Optimal. Leaf size=63 \[ \frac {2}{7} A b^2 x^{7/2}+\frac {2}{15} c x^{15/2} (A c+2 b B)+\frac {2}{11} b x^{11/2} (2 A c+b B)+\frac {2}{19} B c^2 x^{19/2} \]
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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}{7} A b^2 x^{7/2}+\frac {2}{15} c x^{15/2} (A c+2 b B)+\frac {2}{11} b x^{11/2} (2 A c+b B)+\frac {2}{19} B c^2 x^{19/2} \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 )^2}{x^{3/2}} \, dx &=\int x^{5/2} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^{5/2}+b (b B+2 A c) x^{9/2}+c (2 b B+A c) x^{13/2}+B c^2 x^{17/2}\right ) \, dx\\ &=\frac {2}{7} A b^2 x^{7/2}+\frac {2}{11} b (b B+2 A c) x^{11/2}+\frac {2}{15} c (2 b B+A c) x^{15/2}+\frac {2}{19} B c^2 x^{19/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 1.00 \begin {gather*} \frac {2}{7} A b^2 x^{7/2}+\frac {2}{15} c x^{15/2} (A c+2 b B)+\frac {2}{11} b x^{11/2} (2 A c+b B)+\frac {2}{19} B c^2 x^{19/2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 69, normalized size = 1.10 \begin {gather*} \frac {2 \left (3135 A b^2 x^{7/2}+3990 A b c x^{11/2}+1463 A c^2 x^{15/2}+1995 b^2 B x^{11/2}+2926 b B c x^{15/2}+1155 B c^2 x^{19/2}\right )}{21945} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 56, normalized size = 0.89 \begin {gather*} \frac {2}{21945} \, {\left (1155 \, B c^{2} x^{9} + 1463 \, {\left (2 \, B b c + A c^{2}\right )} x^{7} + 3135 \, A b^{2} x^{3} + 1995 \, {\left (B b^{2} + 2 \, A b c\right )} x^{5}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 53, normalized size = 0.84 \begin {gather*} \frac {2}{19} \, B c^{2} x^{\frac {19}{2}} + \frac {4}{15} \, B b c x^{\frac {15}{2}} + \frac {2}{15} \, A c^{2} x^{\frac {15}{2}} + \frac {2}{11} \, B b^{2} x^{\frac {11}{2}} + \frac {4}{11} \, A b c x^{\frac {11}{2}} + \frac {2}{7} \, A b^{2} x^{\frac {7}{2}} \end {gather*}
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 \begin {gather*} \frac {2 \left (1155 B \,c^{2} x^{6}+1463 A \,c^{2} x^{4}+2926 B b c \,x^{4}+3990 A b c \,x^{2}+1995 B \,b^{2} x^{2}+3135 b^{2} A \right ) x^{\frac {7}{2}}}{21945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{19} \, B c^{2} x^{\frac {19}{2}} + \frac {2}{15} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {15}{2}} + \frac {2}{7} \, A b^{2} x^{\frac {7}{2}} + \frac {2}{11} \, {\left (B b^{2} + 2 \, A b c\right )} x^{\frac {11}{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^{11/2}\,\left (\frac {2\,B\,b^2}{11}+\frac {4\,A\,c\,b}{11}\right )+x^{15/2}\,\left (\frac {2\,A\,c^2}{15}+\frac {4\,B\,b\,c}{15}\right )+\frac {2\,A\,b^2\,x^{7/2}}{7}+\frac {2\,B\,c^2\,x^{19/2}}{19} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.48, size = 80, normalized size = 1.27 \begin {gather*} \frac {2 A b^{2} x^{\frac {7}{2}}}{7} + \frac {4 A b c x^{\frac {11}{2}}}{11} + \frac {2 A c^{2} x^{\frac {15}{2}}}{15} + \frac {2 B b^{2} x^{\frac {11}{2}}}{11} + \frac {4 B b c x^{\frac {15}{2}}}{15} + \frac {2 B c^{2} x^{\frac {19}{2}}}{19} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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