Optimal. Leaf size=133 \[ -\frac {(1-x) \sqrt {\frac {2 x}{x^2+1}+1} (x+1)^3}{3 \left (x^2+1\right )}-\frac {4}{3} (1-2 x) \sqrt {\frac {2 x}{x^2+1}+1} (x+1)-\frac {(3 x+4) \left (x^2+1\right ) \sqrt {\frac {2 x}{x^2+1}+1}}{x+1}+\frac {5 \sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1} \sinh ^{-1}(x)}{x+1} \]
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Rubi [A] time = 0.07, antiderivative size = 133, 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, 819, 780, 215} \[ -\frac {(1-x) \sqrt {\frac {2 x}{x^2+1}+1} (x+1)^3}{3 \left (x^2+1\right )}-\frac {4}{3} (1-2 x) \sqrt {\frac {2 x}{x^2+1}+1} (x+1)-\frac {(3 x+4) \left (x^2+1\right ) \sqrt {\frac {2 x}{x^2+1}+1}}{x+1}+\frac {5 \sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1} \sinh ^{-1}(x)}{x+1} \]
Antiderivative was successfully verified.
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Rule 215
Rule 739
Rule 780
Rule 819
Rule 970
Rule 6723
Rubi steps
\begin {align*} \int \left (1+\frac {2 x}{1+x^2}\right )^{5/2} \, 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 )^{5/2}}{\left (1+x^2\right )^{5/2}} \, 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)^5}{\left (1+x^2\right )^{5/2}} \, dx}{16 (2+2 x)}\\ &=-\frac {(1-x) (1+x)^3 \sqrt {1+\frac {2 x}{1+x^2}}}{3 \left (1+x^2\right )}+\frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {(24-8 x) (2+2 x)^3}{\left (1+x^2\right )^{3/2}} \, dx}{48 (2+2 x)}\\ &=-\frac {4}{3} (1-2 x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}-\frac {(1-x) (1+x)^3 \sqrt {1+\frac {2 x}{1+x^2}}}{3 \left (1+x^2\right )}+\frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {(96-288 x) (2+2 x)}{\sqrt {1+x^2}} \, dx}{48 (2+2 x)}\\ &=-\frac {4}{3} (1-2 x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}-\frac {(1-x) (1+x)^3 \sqrt {1+\frac {2 x}{1+x^2}}}{3 \left (1+x^2\right )}-\frac {(4+3 x) \left (1+x^2\right ) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}+\frac {\left (10 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {1}{\sqrt {1+x^2}} \, dx}{2+2 x}\\ &=-\frac {4}{3} (1-2 x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}-\frac {(1-x) (1+x)^3 \sqrt {1+\frac {2 x}{1+x^2}}}{3 \left (1+x^2\right )}-\frac {(4+3 x) \left (1+x^2\right ) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}+\frac {5 \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 = 0.10, size = 64, normalized size = 0.48 \[ \frac {(x+1) \left (3 x^4-8 x^3-18 x^2+15 \left (x^2+1\right )^{3/2} \sinh ^{-1}(x)-12 x-17\right )}{3 \sqrt {\frac {(x+1)^2}{x^2+1}} \left (x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 117, normalized size = 0.88 \[ -\frac {8 \, x^{3} + 8 \, x^{2} + 15 \, {\left (x^{3} + x^{2} + x + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} + 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + x}{x + 1}\right ) - {\left (3 \, x^{4} - 8 \, x^{3} - 18 \, x^{2} - 12 \, x - 17\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + 8 \, x + 8}{3 \, {\left (x^{3} + x^{2} + x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 86, normalized size = 0.65 \[ {\left (\sqrt {2} + 5 \, \log \left (\sqrt {2} + 1\right )\right )} \mathrm {sgn}\left (x + 1\right ) - 5 \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \mathrm {sgn}\left (x + 1\right ) + \frac {{\left ({\left ({\left (3 \, x \mathrm {sgn}\left (x + 1\right ) - 8 \, \mathrm {sgn}\left (x + 1\right )\right )} x - 18 \, \mathrm {sgn}\left (x + 1\right )\right )} x - 12 \, \mathrm {sgn}\left (x + 1\right )\right )} x - 17 \, \mathrm {sgn}\left (x + 1\right )}{3 \, {\left (x^{2} + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 62, normalized size = 0.47 \[ \frac {\left (\frac {x^{2}+2 x +1}{x^{2}+1}\right )^{\frac {5}{2}} \left (x^{2}+1\right ) \left (3 x^{4}-8 x^{3}-18 x^{2}-12 x +15 \left (x^{2}+1\right )^{\frac {3}{2}} \arcsinh \relax (x )-17\right )}{3 \left (x +1\right )^{5}} \]
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 \[ \int {\left (\frac {2 \, x}{x^{2} + 1} + 1\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (\frac {2\,x}{x^2+1}+1\right )}^{5/2} \,d x \]
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 \[ \int \left (\frac {2 x}{x^{2} + 1} + 1\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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