Optimal. Leaf size=51 \[ \frac{6}{25} b^2 c x^{25/2}+\frac{2}{21} b^3 x^{21/2}+\frac{6}{29} b c^2 x^{29/2}+\frac{2}{33} c^3 x^{33/2} \]
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Rubi [A] time = 0.0212143, 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} \[ \frac{6}{25} b^2 c x^{25/2}+\frac{2}{21} b^3 x^{21/2}+\frac{6}{29} b c^2 x^{29/2}+\frac{2}{33} c^3 x^{33/2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 270
Rubi steps
\begin{align*} \int x^{7/2} \left (b x^2+c x^4\right )^3 \, dx &=\int x^{19/2} \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^{19/2}+3 b^2 c x^{23/2}+3 b c^2 x^{27/2}+c^3 x^{31/2}\right ) \, dx\\ &=\frac{2}{21} b^3 x^{21/2}+\frac{6}{25} b^2 c x^{25/2}+\frac{6}{29} b c^2 x^{29/2}+\frac{2}{33} c^3 x^{33/2}\\ \end{align*}
Mathematica [A] time = 0.0138796, size = 51, normalized size = 1. \[ \frac{6}{25} b^2 c x^{25/2}+\frac{2}{21} b^3 x^{21/2}+\frac{6}{29} b c^2 x^{29/2}+\frac{2}{33} c^3 x^{33/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 38, normalized size = 0.8 \begin{align*}{\frac{10150\,{c}^{3}{x}^{6}+34650\,b{c}^{2}{x}^{4}+40194\,{b}^{2}c{x}^{2}+15950\,{b}^{3}}{167475}{x}^{{\frac{21}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968687, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{33} \, c^{3} x^{\frac{33}{2}} + \frac{6}{29} \, b c^{2} x^{\frac{29}{2}} + \frac{6}{25} \, b^{2} c x^{\frac{25}{2}} + \frac{2}{21} \, b^{3} x^{\frac{21}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26853, size = 119, normalized size = 2.33 \begin{align*} \frac{2}{167475} \,{\left (5075 \, c^{3} x^{16} + 17325 \, b c^{2} x^{14} + 20097 \, b^{2} c x^{12} + 7975 \, b^{3} x^{10}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 112.55, size = 49, normalized size = 0.96 \begin{align*} \frac{2 b^{3} x^{\frac{21}{2}}}{21} + \frac{6 b^{2} c x^{\frac{25}{2}}}{25} + \frac{6 b c^{2} x^{\frac{29}{2}}}{29} + \frac{2 c^{3} x^{\frac{33}{2}}}{33} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15837, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{33} \, c^{3} x^{\frac{33}{2}} + \frac{6}{29} \, b c^{2} x^{\frac{29}{2}} + \frac{6}{25} \, b^{2} c x^{\frac{25}{2}} + \frac{2}{21} \, b^{3} x^{\frac{21}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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