Optimal. Leaf size=113 \[ \frac{2 a^2 x^3 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^3 \left (c x^2\right )^{3/2}}+\frac{2 x^3 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^3 \left (c x^2\right )^{3/2}}-\frac{4 a x^3 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^3 \left (c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0397586, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {368, 43} \[ \frac{2 a^2 x^3 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^3 \left (c x^2\right )^{3/2}}+\frac{2 x^3 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^3 \left (c x^2\right )^{3/2}}-\frac{4 a x^3 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 368
Rule 43
Rubi steps
\begin{align*} \int x^2 \sqrt{a+b \sqrt{c x^2}} \, dx &=\frac{x^3 \operatorname{Subst}\left (\int x^2 \sqrt{a+b x} \, dx,x,\sqrt{c x^2}\right )}{\left (c x^2\right )^{3/2}}\\ &=\frac{x^3 \operatorname{Subst}\left (\int \left (\frac{a^2 \sqrt{a+b x}}{b^2}-\frac{2 a (a+b x)^{3/2}}{b^2}+\frac{(a+b x)^{5/2}}{b^2}\right ) \, dx,x,\sqrt{c x^2}\right )}{\left (c x^2\right )^{3/2}}\\ &=\frac{2 a^2 x^3 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^3 \left (c x^2\right )^{3/2}}-\frac{4 a x^3 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^3 \left (c x^2\right )^{3/2}}+\frac{2 x^3 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^3 \left (c x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0290982, size = 64, normalized size = 0.57 \[ \frac{2 x^3 \left (a+b \sqrt{c x^2}\right )^{3/2} \left (8 a^2-12 a b \sqrt{c x^2}+15 b^2 c x^2\right )}{105 b^3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 55, normalized size = 0.5 \begin{align*} -{\frac{2\,{x}^{3}}{105\,{b}^{3}} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}} \left ( -15\,{x}^{2}{b}^{2}c+12\,\sqrt{c{x}^{2}}ab-8\,{a}^{2} \right ) \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06851, size = 522, normalized size = 4.62 \begin{align*} \frac{{\left ({\left (31 \, c^{8} + 3784 \, c^{7} + 91078 \, c^{6} + 622632 \, c^{5} + 1266003 \, c^{4} + 635688 \, c^{3} + 34992 \, c^{2} +{\left (c^{8} + 440 \, c^{7} + 21986 \, c^{6} + 276544 \, c^{5} + 1038501 \, c^{4} + 1095120 \, c^{3} + 221616 \, c^{2}\right )} \sqrt{c}\right )} b^{3} x^{3} +{\left (c^{8} + 382 \, c^{7} + 15946 \, c^{6} + 158172 \, c^{5} + 425925 \, c^{4} + 266814 \, c^{3} + 17496 \, c^{2} +{\left (29 \, c^{7} + 3020 \, c^{6} + 59186 \, c^{5} + 306288 \, c^{4} + 414153 \, c^{3} + 102060 \, c^{2}\right )} \sqrt{c}\right )} a b^{2} x^{2} - 2 \,{\left (c^{7} + 354 \, c^{6} + 13280 \, c^{5} + 112266 \, c^{4} + 231903 \, c^{3} + 84564 \, c^{2} + 2 \,{\left (14 \, c^{6} + 1333 \, c^{5} + 22953 \, c^{4} + 97011 \, c^{3} + 91125 \, c^{2} + 8748 \, c\right )} \sqrt{c}\right )} a^{2} b x + 2 \,{\left (c^{6} + 354 \, c^{5} + 13280 \, c^{4} + 112266 \, c^{3} + 231903 \, c^{2} + 2 \,{\left (14 \, c^{5} + 1333 \, c^{4} + 22953 \, c^{3} + 97011 \, c^{2} + 91125 \, c + 8748\right )} \sqrt{c} + 84564 \, c\right )} a^{3}\right )} \sqrt{b \sqrt{c} x + a}}{{\left (c^{9} + 533 \, c^{8} + 33338 \, c^{7} + 549778 \, c^{6} + 2906397 \, c^{5} + 4893129 \, c^{4} + 2128680 \, c^{3} + 104976 \, c^{2} + 2 \,{\left (17 \, c^{8} + 2552 \, c^{7} + 78518 \, c^{6} + 726132 \, c^{5} + 2190753 \, c^{4} + 1960524 \, c^{3} + 349920 \, c^{2}\right )} \sqrt{c}\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30185, size = 154, normalized size = 1.36 \begin{align*} \frac{2 \,{\left (15 \, b^{3} c^{2} x^{4} - 4 \, a^{2} b c x^{2} +{\left (3 \, a b^{2} c x^{2} + 8 \, a^{3}\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{105 \, b^{3} c^{2} x} \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^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17032, size = 66, normalized size = 0.58 \begin{align*} \frac{2 \,{\left (15 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{2}\right )}}{105 \, b^{3} c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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