}+1} (x+1)-\frac {x \left (x^2+1\right ) \sqrt {\frac {2 x}{x^2+1}+1}}{x+1}+\frac {3 \sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1} \sinh ^{-1}(x)}{x+1} \]
Antiderivative was successfully verified.
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Rule 215
Rule 388
Rule 517
Rule 739
Rule 970
Rule 6723
Rubi steps
\begin {align*} \int \left (1+\frac {2 x}{1+x^2}\right )^{3/2} \, dx &=\frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {\left (1+2 x+x^2\right )^{3/2}}{\left (1+x^2\right )^{3/2}} \, dx}{\sqrt {1+2 x+x^2}}\\ &=\frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {(2+2 x)^3}{\left (1+x^2\right )^{3/2}} \, dx}{4 (2+2 x)}\\ &=-(1-x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}+\frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {(8-8 x) (2+2 x)}{\sqrt {1+x^2}} \, dx}{4 (2+2 x)}\\ &=-(1-x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}+\frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {16-16 x^2}{\sqrt {1+x^2}} \, dx}{4 (2+2 x)}\\ &=-(1-x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}-\frac {x \left (1+x^2\right ) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}+\frac {\left (6 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {1}{\sqrt {1+x^2}} \, dx}{2+2 x}\\ &=-(1-x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}-\frac {x \left (1+x^2\right ) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}+\frac {3 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}} \sinh ^{-1}(x)}{1+x}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.49 \[ \frac {\sqrt {\frac {(x+1)^2}{x^2+1}} \left (x^2+3 \sqrt {x^2+1} \sinh ^{-1}(x)-2 x-1\right )}{x+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 83, normalized size = 0.92 \[ -\frac {3 \, {\left (x + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} + 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + x}{x + 1}\right ) - {\left (x^{2} - 2 \, x - 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + 2 \, x + 2}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 67, normalized size = 0.74 \[ -{\left (\sqrt {2} - 3 \, \log \left (\sqrt {2} + 1\right )\right )} \mathrm {sgn}\left (x + 1\right ) - 3 \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \mathrm {sgn}\left (x + 1\right ) + \frac {{\left (x \mathrm {sgn}\left (x + 1\right ) - 2 \, \mathrm {sgn}\left (x + 1\right )\right )} x - \mathrm {sgn}\left (x + 1\right )}{\sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 49, normalized size = 0.54 \[ \frac {\left (\frac {x^{2}+2 x +1}{x^{2}+1}\right )^{\frac {3}{2}} \left (x^{2}+1\right ) \left (x^{2}-2 x +3 \sqrt {x^{2}+1}\, \arcsinh \relax (x )-1\right )}{\left (x +1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\frac {2 \, x}{x^{2} + 1} + 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (\frac {2\,x}{x^2+1}+1\right )}^{3/2} \,d x \]
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 \[ \int \left (\frac {2 x}{x^{2} + 1} + 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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