Optimal. Leaf size=56 \[ \frac{2 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^2 c}-\frac{2 a \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^2 c} \]
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Rubi [A] time = 0.0250762, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {368, 43} \[ \frac{2 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^2 c}-\frac{2 a \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^2 c} \]
Antiderivative was successfully verified.
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Rule 368
Rule 43
Rubi steps
\begin{align*} \int x \sqrt{a+b \sqrt{c x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int x \sqrt{a+b x} \, dx,x,\sqrt{c x^2}\right )}{c}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b x}}{b}+\frac{(a+b x)^{3/2}}{b}\right ) \, dx,x,\sqrt{c x^2}\right )}{c}\\ &=-\frac{2 a \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^2 c}+\frac{2 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^2 c}\\ \end{align*}
Mathematica [A] time = 0.0159931, size = 43, normalized size = 0.77 \[ \frac{2 \left (a+b \sqrt{c x^2}\right )^{3/2} \left (3 b \sqrt{c x^2}-2 a\right )}{15 b^2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 36, normalized size = 0.6 \begin{align*}{\frac{2}{15\,{b}^{2}c} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}} \left ( -2\,a+3\,b\sqrt{c{x}^{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931034, size = 58, normalized size = 1.04 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{15 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15857, size = 105, normalized size = 1.88 \begin{align*} \frac{2 \,{\left (3 \, b^{2} c x^{2} + \sqrt{c x^{2}} a b - 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{15 \, b^{2} c} \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 x \sqrt{a + b \sqrt{c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12015, size = 46, normalized size = 0.82 \begin{align*} \frac{2 \,{\left (3 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a\right )}}{15 \, b^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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