Optimal. Leaf size=85 \[ \frac{2}{11} b^2 x^{11/2} (3 A c+b B)+\frac{2}{7} A b^3 x^{7/2}+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{2}{5} b c x^{15/2} (A c+b B)+\frac{2}{23} B c^3 x^{23/2} \]
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Rubi [A] time = 0.0511572, antiderivative size = 85, 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} \[ \frac{2}{11} b^2 x^{11/2} (3 A c+b B)+\frac{2}{7} A b^3 x^{7/2}+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{2}{5} b c x^{15/2} (A c+b B)+\frac{2}{23} B c^3 x^{23/2} \]
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 )^3}{x^{7/2}} \, dx &=\int x^{5/2} \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3 x^{5/2}+b^2 (b B+3 A c) x^{9/2}+3 b c (b B+A c) x^{13/2}+c^2 (3 b B+A c) x^{17/2}+B c^3 x^{21/2}\right ) \, dx\\ &=\frac{2}{7} A b^3 x^{7/2}+\frac{2}{11} b^2 (b B+3 A c) x^{11/2}+\frac{2}{5} b c (b B+A c) x^{15/2}+\frac{2}{19} c^2 (3 b B+A c) x^{19/2}+\frac{2}{23} B c^3 x^{23/2}\\ \end{align*}
Mathematica [A] time = 0.0372151, size = 85, normalized size = 1. \[ \frac{2}{11} b^2 x^{11/2} (3 A c+b B)+\frac{2}{7} A b^3 x^{7/2}+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{2}{5} b c x^{15/2} (A c+b B)+\frac{2}{23} B c^3 x^{23/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 80, normalized size = 0.9 \begin{align*}{\frac{14630\,B{c}^{3}{x}^{8}+17710\,A{c}^{3}{x}^{6}+53130\,B{x}^{6}b{c}^{2}+67298\,Ab{c}^{2}{x}^{4}+67298\,B{x}^{4}{b}^{2}c+91770\,A{b}^{2}c{x}^{2}+30590\,B{x}^{2}{b}^{3}+48070\,A{b}^{3}}{168245}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09849, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{23} \, B c^{3} x^{\frac{23}{2}} + \frac{2}{19} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{19}{2}} + \frac{2}{5} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{15}{2}} + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{11}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86223, size = 201, normalized size = 2.36 \begin{align*} \frac{2}{168245} \,{\left (7315 \, B c^{3} x^{11} + 8855 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + 33649 \,{\left (B b^{2} c + A b c^{2}\right )} x^{7} + 24035 \, A b^{3} x^{3} + 15295 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{5}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 112.692, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A b^{3} x^{\frac{7}{2}}}{7} + \frac{6 A b^{2} c x^{\frac{11}{2}}}{11} + \frac{2 A b c^{2} x^{\frac{15}{2}}}{5} + \frac{2 A c^{3} x^{\frac{19}{2}}}{19} + \frac{2 B b^{3} x^{\frac{11}{2}}}{11} + \frac{2 B b^{2} c x^{\frac{15}{2}}}{5} + \frac{6 B b c^{2} x^{\frac{19}{2}}}{19} + \frac{2 B c^{3} x^{\frac{23}{2}}}{23} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13731, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{23} \, B c^{3} x^{\frac{23}{2}} + \frac{6}{19} \, B b c^{2} x^{\frac{19}{2}} + \frac{2}{19} \, A c^{3} x^{\frac{19}{2}} + \frac{2}{5} \, B b^{2} c x^{\frac{15}{2}} + \frac{2}{5} \, A b c^{2} x^{\frac{15}{2}} + \frac{2}{11} \, B b^{3} x^{\frac{11}{2}} + \frac{6}{11} \, A b^{2} c x^{\frac{11}{2}} + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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