Optimal. Leaf size=77 \[ \frac {2}{7} a^2 A x^{7/2}+\frac {2}{9} a^2 B x^{9/2}+\frac {4}{11} a A c x^{11/2}+\frac {4}{13} a B c x^{13/2}+\frac {2}{15} A c^2 x^{15/2}+\frac {2}{17} B c^2 x^{17/2} \]
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Rubi [A] time = 0.03, 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}{7} a^2 A x^{7/2}+\frac {2}{9} a^2 B x^{9/2}+\frac {4}{11} a A c x^{11/2}+\frac {4}{13} a B c x^{13/2}+\frac {2}{15} A c^2 x^{15/2}+\frac {2}{17} B c^2 x^{17/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin {align*} \int x^{5/2} (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A x^{5/2}+a^2 B x^{7/2}+2 a A c x^{9/2}+2 a B c x^{11/2}+A c^2 x^{13/2}+B c^2 x^{15/2}\right ) \, dx\\ &=\frac {2}{7} a^2 A x^{7/2}+\frac {2}{9} a^2 B x^{9/2}+\frac {4}{11} a A c x^{11/2}+\frac {4}{13} a B c x^{13/2}+\frac {2}{15} A c^2 x^{15/2}+\frac {2}{17} B c^2 x^{17/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 60, normalized size = 0.78 \begin {gather*} \frac {2}{63} a^2 x^{7/2} (9 A+7 B x)+\frac {4}{143} a c x^{11/2} (13 A+11 B x)+\frac {2}{255} c^2 x^{15/2} (17 A+15 B x) \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 (109395 a^2 A x^{7/2}+85085 a^2 B x^{9/2}+139230 a A c x^{11/2}+117810 a B c x^{13/2}+51051 A c^2 x^{15/2}+45045 B c^2 x^{17/2}\right )}{765765} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 58, normalized size = 0.75 \begin {gather*} \frac {2}{765765} \, {\left (45045 \, B c^{2} x^{8} + 51051 \, A c^{2} x^{7} + 117810 \, B a c x^{6} + 139230 \, A a c x^{5} + 85085 \, B a^{2} x^{4} + 109395 \, A a^{2} x^{3}\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}{17} \, B c^{2} x^{\frac {17}{2}} + \frac {2}{15} \, A c^{2} x^{\frac {15}{2}} + \frac {4}{13} \, B a c x^{\frac {13}{2}} + \frac {4}{11} \, A a c x^{\frac {11}{2}} + \frac {2}{9} \, B a^{2} x^{\frac {9}{2}} + \frac {2}{7} \, A a^{2} x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 54, normalized size = 0.70 \begin {gather*} \frac {2 \left (45045 B \,c^{2} x^{5}+51051 A \,c^{2} x^{4}+117810 B a c \,x^{3}+139230 A a c \,x^{2}+85085 B \,a^{2} x +109395 A \,a^{2}\right ) x^{\frac {7}{2}}}{765765} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 53, normalized size = 0.69 \begin {gather*} \frac {2}{17} \, B c^{2} x^{\frac {17}{2}} + \frac {2}{15} \, A c^{2} x^{\frac {15}{2}} + \frac {4}{13} \, B a c x^{\frac {13}{2}} + \frac {4}{11} \, A a c x^{\frac {11}{2}} + \frac {2}{9} \, B a^{2} x^{\frac {9}{2}} + \frac {2}{7} \, A a^{2} x^{\frac {7}{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^{7/2}}{7}+\frac {2\,B\,a^2\,x^{9/2}}{9}+\frac {2\,A\,c^2\,x^{15/2}}{15}+\frac {2\,B\,c^2\,x^{17/2}}{17}+\frac {4\,A\,a\,c\,x^{11/2}}{11}+\frac {4\,B\,a\,c\,x^{13/2}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.50, size = 80, normalized size = 1.04 \begin {gather*} \frac {2 A a^{2} x^{\frac {7}{2}}}{7} + \frac {4 A a c x^{\frac {11}{2}}}{11} + \frac {2 A c^{2} x^{\frac {15}{2}}}{15} + \frac {2 B a^{2} x^{\frac {9}{2}}}{9} + \frac {4 B a c x^{\frac {13}{2}}}{13} + \frac {2 B c^{2} x^{\frac {17}{2}}}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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