Optimal. Leaf size=106 \[ 2 \sqrt{a+b \sqrt{x}+c x}-2 \sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \sqrt{x}}{2 \sqrt{a} \sqrt{a+b \sqrt{x}+c x}}\right )+\frac{b \tanh ^{-1}\left (\frac{b+2 c \sqrt{x}}{2 \sqrt{c} \sqrt{a+b \sqrt{x}+c x}}\right )}{\sqrt{c}} \]
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Rubi [A] time = 0.0915753, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1357, 734, 843, 621, 206, 724} \[ 2 \sqrt{a+b \sqrt{x}+c x}-2 \sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \sqrt{x}}{2 \sqrt{a} \sqrt{a+b \sqrt{x}+c x}}\right )+\frac{b \tanh ^{-1}\left (\frac{b+2 c \sqrt{x}}{2 \sqrt{c} \sqrt{a+b \sqrt{x}+c x}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 734
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \sqrt{x}+c x}}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x} \, dx,x,\sqrt{x}\right )\\ &=2 \sqrt{a+b \sqrt{x}+c x}-\operatorname{Subst}\left (\int \frac{-2 a-b x}{x \sqrt{a+b x+c x^2}} \, dx,x,\sqrt{x}\right )\\ &=2 \sqrt{a+b \sqrt{x}+c x}+(2 a) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\sqrt{x}\right )+b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\sqrt{x}\right )\\ &=2 \sqrt{a+b \sqrt{x}+c x}-(4 a) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b \sqrt{x}}{\sqrt{a+b \sqrt{x}+c x}}\right )+(2 b) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c \sqrt{x}}{\sqrt{a+b \sqrt{x}+c x}}\right )\\ &=2 \sqrt{a+b \sqrt{x}+c x}-2 \sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \sqrt{x}}{2 \sqrt{a} \sqrt{a+b \sqrt{x}+c x}}\right )+\frac{b \tanh ^{-1}\left (\frac{b+2 c \sqrt{x}}{2 \sqrt{c} \sqrt{a+b \sqrt{x}+c x}}\right )}{\sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0577812, size = 106, normalized size = 1. \[ 2 \sqrt{a+b \sqrt{x}+c x}-2 \sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \sqrt{x}}{2 \sqrt{a} \sqrt{a+b \sqrt{x}+c x}}\right )+\frac{b \tanh ^{-1}\left (\frac{b+2 c \sqrt{x}}{2 \sqrt{c} \sqrt{a+b \sqrt{x}+c x}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 84, normalized size = 0.8 \begin{align*} 2\,\sqrt{a+cx+b\sqrt{x}}+{b\ln \left ({ \left ({\frac{b}{2}}+c\sqrt{x} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{a+cx+b\sqrt{x}} \right ){\frac{1}{\sqrt{c}}}}-2\,\sqrt{a}\ln \left ({\frac{2\,a+b\sqrt{x}+2\,\sqrt{a}\sqrt{a+cx+b\sqrt{x}}}{\sqrt{x}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x + b \sqrt{x} + a}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \sqrt{x} + c x}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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