Optimal. Leaf size=77 \[ \frac {2}{5} a^2 A x^{5/2}+\frac {2}{7} a^2 B x^{7/2}+\frac {4}{9} a A c x^{9/2}+\frac {4}{11} a B c x^{11/2}+\frac {2}{13} A c^2 x^{13/2}+\frac {2}{15} B c^2 x^{15/2} \]
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Rubi [A] time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {766} \begin {gather*} \frac {2}{5} a^2 A x^{5/2}+\frac {2}{7} a^2 B x^{7/2}+\frac {4}{9} a A c x^{9/2}+\frac {4}{11} a B c x^{11/2}+\frac {2}{13} A c^2 x^{13/2}+\frac {2}{15} B c^2 x^{15/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin {align*} \int x^{3/2} (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A x^{3/2}+a^2 B x^{5/2}+2 a A c x^{7/2}+2 a B c x^{9/2}+A c^2 x^{11/2}+B c^2 x^{13/2}\right ) \, dx\\ &=\frac {2}{5} a^2 A x^{5/2}+\frac {2}{7} a^2 B x^{7/2}+\frac {4}{9} a A c x^{9/2}+\frac {4}{11} a B c x^{11/2}+\frac {2}{13} A c^2 x^{13/2}+\frac {2}{15} B c^2 x^{15/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 0.70 \begin {gather*} \frac {2 x^{5/2} \left (1287 a^2 (7 A+5 B x)+910 a c x^2 (11 A+9 B x)+231 c^2 x^4 (15 A+13 B x)\right )}{45045} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 69, normalized size = 0.90 \begin {gather*} \frac {2 \left (9009 a^2 A x^{5/2}+6435 a^2 B x^{7/2}+10010 a A c x^{9/2}+8190 a B c x^{11/2}+3465 A c^2 x^{13/2}+3003 B c^2 x^{15/2}\right )}{45045} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 58, normalized size = 0.75 \begin {gather*} \frac {2}{45045} \, {\left (3003 \, B c^{2} x^{7} + 3465 \, A c^{2} x^{6} + 8190 \, B a c x^{5} + 10010 \, A a c x^{4} + 6435 \, B a^{2} x^{3} + 9009 \, A a^{2} x^{2}\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 = 0.69 \begin {gather*} \frac {2}{15} \, B c^{2} x^{\frac {15}{2}} + \frac {2}{13} \, A c^{2} x^{\frac {13}{2}} + \frac {4}{11} \, B a c x^{\frac {11}{2}} + \frac {4}{9} \, A a c x^{\frac {9}{2}} + \frac {2}{7} \, B a^{2} x^{\frac {7}{2}} + \frac {2}{5} \, A a^{2} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 54, normalized size = 0.70 \begin {gather*} \frac {2 \left (3003 B \,c^{2} x^{5}+3465 A \,c^{2} x^{4}+8190 B a c \,x^{3}+10010 A a c \,x^{2}+6435 B \,a^{2} x +9009 A \,a^{2}\right ) x^{\frac {5}{2}}}{45045} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 53, normalized size = 0.69 \begin {gather*} \frac {2}{15} \, B c^{2} x^{\frac {15}{2}} + \frac {2}{13} \, A c^{2} x^{\frac {13}{2}} + \frac {4}{11} \, B a c x^{\frac {11}{2}} + \frac {4}{9} \, A a c x^{\frac {9}{2}} + \frac {2}{7} \, B a^{2} x^{\frac {7}{2}} + \frac {2}{5} \, A a^{2} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 53, normalized size = 0.69 \begin {gather*} \frac {2\,A\,a^2\,x^{5/2}}{5}+\frac {2\,B\,a^2\,x^{7/2}}{7}+\frac {2\,A\,c^2\,x^{13/2}}{13}+\frac {2\,B\,c^2\,x^{15/2}}{15}+\frac {4\,A\,a\,c\,x^{9/2}}{9}+\frac {4\,B\,a\,c\,x^{11/2}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.26, size = 80, normalized size = 1.04 \begin {gather*} \frac {2 A a^{2} x^{\frac {5}{2}}}{5} + \frac {4 A a c x^{\frac {9}{2}}}{9} + \frac {2 A c^{2} x^{\frac {13}{2}}}{13} + \frac {2 B a^{2} x^{\frac {7}{2}}}{7} + \frac {4 B a c x^{\frac {11}{2}}}{11} + \frac {2 B c^{2} x^{\frac {15}{2}}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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