Optimal. Leaf size=85 \[ \frac {2}{13} A b^3 x^{13/2}+\frac {2}{15} b^2 x^{15/2} (3 A c+b B)+\frac {2}{19} c^2 x^{19/2} (A c+3 b B)+\frac {6}{17} b c x^{17/2} (A c+b B)+\frac {2}{21} B c^3 x^{21/2} \]
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Rubi [A] time = 0.05, 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 = {765} \begin {gather*} \frac {2}{15} b^2 x^{15/2} (3 A c+b B)+\frac {2}{13} A b^3 x^{13/2}+\frac {2}{19} c^2 x^{19/2} (A c+3 b B)+\frac {6}{17} b c x^{17/2} (A c+b B)+\frac {2}{21} B c^3 x^{21/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int x^{5/2} (A+B x) \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^{11/2}+b^2 (b B+3 A c) x^{13/2}+3 b c (b B+A c) x^{15/2}+c^2 (3 b B+A c) x^{17/2}+B c^3 x^{19/2}\right ) \, dx\\ &=\frac {2}{13} A b^3 x^{13/2}+\frac {2}{15} b^2 (b B+3 A c) x^{15/2}+\frac {6}{17} b c (b B+A c) x^{17/2}+\frac {2}{19} c^2 (3 b B+A c) x^{19/2}+\frac {2}{21} B c^3 x^{21/2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 0.82 \begin {gather*} \frac {2 \left (B x^{13/2} (b+c x)^4-\frac {x^{13/2} \left (1615 b^3+4199 b^2 c x+3705 b c^2 x^2+1105 c^3 x^3\right ) (13 b B-21 A c)}{20995}\right )}{21 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (33915 A b^3 x^{13/2}+88179 A b^2 c x^{15/2}+77805 A b c^2 x^{17/2}+23205 A c^3 x^{19/2}+29393 b^3 B x^{15/2}+77805 b^2 B c x^{17/2}+69615 b B c^2 x^{19/2}+20995 B c^3 x^{21/2}\right )}{440895} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{440895} \, {\left (20995 \, B c^{3} x^{10} + 33915 \, A b^{3} x^{6} + 23205 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + 77805 \, {\left (B b^{2} c + A b c^{2}\right )} x^{8} + 29393 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{7}\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 = 0.91 \begin {gather*} \frac {2}{21} \, B c^{3} x^{\frac {21}{2}} + \frac {6}{19} \, B b c^{2} x^{\frac {19}{2}} + \frac {2}{19} \, A c^{3} x^{\frac {19}{2}} + \frac {6}{17} \, B b^{2} c x^{\frac {17}{2}} + \frac {6}{17} \, A b c^{2} x^{\frac {17}{2}} + \frac {2}{15} \, B b^{3} x^{\frac {15}{2}} + \frac {2}{5} \, A b^{2} c x^{\frac {15}{2}} + \frac {2}{13} \, A b^{3} x^{\frac {13}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 76, normalized size = 0.89 \begin {gather*} \frac {2 \left (20995 B \,c^{3} x^{4}+23205 A \,c^{3} x^{3}+69615 B b \,c^{2} x^{3}+77805 A b \,c^{2} x^{2}+77805 B \,b^{2} c \,x^{2}+88179 A \,b^{2} c x +29393 B \,b^{3} x +33915 A \,b^{3}\right ) x^{\frac {13}{2}}}{440895} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{21} \, B c^{3} x^{\frac {21}{2}} + \frac {2}{13} \, A b^{3} x^{\frac {13}{2}} + \frac {2}{19} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {19}{2}} + \frac {6}{17} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {17}{2}} + \frac {2}{15} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {15}{2}} \end {gather*}
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 \begin {gather*} x^{15/2}\,\left (\frac {2\,B\,b^3}{15}+\frac {2\,A\,c\,b^2}{5}\right )+x^{19/2}\,\left (\frac {2\,A\,c^3}{19}+\frac {6\,B\,b\,c^2}{19}\right )+\frac {2\,A\,b^3\,x^{13/2}}{13}+\frac {2\,B\,c^3\,x^{21/2}}{21}+\frac {6\,b\,c\,x^{17/2}\,\left (A\,c+B\,b\right )}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.92, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {13}{2}}}{13} + \frac {2 A b^{2} c x^{\frac {15}{2}}}{5} + \frac {6 A b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 A c^{3} x^{\frac {19}{2}}}{19} + \frac {2 B b^{3} x^{\frac {15}{2}}}{15} + \frac {6 B b^{2} c x^{\frac {17}{2}}}{17} + \frac {6 B b c^{2} x^{\frac {19}{2}}}{19} + \frac {2 B c^{3} x^{\frac {21}{2}}}{21} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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