Optimal. Leaf size=63 \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{21} c x^{21/2} (A c+2 b B)+\frac{2}{17} b x^{17/2} (2 A c+b B)+\frac{2}{25} B c^2 x^{25/2} \]
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Rubi [A] time = 0.0380087, 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} \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{21} c x^{21/2} (A c+2 b B)+\frac{2}{17} b x^{17/2} (2 A c+b B)+\frac{2}{25} B c^2 x^{25/2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 448
Rubi steps
\begin{align*} \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx &=\int x^{11/2} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^{11/2}+b (b B+2 A c) x^{15/2}+c (2 b B+A c) x^{19/2}+B c^2 x^{23/2}\right ) \, dx\\ &=\frac{2}{13} A b^2 x^{13/2}+\frac{2}{17} b (b B+2 A c) x^{17/2}+\frac{2}{21} c (2 b B+A c) x^{21/2}+\frac{2}{25} B c^2 x^{25/2}\\ \end{align*}
Mathematica [A] time = 0.0294782, size = 63, normalized size = 1. \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{21} c x^{21/2} (A c+2 b B)+\frac{2}{17} b x^{17/2} (2 A c+b B)+\frac{2}{25} B c^2 x^{25/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{9282\,B{c}^{2}{x}^{6}+11050\,A{c}^{2}{x}^{4}+22100\,B{x}^{4}bc+27300\,Abc{x}^{2}+13650\,B{x}^{2}{b}^{2}+17850\,A{b}^{2}}{116025}{x}^{{\frac{13}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13105, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{25} \, B c^{2} x^{\frac{25}{2}} + \frac{2}{21} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{21}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} + \frac{2}{17} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{17}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53459, size = 151, normalized size = 2.4 \begin{align*} \frac{2}{116025} \,{\left (4641 \, B c^{2} x^{12} + 5525 \,{\left (2 \, B b c + A c^{2}\right )} x^{10} + 8925 \, A b^{2} x^{6} + 6825 \,{\left (B b^{2} + 2 \, A b c\right )} x^{8}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 27.9737, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{4 A b c x^{\frac{17}{2}}}{17} + \frac{2 A c^{2} x^{\frac{21}{2}}}{21} + \frac{2 B b^{2} x^{\frac{17}{2}}}{17} + \frac{4 B b c x^{\frac{21}{2}}}{21} + \frac{2 B c^{2} x^{\frac{25}{2}}}{25} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21286, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{25} \, B c^{2} x^{\frac{25}{2}} + \frac{4}{21} \, B b c x^{\frac{21}{2}} + \frac{2}{21} \, A c^{2} x^{\frac{21}{2}} + \frac{2}{17} \, B b^{2} x^{\frac{17}{2}} + \frac{4}{17} \, A b c x^{\frac{17}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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