Optimal. Leaf size=51 \[ \frac{6}{19} b^2 c x^{19/2}+\frac{2}{15} b^3 x^{15/2}+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2} \]
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Rubi [A] time = 0.0191347, antiderivative size = 51, 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 = {1584, 270} \[ \frac{6}{19} b^2 c x^{19/2}+\frac{2}{15} b^3 x^{15/2}+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 270
Rubi steps
\begin{align*} \int \sqrt{x} \left (b x^2+c x^4\right )^3 \, dx &=\int x^{13/2} \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^{13/2}+3 b^2 c x^{17/2}+3 b c^2 x^{21/2}+c^3 x^{25/2}\right ) \, dx\\ &=\frac{2}{15} b^3 x^{15/2}+\frac{6}{19} b^2 c x^{19/2}+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2}\\ \end{align*}
Mathematica [A] time = 0.0100261, size = 41, normalized size = 0.8 \[ \frac{2 x^{15/2} \left (9315 b^2 c x^2+3933 b^3+7695 b c^2 x^4+2185 c^3 x^6\right )}{58995} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 38, normalized size = 0.8 \begin{align*}{\frac{4370\,{c}^{3}{x}^{6}+15390\,b{c}^{2}{x}^{4}+18630\,{b}^{2}c{x}^{2}+7866\,{b}^{3}}{58995}{x}^{{\frac{15}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984155, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{27} \, c^{3} x^{\frac{27}{2}} + \frac{6}{23} \, b c^{2} x^{\frac{23}{2}} + \frac{6}{19} \, b^{2} c x^{\frac{19}{2}} + \frac{2}{15} \, b^{3} x^{\frac{15}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29486, size = 112, normalized size = 2.2 \begin{align*} \frac{2}{58995} \,{\left (2185 \, c^{3} x^{13} + 7695 \, b c^{2} x^{11} + 9315 \, b^{2} c x^{9} + 3933 \, b^{3} x^{7}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.64515, size = 49, normalized size = 0.96 \begin{align*} \frac{2 b^{3} x^{\frac{15}{2}}}{15} + \frac{6 b^{2} c x^{\frac{19}{2}}}{19} + \frac{6 b c^{2} x^{\frac{23}{2}}}{23} + \frac{2 c^{3} x^{\frac{27}{2}}}{27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13674, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{27} \, c^{3} x^{\frac{27}{2}} + \frac{6}{23} \, b c^{2} x^{\frac{23}{2}} + \frac{6}{19} \, b^{2} c x^{\frac{19}{2}} + \frac{2}{15} \, b^{3} x^{\frac{15}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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