Optimal. Leaf size=56 \[ \frac{4 \left (a+b \sqrt{c x^3}\right )^{5/2}}{15 b^2 c}-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^2 c} \]
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Rubi [A] time = 0.0352788, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {369, 266, 43} \[ \frac{4 \left (a+b \sqrt{c x^3}\right )^{5/2}}{15 b^2 c}-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^2 c} \]
Antiderivative was successfully verified.
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Rule 369
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \sqrt{a+b \sqrt{c x^3}} \, dx &=\operatorname{Subst}\left (\int x^2 \sqrt{a+b \sqrt{c} x^{3/2}} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{3} \operatorname{Subst}\left (\int x \sqrt{a+b \sqrt{c} x} \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{3} \operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b \sqrt{c} x}}{b \sqrt{c}}+\frac{\left (a+b \sqrt{c} x\right )^{3/2}}{b \sqrt{c}}\right ) \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^2 c}+\frac{4 \left (a+b \sqrt{c x^3}\right )^{5/2}}{15 b^2 c}\\ \end{align*}
Mathematica [A] time = 0.0326489, size = 43, normalized size = 0.77 \[ \frac{4 \left (a+b \sqrt{c x^3}\right )^{3/2} \left (3 b \sqrt{c x^3}-2 a\right )}{45 b^2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.182, size = 65, normalized size = 1.2 \begin{align*}{\frac{4}{45\,{b}^{2}c}\sqrt{a+b\sqrt{c{x}^{3}}} \left ( 3\,{x}^{3}c\sqrt{c{x}^{3}}{b}^{2}+a{x}^{3}cb-2\,{a}^{2}\sqrt{c{x}^{3}} \right ){\frac{1}{\sqrt{c{x}^{3}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947457, size = 58, normalized size = 1.04 \begin{align*} \frac{4 \,{\left (\frac{3 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{45 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91551, size = 105, normalized size = 1.88 \begin{align*} \frac{4 \,{\left (3 \, b^{2} c x^{3} + \sqrt{c x^{3}} a b - 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{3}} b + a}}{45 \, 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^{2} \sqrt{a + b \sqrt{c x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18973, size = 89, normalized size = 1.59 \begin{align*} \frac{4 \,{\left (\frac{2 \, \sqrt{a c} a^{2}}{b^{2}} - \frac{5 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{3}{2}} a c - 3 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{5}{2}}}{b^{2} c^{2}}\right )}{\left | c \right |}}{45 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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