Optimal. Leaf size=85 \[ \frac {2}{11} A b^3 x^{11/2}+\frac {2}{13} b^2 x^{13/2} (3 A c+b B)+\frac {2}{17} c^2 x^{17/2} (A c+3 b B)+\frac {2}{5} b c x^{15/2} (A c+b B)+\frac {2}{19} B c^3 x^{19/2} \]
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Rubi [A] time = 0.04, 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}{13} b^2 x^{13/2} (3 A c+b B)+\frac {2}{11} A b^3 x^{11/2}+\frac {2}{17} c^2 x^{17/2} (A c+3 b B)+\frac {2}{5} b c x^{15/2} (A c+b B)+\frac {2}{19} B c^3 x^{19/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int x^{3/2} (A+B x) \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^{9/2}+b^2 (b B+3 A c) x^{11/2}+3 b c (b B+A c) x^{13/2}+c^2 (3 b B+A c) x^{15/2}+B c^3 x^{17/2}\right ) \, dx\\ &=\frac {2}{11} A b^3 x^{11/2}+\frac {2}{13} b^2 (b B+3 A c) x^{13/2}+\frac {2}{5} b c (b B+A c) x^{15/2}+\frac {2}{17} c^2 (3 b B+A c) x^{17/2}+\frac {2}{19} B c^3 x^{19/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 70, normalized size = 0.82 \begin {gather*} \frac {2 \left (B x^{11/2} (b+c x)^4-\frac {x^{11/2} \left (1105 b^3+2805 b^2 c x+2431 b c^2 x^2+715 c^3 x^3\right ) (11 b B-19 A c)}{12155}\right )}{19 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 (20995 A b^3 x^{11/2}+53295 A b^2 c x^{13/2}+46189 A b c^2 x^{15/2}+13585 A c^3 x^{17/2}+17765 b^3 B x^{13/2}+46189 b^2 B c x^{15/2}+40755 b B c^2 x^{17/2}+12155 B c^3 x^{19/2}\right )}{230945} \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}{230945} \, {\left (12155 \, B c^{3} x^{9} + 20995 \, A b^{3} x^{5} + 13585 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{8} + 46189 \, {\left (B b^{2} c + A b c^{2}\right )} x^{7} + 17765 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6}\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 = 77, normalized size = 0.91 \begin {gather*} \frac {2}{19} \, B c^{3} x^{\frac {19}{2}} + \frac {6}{17} \, B b c^{2} x^{\frac {17}{2}} + \frac {2}{17} \, A c^{3} x^{\frac {17}{2}} + \frac {2}{5} \, B b^{2} c x^{\frac {15}{2}} + \frac {2}{5} \, A b c^{2} x^{\frac {15}{2}} + \frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {6}{13} \, A b^{2} c x^{\frac {13}{2}} + \frac {2}{11} \, A b^{3} x^{\frac {11}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 76, normalized size = 0.89 \begin {gather*} \frac {2 \left (12155 B \,c^{3} x^{4}+13585 A \,c^{3} x^{3}+40755 B b \,c^{2} x^{3}+46189 A b \,c^{2} x^{2}+46189 B \,b^{2} c \,x^{2}+53295 A \,b^{2} c x +17765 B \,b^{3} x +20995 A \,b^{3}\right ) x^{\frac {11}{2}}}{230945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{19} \, B c^{3} x^{\frac {19}{2}} + \frac {2}{11} \, A b^{3} x^{\frac {11}{2}} + \frac {2}{17} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {17}{2}} + \frac {2}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {15}{2}} + \frac {2}{13} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {13}{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^{13/2}\,\left (\frac {2\,B\,b^3}{13}+\frac {6\,A\,c\,b^2}{13}\right )+x^{17/2}\,\left (\frac {2\,A\,c^3}{17}+\frac {6\,B\,b\,c^2}{17}\right )+\frac {2\,A\,b^3\,x^{11/2}}{11}+\frac {2\,B\,c^3\,x^{19/2}}{19}+\frac {2\,b\,c\,x^{15/2}\,\left (A\,c+B\,b\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.27, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {11}{2}}}{11} + \frac {6 A b^{2} c x^{\frac {13}{2}}}{13} + \frac {2 A b c^{2} x^{\frac {15}{2}}}{5} + \frac {2 A c^{3} x^{\frac {17}{2}}}{17} + \frac {2 B b^{3} x^{\frac {13}{2}}}{13} + \frac {2 B b^{2} c x^{\frac {15}{2}}}{5} + \frac {6 B b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 B c^{3} x^{\frac {19}{2}}}{19} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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