Optimal. Leaf size=169 \[ \frac{5 b^3 c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{32 a^3 \sqrt{\frac{c}{x}}}-\frac{5 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{32 a^{7/2}}-\frac{5 b^2 c x \sqrt{a+b \sqrt{\frac{c}{x}}}}{48 a^2}+\frac{b c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{12 a \left (\frac{c}{x}\right )^{3/2}}+\frac{1}{2} x^2 \sqrt{a+b \sqrt{\frac{c}{x}}} \]
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Rubi [A] time = 0.101512, antiderivative size = 172, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {369, 266, 47, 51, 63, 208} \[ \frac{5 b^3 c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{32 a^3 \sqrt{\frac{c}{x}}}-\frac{5 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{32 a^{7/2}}-\frac{5 b^2 c x \sqrt{a+b \sqrt{\frac{c}{x}}}}{48 a^2}+\frac{b x^3 \left (\frac{c}{x}\right )^{3/2} \sqrt{a+b \sqrt{\frac{c}{x}}}}{12 a c}+\frac{1}{2} x^2 \sqrt{a+b \sqrt{\frac{c}{x}}} \]
Antiderivative was successfully verified.
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Rule 369
Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+b \sqrt{\frac{c}{x}}} x \, dx &=\operatorname{Subst}\left (\int \sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}} 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^5} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{1}{2} \sqrt{a+b \sqrt{\frac{c}{x}}} x^2-\operatorname{Subst}\left (\frac{1}{4} \left (b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \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{1}{2} \sqrt{a+b \sqrt{\frac{c}{x}}} x^2+\frac{b \sqrt{a+b \sqrt{\frac{c}{x}}} \left (\frac{c}{x}\right )^{3/2} x^3}{12 a c}+\operatorname{Subst}\left (\frac{\left (5 b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{24 a},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\frac{5 b^2 c \sqrt{a+b \sqrt{\frac{c}{x}}} x}{48 a^2}+\frac{1}{2} \sqrt{a+b \sqrt{\frac{c}{x}}} x^2+\frac{b \sqrt{a+b \sqrt{\frac{c}{x}}} \left (\frac{c}{x}\right )^{3/2} x^3}{12 a c}-\operatorname{Subst}\left (\frac{\left (5 b^3 c^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{32 a^2},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{5 b^3 c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{32 a^3 \sqrt{\frac{c}{x}}}-\frac{5 b^2 c \sqrt{a+b \sqrt{\frac{c}{x}}} x}{48 a^2}+\frac{1}{2} \sqrt{a+b \sqrt{\frac{c}{x}}} x^2+\frac{b \sqrt{a+b \sqrt{\frac{c}{x}}} \left (\frac{c}{x}\right )^{3/2} x^3}{12 a c}+\operatorname{Subst}\left (\frac{\left (5 b^4 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{64 a^3},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{5 b^3 c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{32 a^3 \sqrt{\frac{c}{x}}}-\frac{5 b^2 c \sqrt{a+b \sqrt{\frac{c}{x}}} x}{48 a^2}+\frac{1}{2} \sqrt{a+b \sqrt{\frac{c}{x}}} x^2+\frac{b \sqrt{a+b \sqrt{\frac{c}{x}}} \left (\frac{c}{x}\right )^{3/2} x^3}{12 a c}+\operatorname{Subst}\left (\frac{\left (5 b^3 c^{3/2}\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 )}{32 a^3},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{5 b^3 c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{32 a^3 \sqrt{\frac{c}{x}}}-\frac{5 b^2 c \sqrt{a+b \sqrt{\frac{c}{x}}} x}{48 a^2}+\frac{1}{2} \sqrt{a+b \sqrt{\frac{c}{x}}} x^2+\frac{b \sqrt{a+b \sqrt{\frac{c}{x}}} \left (\frac{c}{x}\right )^{3/2} x^3}{12 a c}-\frac{5 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{32 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.128274, size = 111, normalized size = 0.66 \[ \frac{\sqrt{a} x \sqrt{a+b \sqrt{\frac{c}{x}}} \left (8 a^2 b x \sqrt{\frac{c}{x}}+48 a^3 x-10 a b^2 c+15 b^3 c \sqrt{\frac{c}{x}}\right )-15 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{96 a^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 211, normalized size = 1.3 \begin{align*} -{\frac{1}{192}\sqrt{a+b\sqrt{{\frac{c}{x}}}}\sqrt{x} \left ( 15\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){c}^{2}a{b}^{4}-30\,{a}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x} \left ({\frac{c}{x}} \right ) ^{3/2}{x}^{3/2}{b}^{3}-60\,{a}^{5/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}c\sqrt{x}{b}^{2}+80\,{a}^{5/2} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{{\frac{c}{x}}}\sqrt{x}b-96\,\sqrt{x} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{7/2} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{a}^{-{\frac{9}{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 = 1.51508, size = 506, normalized size = 2.99 \begin{align*} \left [\frac{15 \, \sqrt{a} b^{4} c^{2} \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 (10 \, a^{2} b^{2} c x - 48 \, a^{4} x^{2} -{\left (15 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{192 \, a^{4}}, \frac{15 \, \sqrt{-a} b^{4} c^{2} \arctan \left (\frac{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}{a}\right ) -{\left (10 \, a^{2} b^{2} c x - 48 \, a^{4} x^{2} -{\left (15 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{96 \, a^{4}}\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 x \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(-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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