Optimal. Leaf size=42 \[ \frac{4 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^2}-\frac{4 a \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.0169726, antiderivative size = 42, 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} \[ \frac{4 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^2}-\frac{4 a \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \sqrt{a+b \sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int x \sqrt{a+b x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b x}}{b}+\frac{(a+b x)^{3/2}}{b}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 a \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2}+\frac{4 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0159094, size = 32, normalized size = 0.76 \[ \frac{4 \left (a+b \sqrt{x}\right )^{3/2} \left (3 b \sqrt{x}-2 a\right )}{15 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 30, normalized size = 0.7 \begin{align*} 4\,{\frac{1/5\, \left ( a+b\sqrt{x} \right ) ^{5/2}-1/3\,a \left ( a+b\sqrt{x} \right ) ^{3/2}}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973247, size = 41, normalized size = 0.98 \begin{align*} \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}}}{5 \, b^{2}} - \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33174, size = 84, normalized size = 2. \begin{align*} \frac{4 \,{\left (3 \, b^{2} x + a b \sqrt{x} - 2 \, a^{2}\right )} \sqrt{b \sqrt{x} + a}}{15 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.43063, size = 272, normalized size = 6.48 \begin{align*} - \frac{8 a^{\frac{9}{2}} x^{2} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{9}{2}} x^{2}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} - \frac{4 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{7}{2}} b x^{\frac{5}{2}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{16 a^{\frac{5}{2}} b^{2} x^{3} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{12 a^{\frac{3}{2}} b^{3} x^{\frac{7}{2}} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 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 = 1.09986, size = 39, normalized size = 0.93 \begin{align*} \frac{4 \,{\left (3 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a\right )}}{15 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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