Optimal. Leaf size=40 \[ \frac{4 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2}-\frac{4 a \sqrt{a+b \sqrt{x}}}{b^2} \]
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Rubi [A] time = 0.0170122, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {190, 43} \[ \frac{4 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2}-\frac{4 a \sqrt{a+b \sqrt{x}}}{b^2} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \sqrt{x}}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 a \sqrt{a+b \sqrt{x}}}{b^2}+\frac{4 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.0134767, size = 31, normalized size = 0.78 \[ \frac{4 \left (b \sqrt{x}-2 a\right ) \sqrt{a+b \sqrt{x}}}{3 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 30, normalized size = 0.8 \begin{align*} 4\,{\frac{1/3\, \left ( a+b\sqrt{x} \right ) ^{3/2}-a\sqrt{a+b\sqrt{x}}}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948876, size = 41, normalized size = 1.02 \begin{align*} \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}}}{3 \, b^{2}} - \frac{4 \, \sqrt{b \sqrt{x} + a} a}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31325, size = 63, normalized size = 1.58 \begin{align*} \frac{4 \, \sqrt{b \sqrt{x} + a}{\left (b \sqrt{x} - 2 \, a\right )}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.98405, size = 219, normalized size = 5.48 \begin{align*} - \frac{8 a^{\frac{7}{2}} x^{2} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{7}{2}} x^{2}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} - \frac{4 a^{\frac{5}{2}} b x^{\frac{5}{2}} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{5}{2}} b x^{\frac{5}{2}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} + \frac{4 a^{\frac{3}{2}} b^{2} x^{3} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11293, size = 36, normalized size = 0.9 \begin{align*} \frac{4 \,{\left ({\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b \sqrt{x} + a} a\right )}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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