Optimal. Leaf size=94 \[ \frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{9} \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )-\frac {2 \sqrt {1+x} \sqrt {1-x+x^2} \tanh ^{-1}\left (\sqrt {1+x^3}\right )}{3 \sqrt {1+x^3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {929, 272, 52,
65, 213} \begin {gather*} \frac {2}{3} \sqrt {x+1} \sqrt {x^2-x+1}+\frac {2}{9} \sqrt {x+1} \sqrt {x^2-x+1} \left (x^3+1\right )-\frac {2 \sqrt {x+1} \sqrt {x^2-x+1} \tanh ^{-1}\left (\sqrt {x^3+1}\right )}{3 \sqrt {x^3+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 213
Rule 272
Rule 929
Rubi steps
\begin {align*} \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x} \, dx &=\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {\left (1+x^3\right )^{3/2}}{x} \, dx}{\sqrt {1+x^3}}\\ &=\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \text {Subst}\left (\int \frac {(1+x)^{3/2}}{x} \, dx,x,x^3\right )}{3 \sqrt {1+x^3}}\\ &=\frac {2}{9} \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )+\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^3\right )}{3 \sqrt {1+x^3}}\\ &=\frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{9} \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )+\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )}{3 \sqrt {1+x^3}}\\ &=\frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{9} \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )+\frac {\left (2 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )}{3 \sqrt {1+x^3}}\\ &=\frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{9} \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )-\frac {2 \sqrt {1+x} \sqrt {1-x+x^2} \tanh ^{-1}\left (\sqrt {1+x^3}\right )}{3 \sqrt {1+x^3}}\\ \end {align*}
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Mathematica [A]
time = 10.05, size = 53, normalized size = 0.56 \begin {gather*} \frac {2}{9} \left (\sqrt {1+x} \sqrt {1-x+x^2} \left (4+x^3\right )-3 \tanh ^{-1}\left (\sqrt {1+x} \sqrt {1-x+x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 57, normalized size = 0.61
method | result | size |
default | \(-\frac {2 \sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (-x^{3} \sqrt {x^{3}+1}+3 \arctanh \left (\sqrt {x^{3}+1}\right )-4 \sqrt {x^{3}+1}\right )}{9 \sqrt {x^{3}+1}}\) | \(57\) |
elliptic | \(\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (\frac {2 x^{3} \sqrt {x^{3}+1}}{9}+\frac {8 \sqrt {x^{3}+1}}{9}-\frac {2 \arctanh \left (\sqrt {x^{3}+1}\right )}{3}\right )}{x^{3}+1}\) | \(70\) |
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 = 2.69, size = 65, normalized size = 0.69 \begin {gather*} \frac {2}{9} \, {\left (x^{3} + 4\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} - \frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} - 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 \frac {\left (x + 1\right )^{\frac {3}{2}} \left (x^{2} - x + 1\right )^{\frac {3}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \frac {{\left (x+1\right )}^{3/2}\,{\left (x^2-x+1\right )}^{3/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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