Optimal. Leaf size=62 \[ \frac {b x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {a \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rubi [A] time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 43} \begin {gather*} \frac {b x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {a \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{x} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (b^2+\frac {a b}{x}\right ) \, dx}{a b+b^2 x}\\ &=\frac {b x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {a \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.44 \begin {gather*} \frac {\sqrt {(a+b x)^2} (a \log (x)+b x)}{a+b x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.22, size = 181, normalized size = 2.92 \begin {gather*} \frac {1}{2} \sqrt {a^2+2 a b x+b^2 x^2}+\frac {1}{2} a \log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right )-\frac {a \left (\sqrt {b^2}+b\right ) \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )}{2 b}-\frac {a \sqrt {b^2} \log \left (b \sqrt {a^2+2 a b x+b^2 x^2}-a b-\sqrt {b^2} b x\right )}{2 b}-\frac {1}{2} \sqrt {b^2} x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 8, normalized size = 0.13 \begin {gather*} b x + a \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 21, normalized size = 0.34 \begin {gather*} b x \mathrm {sgn}\left (b x + a\right ) + a \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 19, normalized size = 0.31 \begin {gather*} \left (a \ln \left (b x \right )+b x +a \right ) \mathrm {csgn}\left (b x +a \right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.45, size = 82, normalized size = 1.32 \begin {gather*} \left (-1\right )^{2 \, b^{2} x + 2 \, a b} a \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} a \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 98, normalized size = 1.58 \begin {gather*} \sqrt {a^2+2\,a\,b\,x+b^2\,x^2}-\ln \left (a\,b+\frac {a^2}{x}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x}\right )\,\sqrt {a^2}+\frac {a\,b\,\ln \left (a\,b+\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {b^2}+b^2\,x\right )}{\sqrt {b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 7, normalized size = 0.11 \begin {gather*} a \log {\relax (x )} + b x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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