Optimal. Leaf size=109 \[ \frac {2}{9} a^3 A x^{9/2}+\frac {2}{11} a^3 B x^{11/2}+\frac {6}{13} a^2 A c x^{13/2}+\frac {2}{5} a^2 B c x^{15/2}+\frac {6}{17} a A c^2 x^{17/2}+\frac {6}{19} a B c^2 x^{19/2}+\frac {2}{21} A c^3 x^{21/2}+\frac {2}{23} B c^3 x^{23/2} \]
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Rubi [A] time = 0.04, antiderivative size = 109, 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 {6}{13} a^2 A c x^{13/2}+\frac {2}{9} a^3 A x^{9/2}+\frac {2}{5} a^2 B c x^{15/2}+\frac {2}{11} a^3 B x^{11/2}+\frac {6}{17} a A c^2 x^{17/2}+\frac {6}{19} a B c^2 x^{19/2}+\frac {2}{21} A c^3 x^{21/2}+\frac {2}{23} B c^3 x^{23/2} \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 )^3 \, dx &=\int \left (a^3 A x^{7/2}+a^3 B x^{9/2}+3 a^2 A c x^{11/2}+3 a^2 B c x^{13/2}+3 a A c^2 x^{15/2}+3 a B c^2 x^{17/2}+A c^3 x^{19/2}+B c^3 x^{21/2}\right ) \, dx\\ &=\frac {2}{9} a^3 A x^{9/2}+\frac {2}{11} a^3 B x^{11/2}+\frac {6}{13} a^2 A c x^{13/2}+\frac {2}{5} a^2 B c x^{15/2}+\frac {6}{17} a A c^2 x^{17/2}+\frac {6}{19} a B c^2 x^{19/2}+\frac {2}{21} A c^3 x^{21/2}+\frac {2}{23} B c^3 x^{23/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 83, normalized size = 0.76 \begin {gather*} \frac {2}{99} a^3 x^{9/2} (11 A+9 B x)+\frac {2}{65} a^2 c x^{13/2} (15 A+13 B x)+\frac {6}{323} a c^2 x^{17/2} (19 A+17 B x)+\frac {2}{483} c^3 x^{21/2} (23 A+21 B x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 97, normalized size = 0.89 \begin {gather*} \frac {2 \left (37182145 a^3 A x^{9/2}+30421755 a^3 B x^{11/2}+77224455 a^2 A c x^{13/2}+66927861 a^2 B c x^{15/2}+59053995 a A c^2 x^{17/2}+52837785 a B c^2 x^{19/2}+15935205 A c^3 x^{21/2}+14549535 B c^3 x^{23/2}\right )}{334639305} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 82, normalized size = 0.75 \begin {gather*} \frac {2}{334639305} \, {\left (14549535 \, B c^{3} x^{11} + 15935205 \, A c^{3} x^{10} + 52837785 \, B a c^{2} x^{9} + 59053995 \, A a c^{2} x^{8} + 66927861 \, B a^{2} c x^{7} + 77224455 \, A a^{2} c x^{6} + 30421755 \, B a^{3} x^{5} + 37182145 \, A a^{3} x^{4}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 77, normalized size = 0.71 \begin {gather*} \frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{21} \, A c^{3} x^{\frac {21}{2}} + \frac {6}{19} \, B a c^{2} x^{\frac {19}{2}} + \frac {6}{17} \, A a c^{2} x^{\frac {17}{2}} + \frac {2}{5} \, B a^{2} c x^{\frac {15}{2}} + \frac {6}{13} \, A a^{2} c x^{\frac {13}{2}} + \frac {2}{11} \, B a^{3} x^{\frac {11}{2}} + \frac {2}{9} \, A a^{3} 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 = 78, normalized size = 0.72 \begin {gather*} \frac {2 \left (14549535 B \,c^{3} x^{7}+15935205 A \,c^{3} x^{6}+52837785 B a \,c^{2} x^{5}+59053995 A a \,c^{2} x^{4}+66927861 B \,a^{2} c \,x^{3}+77224455 A \,a^{2} c \,x^{2}+30421755 B \,a^{3} x +37182145 A \,a^{3}\right ) x^{\frac {9}{2}}}{334639305} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 77, normalized size = 0.71 \begin {gather*} \frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{21} \, A c^{3} x^{\frac {21}{2}} + \frac {6}{19} \, B a c^{2} x^{\frac {19}{2}} + \frac {6}{17} \, A a c^{2} x^{\frac {17}{2}} + \frac {2}{5} \, B a^{2} c x^{\frac {15}{2}} + \frac {6}{13} \, A a^{2} c x^{\frac {13}{2}} + \frac {2}{11} \, B a^{3} x^{\frac {11}{2}} + \frac {2}{9} \, A a^{3} x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 77, normalized size = 0.71 \begin {gather*} \frac {2\,A\,a^3\,x^{9/2}}{9}+\frac {2\,B\,a^3\,x^{11/2}}{11}+\frac {2\,A\,c^3\,x^{21/2}}{21}+\frac {2\,B\,c^3\,x^{23/2}}{23}+\frac {6\,A\,a^2\,c\,x^{13/2}}{13}+\frac {6\,A\,a\,c^2\,x^{17/2}}{17}+\frac {2\,B\,a^2\,c\,x^{15/2}}{5}+\frac {6\,B\,a\,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 = 27.94, size = 114, normalized size = 1.05 \begin {gather*} \frac {2 A a^{3} x^{\frac {9}{2}}}{9} + \frac {6 A a^{2} c x^{\frac {13}{2}}}{13} + \frac {6 A a c^{2} x^{\frac {17}{2}}}{17} + \frac {2 A c^{3} x^{\frac {21}{2}}}{21} + \frac {2 B a^{3} x^{\frac {11}{2}}}{11} + \frac {2 B a^{2} c x^{\frac {15}{2}}}{5} + \frac {6 B a c^{2} x^{\frac {19}{2}}}{19} + \frac {2 B c^{3} x^{\frac {23}{2}}}{23} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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