Optimal. Leaf size=97 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{\sqrt{a-b \sqrt{c}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{\sqrt{a+b \sqrt{c}}} \]
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Rubi [A] time = 0.0850731, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {371, 1398, 827, 1166, 207} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{\sqrt{a-b \sqrt{c}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{\sqrt{a+b \sqrt{c}}} \]
Antiderivative was successfully verified.
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Rule 371
Rule 1398
Rule 827
Rule 1166
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{a+b \sqrt{c+d x}}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sqrt{x}} (-c+x)} \, dx,x,c+d x\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x} \left (-c+x^2\right )} \, dx,x,\sqrt{c+d x}\right )\\ &=4 \operatorname{Subst}\left (\int \frac{-a+x^2}{a^2-b^2 c-2 a x^2+x^4} \, dx,x,\sqrt{a+b \sqrt{c+d x}}\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{-a-b \sqrt{c}+x^2} \, dx,x,\sqrt{a+b \sqrt{c+d x}}\right )+2 \operatorname{Subst}\left (\int \frac{1}{-a+b \sqrt{c}+x^2} \, dx,x,\sqrt{a+b \sqrt{c+d x}}\right )\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{\sqrt{a-b \sqrt{c}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{\sqrt{a+b \sqrt{c}}}\\ \end{align*}
Mathematica [A] time = 0.11024, size = 97, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{\sqrt{a-b \sqrt{c}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{\sqrt{a+b \sqrt{c}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 92, normalized size = 1. \begin{align*} 2\,{\frac{1}{\sqrt{-\sqrt{{b}^{2}c}-a}}\arctan \left ({\frac{\sqrt{a+b\sqrt{dx+c}}}{\sqrt{-\sqrt{{b}^{2}c}-a}}} \right ) }+2\,{\frac{1}{\sqrt{\sqrt{{b}^{2}c}-a}}\arctan \left ({\frac{\sqrt{a+b\sqrt{dx+c}}}{\sqrt{\sqrt{{b}^{2}c}-a}}} \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{1}{\sqrt{\sqrt{d x + c} b + a} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.44426, size = 1438, normalized size = 14.82 \begin{align*} \sqrt{-\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} \log \left (4 \,{\left ({\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a\right )} \sqrt{-\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} + 4 \, \sqrt{\sqrt{d x + c} b + a}\right ) - \sqrt{-\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} \log \left (-4 \,{\left ({\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a\right )} \sqrt{-\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} + 4 \, \sqrt{\sqrt{d x + c} b + a}\right ) - \sqrt{\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} \log \left (4 \,{\left ({\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a\right )} \sqrt{\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} + 4 \, \sqrt{\sqrt{d x + c} b + a}\right ) + \sqrt{\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} \log \left (-4 \,{\left ({\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a\right )} \sqrt{\frac{{\left (b^{2} c - a^{2}\right )} \sqrt{\frac{b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} + 4 \, \sqrt{\sqrt{d x + c} b + a}\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 \frac{1}{x \sqrt{a + b \sqrt{c + d 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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