Optimal. Leaf size=85 \[ \frac {2}{7} a^3 A x^{7/2}+\frac {2}{11} a^2 x^{11/2} (a B+3 A b)+\frac {2}{19} b^2 x^{19/2} (3 a B+A b)+\frac {2}{5} a b x^{15/2} (a B+A b)+\frac {2}{23} b^3 B x^{23/2} \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \[ \frac {2}{11} a^2 x^{11/2} (a B+3 A b)+\frac {2}{7} a^3 A x^{7/2}+\frac {2}{19} b^2 x^{19/2} (3 a B+A b)+\frac {2}{5} a b x^{15/2} (a B+A b)+\frac {2}{23} b^3 B x^{23/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx &=\int \left (a^3 A x^{5/2}+a^2 (3 A b+a B) x^{9/2}+3 a b (A b+a B) x^{13/2}+b^2 (A b+3 a B) x^{17/2}+b^3 B x^{21/2}\right ) \, dx\\ &=\frac {2}{7} a^3 A x^{7/2}+\frac {2}{11} a^2 (3 A b+a B) x^{11/2}+\frac {2}{5} a b (A b+a B) x^{15/2}+\frac {2}{19} b^2 (A b+3 a B) x^{19/2}+\frac {2}{23} b^3 B x^{23/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 85, normalized size = 1.00 \[ \frac {2}{7} a^3 A x^{7/2}+\frac {2}{11} a^2 x^{11/2} (a B+3 A b)+\frac {2}{19} b^2 x^{19/2} (3 a B+A b)+\frac {2}{5} a b x^{15/2} (a B+A b)+\frac {2}{23} b^3 B x^{23/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 78, normalized size = 0.92 \[ \frac {2}{168245} \, {\left (7315 \, B b^{3} x^{11} + 8855 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 33649 \, {\left (B a^{2} b + A a b^{2}\right )} x^{7} + 24035 \, A a^{3} x^{3} + 15295 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 77, normalized size = 0.91 \[ \frac {2}{23} \, B b^{3} x^{\frac {23}{2}} + \frac {6}{19} \, B a b^{2} x^{\frac {19}{2}} + \frac {2}{19} \, A b^{3} x^{\frac {19}{2}} + \frac {2}{5} \, B a^{2} b x^{\frac {15}{2}} + \frac {2}{5} \, A a b^{2} x^{\frac {15}{2}} + \frac {2}{11} \, B a^{3} x^{\frac {11}{2}} + \frac {6}{11} \, A a^{2} b x^{\frac {11}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.94 \[ \frac {2 \left (7315 B \,b^{3} x^{8}+8855 x^{6} A \,b^{3}+26565 B a \,b^{2} x^{6}+33649 x^{4} A a \,b^{2}+33649 x^{4} B \,a^{2} b +45885 A \,a^{2} b \,x^{2}+15295 B \,a^{3} x^{2}+24035 A \,a^{3}\right ) x^{\frac {7}{2}}}{168245} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 73, normalized size = 0.86 \[ \frac {2}{23} \, B b^{3} x^{\frac {23}{2}} + \frac {2}{19} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {19}{2}} + \frac {2}{5} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {15}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} + \frac {2}{11} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {11}{2}} \]
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 \[ x^{11/2}\,\left (\frac {2\,B\,a^3}{11}+\frac {6\,A\,b\,a^2}{11}\right )+x^{19/2}\,\left (\frac {2\,A\,b^3}{19}+\frac {6\,B\,a\,b^2}{19}\right )+\frac {2\,A\,a^3\,x^{7/2}}{7}+\frac {2\,B\,b^3\,x^{23/2}}{23}+\frac {2\,a\,b\,x^{15/2}\,\left (A\,b+B\,a\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.63, size = 114, normalized size = 1.34 \[ \frac {2 A a^{3} x^{\frac {7}{2}}}{7} + \frac {6 A a^{2} b x^{\frac {11}{2}}}{11} + \frac {2 A a b^{2} x^{\frac {15}{2}}}{5} + \frac {2 A b^{3} x^{\frac {19}{2}}}{19} + \frac {2 B a^{3} x^{\frac {11}{2}}}{11} + \frac {2 B a^{2} b x^{\frac {15}{2}}}{5} + \frac {6 B a b^{2} x^{\frac {19}{2}}}{19} + \frac {2 B b^{3} x^{\frac {23}{2}}}{23} \]
Verification of antiderivative is not currently implemented for this CAS.
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