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 = 0.04, 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} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 448
Rule 1584
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*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.84 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 69, normalized size = 1.10 \begin {gather*} \frac {2 \left (3933 A b^2 x^{15/2}+6210 A b c x^{19/2}+2565 A c^2 x^{23/2}+3105 b^2 B x^{19/2}+5130 b B c x^{23/2}+2185 B c^2 x^{27/2}\right )}{58995} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 56, normalized size = 0.89 \begin {gather*} \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 {gather*}
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 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 56, normalized size = 0.89 \begin {gather*} \frac {2 \left (2185 B \,c^{2} x^{6}+2565 A \,c^{2} x^{4}+5130 B b c \,x^{4}+6210 A b c \,x^{2}+3105 B \,b^{2} x^{2}+3933 b^{2} A \right ) x^{\frac {15}{2}}}{58995} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 51, normalized size = 0.81 \begin {gather*} \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 {gather*}
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 \begin {gather*} x^{19/2}\,\left (\frac {2\,B\,b^2}{19}+\frac {4\,A\,c\,b}{19}\right )+x^{23/2}\,\left (\frac {2\,A\,c^2}{23}+\frac {4\,B\,b\,c}{23}\right )+\frac {2\,A\,b^2\,x^{15/2}}{15}+\frac {2\,B\,c^2\,x^{27/2}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 39.82, size = 80, normalized size = 1.27 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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