Optimal. Leaf size=66 \[ \frac {2}{3} \sqrt {x+1} \sqrt {x^2-x+1}-\frac {2 \sqrt {x+1} \sqrt {x^2-x+1} \tanh ^{-1}\left (\sqrt {x^3+1}\right )}{3 \sqrt {x^3+1}} \]
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Rubi [A] time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {915, 266, 50, 63, 207} \[ \frac {2}{3} \sqrt {x+1} \sqrt {x^2-x+1}-\frac {2 \sqrt {x+1} \sqrt {x^2-x+1} \tanh ^{-1}\left (\sqrt {x^3+1}\right )}{3 \sqrt {x^3+1}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 207
Rule 266
Rule 915
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x} \, dx &=\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {\sqrt {1+x^3}}{x} \, dx}{\sqrt {1+x^3}}\\ &=\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \operatorname {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 {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \operatorname {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 {\left (2 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \operatorname {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 \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 [C] time = 0.41, size = 197, normalized size = 2.98 \[ \frac {\sqrt {x+1} \left (2 \left (x^2-x+1\right )+\frac {3 i \sqrt {2} \sqrt {\frac {-2 i x+\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt {\frac {2 i x+\sqrt {3}-i}{\sqrt {3}-3 i}} \Pi \left (\frac {3}{2}-\frac {i \sqrt {3}}{2};i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {i (x+1)}{3 i+\sqrt {3}}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i (x+1)}{\sqrt {3}+3 i}}}\right )}{3 \sqrt {x^2-x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 60, normalized size = 0.91 \[ \frac {2}{3} \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} - \frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} - 1\right ) \]
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 \[ \int \frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 43, normalized size = 0.65 \[ -\frac {2 \sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (\arctanh \left (\sqrt {x^{3}+1}\right )-\sqrt {x^{3}+1}\right )}{3 \sqrt {x^{3}+1}} \]
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 \[ \int \frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {x+1}\,\sqrt {x^2-x+1}}{x} \,d x \]
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 \[ \int \frac {\sqrt {x + 1} \sqrt {x^{2} - x + 1}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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