Optimal. Leaf size=77 \[ \frac {2}{9} a^2 A x^{9/2}+\frac {2}{11} a^2 B x^{11/2}+\frac {4}{13} a A c x^{13/2}+\frac {4}{15} a B c x^{15/2}+\frac {2}{17} A c^2 x^{17/2}+\frac {2}{19} B c^2 x^{19/2} \]
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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}{9} a^2 A x^{9/2}+\frac {2}{11} a^2 B x^{11/2}+\frac {4}{13} a A c x^{13/2}+\frac {4}{15} a B c x^{15/2}+\frac {2}{17} A c^2 x^{17/2}+\frac {2}{19} B c^2 x^{19/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin {align*} \int x^{7/2} (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A x^{7/2}+a^2 B x^{9/2}+2 a A c x^{11/2}+2 a B c x^{13/2}+A c^2 x^{15/2}+B c^2 x^{17/2}\right ) \, dx\\ &=\frac {2}{9} a^2 A x^{9/2}+\frac {2}{11} a^2 B x^{11/2}+\frac {4}{13} a A c x^{13/2}+\frac {4}{15} a B c x^{15/2}+\frac {2}{17} A c^2 x^{17/2}+\frac {2}{19} B c^2 x^{19/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 60, normalized size = 0.78 \begin {gather*} \frac {2}{99} a^2 x^{9/2} (11 A+9 B x)+\frac {4}{195} a c x^{13/2} (15 A+13 B x)+\frac {2}{323} c^2 x^{17/2} (19 A+17 B x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 69, normalized size = 0.90 \begin {gather*} \frac {2 \left (230945 a^2 A x^{9/2}+188955 a^2 B x^{11/2}+319770 a A c x^{13/2}+277134 a B c x^{15/2}+122265 A c^2 x^{17/2}+109395 B c^2 x^{19/2}\right )}{2078505} \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}{2078505} \, {\left (109395 \, B c^{2} x^{9} + 122265 \, A c^{2} x^{8} + 277134 \, B a c x^{7} + 319770 \, A a c x^{6} + 188955 \, B a^{2} x^{5} + 230945 \, A a^{2} x^{4}\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 = 53, normalized size = 0.69 \begin {gather*} \frac {2}{19} \, B c^{2} x^{\frac {19}{2}} + \frac {2}{17} \, A c^{2} x^{\frac {17}{2}} + \frac {4}{15} \, B a c x^{\frac {15}{2}} + \frac {4}{13} \, A a c x^{\frac {13}{2}} + \frac {2}{11} \, B a^{2} x^{\frac {11}{2}} + \frac {2}{9} \, A a^{2} x^{\frac {9}{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 (109395 B \,c^{2} x^{5}+122265 A \,c^{2} x^{4}+277134 B a c \,x^{3}+319770 A a c \,x^{2}+188955 B \,a^{2} x +230945 A \,a^{2}\right ) x^{\frac {9}{2}}}{2078505} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 53, normalized size = 0.69 \begin {gather*} \frac {2}{19} \, B c^{2} x^{\frac {19}{2}} + \frac {2}{17} \, A c^{2} x^{\frac {17}{2}} + \frac {4}{15} \, B a c x^{\frac {15}{2}} + \frac {4}{13} \, A a c x^{\frac {13}{2}} + \frac {2}{11} \, B a^{2} x^{\frac {11}{2}} + \frac {2}{9} \, A a^{2} x^{\frac {9}{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^{9/2}}{9}+\frac {2\,B\,a^2\,x^{11/2}}{11}+\frac {2\,A\,c^2\,x^{17/2}}{17}+\frac {2\,B\,c^2\,x^{19/2}}{19}+\frac {4\,A\,a\,c\,x^{13/2}}{13}+\frac {4\,B\,a\,c\,x^{15/2}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.12, size = 80, normalized size = 1.04 \begin {gather*} \frac {2 A a^{2} x^{\frac {9}{2}}}{9} + \frac {4 A a c x^{\frac {13}{2}}}{13} + \frac {2 A c^{2} x^{\frac {17}{2}}}{17} + \frac {2 B a^{2} x^{\frac {11}{2}}}{11} + \frac {4 B a 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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