Optimal. Leaf size=63 \[ \frac{2}{15} A b^2 x^{15/2}+\frac{2}{23} c x^{23/2} (A c+2 b B)+\frac{2}{19} b x^{19/2} (2 A c+b B)+\frac{2}{27} B c^2 x^{27/2} \]
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Rubi [A] time = 0.0388137, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1584, 448} \[ \frac{2}{15} A b^2 x^{15/2}+\frac{2}{23} c x^{23/2} (A c+2 b B)+\frac{2}{19} b x^{19/2} (2 A c+b B)+\frac{2}{27} B c^2 x^{27/2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 448
Rubi steps
\begin{align*} \int x^{5/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx &=\int x^{13/2} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^{13/2}+b (b B+2 A c) x^{17/2}+c (2 b B+A c) x^{21/2}+B c^2 x^{25/2}\right ) \, dx\\ &=\frac{2}{15} A b^2 x^{15/2}+\frac{2}{19} b (b B+2 A c) x^{19/2}+\frac{2}{23} c (2 b B+A c) x^{23/2}+\frac{2}{27} B c^2 x^{27/2}\\ \end{align*}
Mathematica [A] time = 0.0297814, size = 53, normalized size = 0.84 \[ \frac{2 x^{15/2} \left (3933 A b^2+2565 c x^4 (A c+2 b B)+3105 b x^2 (2 A c+b B)+2185 B c^2 x^6\right )}{58995} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{4370\,B{c}^{2}{x}^{6}+5130\,A{c}^{2}{x}^{4}+10260\,B{x}^{4}bc+12420\,Abc{x}^{2}+6210\,B{x}^{2}{b}^{2}+7866\,A{b}^{2}}{58995}{x}^{{\frac{15}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15168, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{27} \, B c^{2} x^{\frac{27}{2}} + \frac{2}{23} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{23}{2}} + \frac{2}{15} \, A b^{2} x^{\frac{15}{2}} + \frac{2}{19} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{19}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70015, size = 150, normalized size = 2.38 \begin{align*} \frac{2}{58995} \,{\left (2185 \, B c^{2} x^{13} + 2565 \,{\left (2 \, B b c + A c^{2}\right )} x^{11} + 3933 \, A b^{2} x^{7} + 3105 \,{\left (B b^{2} + 2 \, A b c\right )} x^{9}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 53.7629, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{15}{2}}}{15} + \frac{4 A b c x^{\frac{19}{2}}}{19} + \frac{2 A c^{2} x^{\frac{23}{2}}}{23} + \frac{2 B b^{2} x^{\frac{19}{2}}}{19} + \frac{4 B b c x^{\frac{23}{2}}}{23} + \frac{2 B c^{2} x^{\frac{27}{2}}}{27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16581, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{27} \, B c^{2} x^{\frac{27}{2}} + \frac{4}{23} \, B b c x^{\frac{23}{2}} + \frac{2}{23} \, A c^{2} x^{\frac{23}{2}} + \frac{2}{19} \, B b^{2} x^{\frac{19}{2}} + \frac{4}{19} \, A b c x^{\frac{19}{2}} + \frac{2}{15} \, A b^{2} x^{\frac{15}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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