Optimal. Leaf size=85 \[ \frac {2}{11} A b^3 x^{11/2}+\frac {2}{15} b^2 x^{15/2} (3 A c+b B)+\frac {2}{23} c^2 x^{23/2} (A c+3 b B)+\frac {6}{19} b c x^{19/2} (A c+b B)+\frac {2}{27} B c^3 x^{27/2} \]
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Rubi [A] time = 0.05, antiderivative size = 85, 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} b^2 x^{15/2} (3 A c+b B)+\frac {2}{11} A b^3 x^{11/2}+\frac {2}{23} c^2 x^{23/2} (A c+3 b B)+\frac {6}{19} b c x^{19/2} (A c+b B)+\frac {2}{27} B c^3 x^{27/2} \end {gather*}
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 )^3}{x^{3/2}} \, dx &=\int x^{9/2} \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3 x^{9/2}+b^2 (b B+3 A c) x^{13/2}+3 b c (b B+A c) x^{17/2}+c^2 (3 b B+A c) x^{21/2}+B c^3 x^{25/2}\right ) \, dx\\ &=\frac {2}{11} A b^3 x^{11/2}+\frac {2}{15} b^2 (b B+3 A c) x^{15/2}+\frac {6}{19} b c (b B+A c) x^{19/2}+\frac {2}{23} c^2 (3 b B+A c) x^{23/2}+\frac {2}{27} B c^3 x^{27/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 85, normalized size = 1.00 \begin {gather*} \frac {2}{11} A b^3 x^{11/2}+\frac {2}{15} b^2 x^{15/2} (3 A c+b B)+\frac {2}{23} c^2 x^{23/2} (A c+3 b B)+\frac {6}{19} b c x^{19/2} (A c+b B)+\frac {2}{27} B c^3 x^{27/2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (58995 A b^3 x^{11/2}+129789 A b^2 c x^{15/2}+102465 A b c^2 x^{19/2}+28215 A c^3 x^{23/2}+43263 b^3 B x^{15/2}+102465 b^2 B c x^{19/2}+84645 b B c^2 x^{23/2}+24035 B c^3 x^{27/2}\right )}{648945} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{648945} \, {\left (24035 \, B c^{3} x^{13} + 28215 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{11} + 102465 \, {\left (B b^{2} c + A b c^{2}\right )} x^{9} + 58995 \, A b^{3} x^{5} + 43263 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{7}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{27} \, B c^{3} x^{\frac {27}{2}} + \frac {6}{23} \, B b c^{2} x^{\frac {23}{2}} + \frac {2}{23} \, A c^{3} x^{\frac {23}{2}} + \frac {6}{19} \, B b^{2} c x^{\frac {19}{2}} + \frac {6}{19} \, A b c^{2} x^{\frac {19}{2}} + \frac {2}{15} \, B b^{3} x^{\frac {15}{2}} + \frac {2}{5} \, A b^{2} c x^{\frac {15}{2}} + \frac {2}{11} \, A b^{3} x^{\frac {11}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 80, normalized size = 0.94 \begin {gather*} \frac {2 \left (24035 B \,c^{3} x^{8}+28215 A \,c^{3} x^{6}+84645 B b \,c^{2} x^{6}+102465 A b \,c^{2} x^{4}+102465 B \,b^{2} c \,x^{4}+129789 A \,b^{2} c \,x^{2}+43263 B \,b^{3} x^{2}+58995 A \,b^{3}\right ) x^{\frac {11}{2}}}{648945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{27} \, B c^{3} x^{\frac {27}{2}} + \frac {2}{23} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {23}{2}} + \frac {6}{19} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {19}{2}} + \frac {2}{11} \, A b^{3} x^{\frac {11}{2}} + \frac {2}{15} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {15}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{15/2}\,\left (\frac {2\,B\,b^3}{15}+\frac {2\,A\,c\,b^2}{5}\right )+x^{23/2}\,\left (\frac {2\,A\,c^3}{23}+\frac {6\,B\,b\,c^2}{23}\right )+\frac {2\,A\,b^3\,x^{11/2}}{11}+\frac {2\,B\,c^3\,x^{27/2}}{27}+\frac {6\,b\,c\,x^{19/2}\,\left (A\,c+B\,b\right )}{19} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 31.77, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {11}{2}}}{11} + \frac {2 A b^{2} c x^{\frac {15}{2}}}{5} + \frac {6 A b c^{2} x^{\frac {19}{2}}}{19} + \frac {2 A c^{3} x^{\frac {23}{2}}}{23} + \frac {2 B b^{3} x^{\frac {15}{2}}}{15} + \frac {6 B b^{2} c x^{\frac {19}{2}}}{19} + \frac {6 B b c^{2} x^{\frac {23}{2}}}{23} + \frac {2 B c^{3} x^{\frac {27}{2}}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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