Optimal. Leaf size=92 \[ -\frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a \sqrt{\frac{c}{x}}}+x \sqrt{a+b \sqrt{\frac{c}{x}}} \]
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Rubi [A] time = 0.0486613, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {255, 190, 47, 51, 63, 208} \[ -\frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a \sqrt{\frac{c}{x}}}+x \sqrt{a+b \sqrt{\frac{c}{x}}} \]
Antiderivative was successfully verified.
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Rule 255
Rule 190
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+b \sqrt{\frac{c}{x}}} \, dx &=\operatorname{Subst}\left (\int \sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \frac{\sqrt{a+b \sqrt{c} x}}{x^3} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\sqrt{a+b \sqrt{\frac{c}{x}}} x-\operatorname{Subst}\left (\frac{1}{2} \left (b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a \sqrt{\frac{c}{x}}}+\sqrt{a+b \sqrt{\frac{c}{x}}} x+\operatorname{Subst}\left (\frac{\left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{4 a},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a \sqrt{\frac{c}{x}}}+\sqrt{a+b \sqrt{\frac{c}{x}}} x+\operatorname{Subst}\left (\frac{\left (b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b \sqrt{c}}+\frac{x^2}{b \sqrt{c}}} \, dx,x,\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}}\right )}{2 a},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a \sqrt{\frac{c}{x}}}+\sqrt{a+b \sqrt{\frac{c}{x}}} x-\frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0656965, size = 79, normalized size = 0.86 \[ \frac{\sqrt{a} x \sqrt{a+b \sqrt{\frac{c}{x}}} \left (2 a+b \sqrt{\frac{c}{x}}\right )-b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 147, normalized size = 1.6 \begin{align*}{\frac{1}{4}\sqrt{a+b\sqrt{{\frac{c}{x}}}}\sqrt{x} \left ( 2\,{a}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{{\frac{c}{x}}}\sqrt{x}b-{b}^{2}c\ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ) a+4\,{a}^{5/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{x} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{a}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5454, size = 373, normalized size = 4.05 \begin{align*} \left [\frac{\sqrt{a} b^{2} c \log \left (-2 \, \sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{a} x \sqrt{\frac{c}{x}} + 2 \, a x \sqrt{\frac{c}{x}} + b c\right ) + 2 \,{\left (a b x \sqrt{\frac{c}{x}} + 2 \, a^{2} x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{4 \, a^{2}}, \frac{\sqrt{-a} b^{2} c \arctan \left (\frac{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}{a}\right ) +{\left (a b x \sqrt{\frac{c}{x}} + 2 \, a^{2} x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{2 \, a^{2}}\right ] \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 \sqrt{a + b \sqrt{\frac{c}{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: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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