Optimal. Leaf size=56 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^2 d}-\frac{4 a \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d} \]
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Rubi [A] time = 0.0327106, antiderivative size = 56, 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} \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^2 d}-\frac{4 a \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d} \]
Antiderivative was successfully verified.
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Rule 247
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \sqrt{a+b \sqrt{c+d x}} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{a+b \sqrt{x}} \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int x \sqrt{a+b x} \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b x}}{b}+\frac{(a+b x)^{3/2}}{b}\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=-\frac{4 a \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^2 d}\\ \end{align*}
Mathematica [A] time = 0.0241736, size = 43, normalized size = 0.77 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{3/2} \left (3 b \sqrt{c+d x}-2 a\right )}{15 b^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 41, normalized size = 0.7 \begin{align*} 4\,{\frac{1/5\, \left ( a+b\sqrt{dx+c} \right ) ^{5/2}-1/3\,a \left ( a+b\sqrt{dx+c} \right ) ^{3/2}}{{b}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10962, size = 58, normalized size = 1.04 \begin{align*} \frac{4 \,{\left (\frac{3 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38948, size = 122, normalized size = 2.18 \begin{align*} \frac{4 \,{\left (3 \, b^{2} d x + 3 \, b^{2} c + \sqrt{d x + c} a b - 2 \, a^{2}\right )} \sqrt{\sqrt{d x + c} b + a}}{15 \, 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 \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 = 1.11971, size = 159, normalized size = 2.84 \begin{align*} \frac{4 \,{\left (3 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )}^{2} b^{2} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 5 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )} a b^{2} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )\right )}{\left | b \right |}}{15 \, b^{6} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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