Optimal. Leaf size=133 \[ -\frac {4}{3} (1-2 x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}-\frac {(1-x) (1+x)^3 \sqrt {1+\frac {2 x}{1+x^2}}}{3 \left (1+x^2\right )}-\frac {(4+3 x) \left (1+x^2\right ) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}+\frac {5 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}} \sinh ^{-1}(x)}{1+x} \]
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Rubi [A]
time = 0.05, 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 = {6855, 984, 753,
833, 794, 221} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 753
Rule 794
Rule 833
Rule 984
Rule 6855
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 = 0.28, size = 74, normalized size = 0.56 \begin {gather*} \frac {(1+x) \left (-17-12 x-18 x^2-8 x^3+3 x^4+15 \left (1+x^2\right )^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right )\right )}{3 \sqrt {\frac {(1+x)^2}{1+x^2}} \left (1+x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 62, normalized size = 0.47
method | result | size |
default | \(\frac {\left (\frac {x^{2}+2 x +1}{x^{2}+1}\right )^{\frac {5}{2}} \left (x^{2}+1\right ) \left (15 \arcsinh \left (x \right ) \left (x^{2}+1\right )^{\frac {3}{2}}+3 x^{4}-8 x^{3}-18 x^{2}-12 x -17\right )}{3 \left (1+x \right )^{5}}\) | \(62\) |
risch | \(\frac {\left (3 x^{4}-8 x^{3}-18 x^{2}-12 x -17\right ) \sqrt {\frac {\left (1+x \right )^{2}}{x^{2}+1}}}{3 \left (x^{2}+1\right ) \left (1+x \right )}+\frac {5 \arcsinh \left (x \right ) \sqrt {x^{2}+1}\, \sqrt {\frac {\left (1+x \right )^{2}}{x^{2}+1}}}{1+x}\) | \(82\) |
trager | \(\frac {\left (3 x^{4}-8 x^{3}-18 x^{2}-12 x -17\right ) \sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}}{3 \left (x^{2}+1\right ) \left (1+x \right )}+5 \ln \left (\frac {\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}\, x^{2}+x^{2}+\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}+x}{1+x}\right )\) | \(117\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 117, normalized size = 0.88 \begin {gather*} -\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 )}} \end {gather*}
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 \begin {gather*} \int \left (\frac {2 x}{x^{2} + 1} + 1\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.03, size = 86, normalized size = 0.65 \begin {gather*} {\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}}} \end {gather*}
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 \begin {gather*} \int {\left (\frac {2\,x}{x^2+1}+1\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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