Optimal. Leaf size=85 \[ \frac {2}{9} A b^3 x^{9/2}+\frac {2}{11} b^2 x^{11/2} (3 A c+b B)+\frac {2}{15} c^2 x^{15/2} (A c+3 b B)+\frac {6}{13} b c x^{13/2} (A c+b B)+\frac {2}{17} B c^3 x^{17/2} \]
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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}{11} b^2 x^{11/2} (3 A c+b B)+\frac {2}{9} A b^3 x^{9/2}+\frac {2}{15} c^2 x^{15/2} (A c+3 b B)+\frac {6}{13} b c x^{13/2} (A c+b B)+\frac {2}{17} B c^3 x^{17/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \sqrt {x} (A+B x) \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^{7/2}+b^2 (b B+3 A c) x^{9/2}+3 b c (b B+A c) x^{11/2}+c^2 (3 b B+A c) x^{13/2}+B c^3 x^{15/2}\right ) \, dx\\ &=\frac {2}{9} A b^3 x^{9/2}+\frac {2}{11} b^2 (b B+3 A c) x^{11/2}+\frac {6}{13} b c (b B+A c) x^{13/2}+\frac {2}{15} c^2 (3 b B+A c) x^{15/2}+\frac {2}{17} B c^3 x^{17/2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 0.82 \begin {gather*} \frac {2 \left (B x^{9/2} (b+c x)^4-\frac {x^{9/2} \left (715 b^3+1755 b^2 c x+1485 b c^2 x^2+429 c^3 x^3\right ) (9 b B-17 A c)}{6435}\right )}{17 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 (12155 A b^3 x^{9/2}+29835 A b^2 c x^{11/2}+25245 A b c^2 x^{13/2}+7293 A c^3 x^{15/2}+9945 b^3 B x^{11/2}+25245 b^2 B c x^{13/2}+21879 b B c^2 x^{15/2}+6435 B c^3 x^{17/2}\right )}{109395} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{109395} \, {\left (6435 \, B c^{3} x^{8} + 12155 \, A b^{3} x^{4} + 7293 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + 25245 \, {\left (B b^{2} c + A b c^{2}\right )} x^{6} + 9945 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{5}\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}{17} \, B c^{3} x^{\frac {17}{2}} + \frac {2}{5} \, B b c^{2} x^{\frac {15}{2}} + \frac {2}{15} \, A c^{3} x^{\frac {15}{2}} + \frac {6}{13} \, B b^{2} c x^{\frac {13}{2}} + \frac {6}{13} \, A b c^{2} x^{\frac {13}{2}} + \frac {2}{11} \, B b^{3} x^{\frac {11}{2}} + \frac {6}{11} \, A b^{2} c x^{\frac {11}{2}} + \frac {2}{9} \, A b^{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 = 76, normalized size = 0.89 \begin {gather*} \frac {2 \left (6435 B \,c^{3} x^{4}+7293 A \,c^{3} x^{3}+21879 B b \,c^{2} x^{3}+25245 A b \,c^{2} x^{2}+25245 B \,b^{2} c \,x^{2}+29835 A \,b^{2} c x +9945 B \,b^{3} x +12155 A \,b^{3}\right ) x^{\frac {9}{2}}}{109395} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{17} \, B c^{3} x^{\frac {17}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} + \frac {2}{15} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {15}{2}} + \frac {6}{13} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {11}{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^{11/2}\,\left (\frac {2\,B\,b^3}{11}+\frac {6\,A\,c\,b^2}{11}\right )+x^{15/2}\,\left (\frac {2\,A\,c^3}{15}+\frac {2\,B\,b\,c^2}{5}\right )+\frac {2\,A\,b^3\,x^{9/2}}{9}+\frac {2\,B\,c^3\,x^{17/2}}{17}+\frac {6\,b\,c\,x^{13/2}\,\left (A\,c+B\,b\right )}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.76, size = 95, normalized size = 1.12 \begin {gather*} \frac {2 A b^{3} x^{\frac {9}{2}}}{9} + \frac {2 B c^{3} x^{\frac {17}{2}}}{17} + \frac {2 x^{\frac {15}{2}} \left (A c^{3} + 3 B b c^{2}\right )}{15} + \frac {2 x^{\frac {13}{2}} \left (3 A b c^{2} + 3 B b^{2} c\right )}{13} + \frac {2 x^{\frac {11}{2}} \left (3 A b^{2} c + B b^{3}\right )}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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