Optimal. Leaf size=85 \[ \frac{2}{15} b^2 x^{15/2} (3 A c+b B)+\frac{2}{13} A b^3 x^{13/2}+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{6}{17} b c x^{17/2} (A c+b B)+\frac{2}{21} B c^3 x^{21/2} \]
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Rubi [A] time = 0.0453359, antiderivative size = 85, normalized size of antiderivative = 1., 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} \[ \frac{2}{15} b^2 x^{15/2} (3 A c+b B)+\frac{2}{13} A b^3 x^{13/2}+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{6}{17} b c x^{17/2} (A c+b B)+\frac{2}{21} B c^3 x^{21/2} \]
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*}
Mathematica [A] time = 0.0469798, size = 70, normalized size = 0.82 \[ \frac{2 \left (B x^{13/2} (b+c x)^4-\frac{x^{13/2} \left (4199 b^2 c x+1615 b^3+3705 b c^2 x^2+1105 c^3 x^3\right ) (13 b B-21 A c)}{20995}\right )}{21 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 76, normalized size = 0.9 \begin{align*}{\frac{41990\,B{c}^{3}{x}^{4}+46410\,A{x}^{3}{c}^{3}+139230\,B{x}^{3}b{c}^{2}+155610\,A{x}^{2}b{c}^{2}+155610\,B{x}^{2}{b}^{2}c+176358\,A{b}^{2}cx+58786\,{b}^{3}Bx+67830\,A{b}^{3}}{440895}{x}^{{\frac{13}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07532, size = 99, normalized size = 1.16 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8355, size = 204, normalized size = 2.4 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.7127, size = 114, normalized size = 1.34 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15122, size = 104, normalized size = 1.22 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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