Optimal. Leaf size=63 \[ \frac {2}{11} A b^2 x^{11/2}+\frac {2}{15} c x^{15/2} (A c+2 b B)+\frac {2}{13} b x^{13/2} (2 A c+b B)+\frac {2}{17} B c^2 x^{17/2} \]
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Rubi [A] time = 0.03, antiderivative size = 63, 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} A b^2 x^{11/2}+\frac {2}{15} c x^{15/2} (A c+2 b B)+\frac {2}{13} b x^{13/2} (2 A c+b B)+\frac {2}{17} B c^2 x^{17/2} \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 )^2 \, dx &=\int \left (A b^2 x^{9/2}+b (b B+2 A c) x^{11/2}+c (2 b B+A c) x^{13/2}+B c^2 x^{15/2}\right ) \, dx\\ &=\frac {2}{11} A b^2 x^{11/2}+\frac {2}{13} b (b B+2 A c) x^{13/2}+\frac {2}{15} c (2 b B+A c) x^{15/2}+\frac {2}{17} B c^2 x^{17/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 0.87 \begin {gather*} \frac {2 x^{11/2} \left (17 A \left (195 b^2+330 b c x+143 c^2 x^2\right )+11 B x \left (255 b^2+442 b c x+195 c^2 x^2\right )\right )}{36465} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 69, normalized size = 1.10 \begin {gather*} \frac {2 \left (3315 A b^2 x^{11/2}+5610 A b c x^{13/2}+2431 A c^2 x^{15/2}+2805 b^2 B x^{13/2}+4862 b B c x^{15/2}+2145 B c^2 x^{17/2}\right )}{36465} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 56, normalized size = 0.89 \begin {gather*} \frac {2}{36465} \, {\left (2145 \, B c^{2} x^{8} + 3315 \, A b^{2} x^{5} + 2431 \, {\left (2 \, B b c + A c^{2}\right )} x^{7} + 2805 \, {\left (B b^{2} + 2 \, A b 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 = 53, normalized size = 0.84 \begin {gather*} \frac {2}{17} \, B c^{2} x^{\frac {17}{2}} + \frac {4}{15} \, B b c x^{\frac {15}{2}} + \frac {2}{15} \, A c^{2} x^{\frac {15}{2}} + \frac {2}{13} \, B b^{2} x^{\frac {13}{2}} + \frac {4}{13} \, A b c x^{\frac {13}{2}} + \frac {2}{11} \, A b^{2} x^{\frac {11}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 52, normalized size = 0.83 \begin {gather*} \frac {2 \left (2145 B \,c^{2} x^{3}+2431 A \,c^{2} x^{2}+4862 B b c \,x^{2}+5610 A b c x +2805 B \,b^{2} x +3315 A \,b^{2}\right ) x^{\frac {11}{2}}}{36465} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{17} \, B c^{2} x^{\frac {17}{2}} + \frac {2}{11} \, A b^{2} x^{\frac {11}{2}} + \frac {2}{15} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {15}{2}} + \frac {2}{13} \, {\left (B b^{2} + 2 \, A b 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 = 0.05, size = 51, normalized size = 0.81 \begin {gather*} x^{13/2}\,\left (\frac {2\,B\,b^2}{13}+\frac {4\,A\,c\,b}{13}\right )+x^{15/2}\,\left (\frac {2\,A\,c^2}{15}+\frac {4\,B\,b\,c}{15}\right )+\frac {2\,A\,b^2\,x^{11/2}}{11}+\frac {2\,B\,c^2\,x^{17/2}}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.36, size = 80, normalized size = 1.27 \begin {gather*} \frac {2 A b^{2} x^{\frac {11}{2}}}{11} + \frac {4 A b c x^{\frac {13}{2}}}{13} + \frac {2 A c^{2} x^{\frac {15}{2}}}{15} + \frac {2 B b^{2} x^{\frac {13}{2}}}{13} + \frac {4 B b c x^{\frac {15}{2}}}{15} + \frac {2 B c^{2} x^{\frac {17}{2}}}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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