Optimal. Leaf size=63 \[ \frac{2}{3} A b^2 x^{3/2}+\frac{2}{11} c x^{11/2} (A c+2 b B)+\frac{2}{7} b x^{7/2} (2 A c+b B)+\frac{2}{15} B c^2 x^{15/2} \]
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Rubi [A] time = 0.0379469, 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} \[ \frac{2}{3} A b^2 x^{3/2}+\frac{2}{11} c x^{11/2} (A c+2 b B)+\frac{2}{7} b x^{7/2} (2 A c+b B)+\frac{2}{15} B c^2 x^{15/2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 448
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx &=\int \sqrt{x} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 \sqrt{x}+b (b B+2 A c) x^{5/2}+c (2 b B+A c) x^{9/2}+B c^2 x^{13/2}\right ) \, dx\\ &=\frac{2}{3} A b^2 x^{3/2}+\frac{2}{7} b (b B+2 A c) x^{7/2}+\frac{2}{11} c (2 b B+A c) x^{11/2}+\frac{2}{15} B c^2 x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.0297101, size = 53, normalized size = 0.84 \[ \frac{2 x^{3/2} \left (385 A b^2+105 c x^4 (A c+2 b B)+165 b x^2 (2 A c+b B)+77 B c^2 x^6\right )}{1155} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 56, normalized size = 0.9 \begin{align*}{\frac{154\,B{c}^{2}{x}^{6}+210\,A{c}^{2}{x}^{4}+420\,B{x}^{4}bc+660\,Abc{x}^{2}+330\,B{x}^{2}{b}^{2}+770\,A{b}^{2}}{1155}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16944, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{11}{2}} + \frac{2}{3} \, A b^{2} x^{\frac{3}{2}} + \frac{2}{7} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46583, size = 136, normalized size = 2.16 \begin{align*} \frac{2}{1155} \,{\left (77 \, B c^{2} x^{7} + 105 \,{\left (2 \, B b c + A c^{2}\right )} x^{5} + 385 \, A b^{2} x + 165 \,{\left (B b^{2} + 2 \, A b c\right )} x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 21.5161, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{3}{2}}}{3} + \frac{4 A b c x^{\frac{7}{2}}}{7} + \frac{2 A c^{2} x^{\frac{11}{2}}}{11} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} + \frac{4 B b c x^{\frac{11}{2}}}{11} + \frac{2 B c^{2} x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19147, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{4}{11} \, B b c x^{\frac{11}{2}} + \frac{2}{11} \, A c^{2} x^{\frac{11}{2}} + \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{4}{7} \, A b c x^{\frac{7}{2}} + \frac{2}{3} \, A b^{2} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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