Optimal. Leaf size=54 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d}-\frac{4 a \sqrt{a+b \sqrt{c+d x}}}{b^2 d} \]
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Rubi [A] time = 0.0314209, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {247, 190, 43} \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d}-\frac{4 a \sqrt{a+b \sqrt{c+d x}}}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 247
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \sqrt{c+d x}}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sqrt{x}}} \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x}} \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=-\frac{4 a \sqrt{a+b \sqrt{c+d x}}}{b^2 d}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d}\\ \end{align*}
Mathematica [A] time = 0.0227483, size = 42, normalized size = 0.78 \[ \frac{4 \left (b \sqrt{c+d x}-2 a\right ) \sqrt{a+b \sqrt{c+d x}}}{3 b^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 41, normalized size = 0.8 \begin{align*} 4\,{\frac{1/3\, \left ( a+b\sqrt{dx+c} \right ) ^{3/2}-a\sqrt{a+b\sqrt{dx+c}}}{{b}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997243, size = 57, normalized size = 1.06 \begin{align*} \frac{4 \,{\left (\frac{{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}}}{b^{2}} - \frac{3 \, \sqrt{\sqrt{d x + c} b + a} a}{b^{2}}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29351, size = 85, normalized size = 1.57 \begin{align*} \frac{4 \, \sqrt{\sqrt{d x + c} b + a}{\left (\sqrt{d x + c} b - 2 \, a\right )}}{3 \, b^{2} d} \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}{\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 [B] time = 1.22179, size = 135, normalized size = 2.5 \begin{align*} \frac{4 \,{\left (\sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 3 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}} a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )\right )}}{3 \, b^{2} d{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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