Optimal. Leaf size=63 \[ \frac {2}{5} A b^2 x^{5/2}+\frac {2}{13} c x^{13/2} (A c+2 b B)+\frac {2}{9} b x^{9/2} (2 A c+b B)+\frac {2}{17} B c^2 x^{17/2} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, 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} \[ \frac {2}{5} A b^2 x^{5/2}+\frac {2}{13} c x^{13/2} (A c+2 b B)+\frac {2}{9} b x^{9/2} (2 A c+b B)+\frac {2}{17} B c^2 x^{17/2} \]
Antiderivative was successfully verified.
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Rule 448
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^{5/2}} \, dx &=\int x^{3/2} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^{3/2}+b (b B+2 A c) x^{7/2}+c (2 b B+A c) x^{11/2}+B c^2 x^{15/2}\right ) \, dx\\ &=\frac {2}{5} A b^2 x^{5/2}+\frac {2}{9} b (b B+2 A c) x^{9/2}+\frac {2}{13} c (2 b B+A c) x^{13/2}+\frac {2}{17} B c^2 x^{17/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.84 \[ \frac {2 x^{5/2} \left (1989 A b^2+765 c x^4 (A c+2 b B)+1105 b x^2 (2 A c+b B)+585 B c^2 x^6\right )}{9945} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 56, normalized size = 0.89 \[ \frac {2}{9945} \, {\left (585 \, B c^{2} x^{8} + 765 \, {\left (2 \, B b c + A c^{2}\right )} x^{6} + 1989 \, A b^{2} x^{2} + 1105 \, {\left (B b^{2} + 2 \, A b c\right )} x^{4}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.84 \[ \frac {2}{17} \, B c^{2} x^{\frac {17}{2}} + \frac {4}{13} \, B b c x^{\frac {13}{2}} + \frac {2}{13} \, A c^{2} x^{\frac {13}{2}} + \frac {2}{9} \, B b^{2} x^{\frac {9}{2}} + \frac {4}{9} \, A b c x^{\frac {9}{2}} + \frac {2}{5} \, A b^{2} x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 56, normalized size = 0.89 \[ \frac {2 \left (585 B \,c^{2} x^{6}+765 A \,c^{2} x^{4}+1530 B b c \,x^{4}+2210 A b c \,x^{2}+1105 B \,b^{2} x^{2}+1989 b^{2} A \right ) x^{\frac {5}{2}}}{9945} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 51, normalized size = 0.81 \[ \frac {2}{17} \, B c^{2} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {13}{2}} + \frac {2}{5} \, A b^{2} x^{\frac {5}{2}} + \frac {2}{9} \, {\left (B b^{2} + 2 \, A b c\right )} x^{\frac {9}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 0.81 \[ x^{9/2}\,\left (\frac {2\,B\,b^2}{9}+\frac {4\,A\,c\,b}{9}\right )+x^{13/2}\,\left (\frac {2\,A\,c^2}{13}+\frac {4\,B\,b\,c}{13}\right )+\frac {2\,A\,b^2\,x^{5/2}}{5}+\frac {2\,B\,c^2\,x^{17/2}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.06, size = 80, normalized size = 1.27 \[ \frac {2 A b^{2} x^{\frac {5}{2}}}{5} + \frac {4 A b c x^{\frac {9}{2}}}{9} + \frac {2 A c^{2} x^{\frac {13}{2}}}{13} + \frac {2 B b^{2} x^{\frac {9}{2}}}{9} + \frac {4 B b c x^{\frac {13}{2}}}{13} + \frac {2 B c^{2} x^{\frac {17}{2}}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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