Optimal. Leaf size=63 \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{15} b x^{15/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]
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Rubi [A] time = 0.0333596, antiderivative size = 63, 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} \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{15} b x^{15/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int x^{7/2} (A+B x) \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 x^{11/2}+b (b B+2 A c) x^{13/2}+c (2 b B+A c) x^{15/2}+B c^2 x^{17/2}\right ) \, dx\\ &=\frac{2}{13} A b^2 x^{13/2}+\frac{2}{15} b (b B+2 A c) x^{15/2}+\frac{2}{17} c (2 b B+A c) x^{17/2}+\frac{2}{19} B c^2 x^{19/2}\\ \end{align*}
Mathematica [A] time = 0.0192609, size = 55, normalized size = 0.87 \[ \frac{2 x^{13/2} \left (19 A \left (255 b^2+442 b c x+195 c^2 x^2\right )+13 B x \left (323 b^2+570 b c x+255 c^2 x^2\right )\right )}{62985} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 52, normalized size = 0.8 \begin{align*}{\frac{6630\,B{c}^{2}{x}^{3}+7410\,A{c}^{2}{x}^{2}+14820\,B{x}^{2}bc+16796\,Abcx+8398\,{b}^{2}Bx+9690\,A{b}^{2}}{62985}{x}^{{\frac{13}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12262, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} + \frac{2}{17} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{17}{2}} + \frac{2}{15} \,{\left (B b^{2} + 2 \, A b 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 = 1.71771, size = 147, normalized size = 2.33 \begin{align*} \frac{2}{62985} \,{\left (3315 \, B c^{2} x^{9} + 4845 \, A b^{2} x^{6} + 3705 \,{\left (2 \, B b c + A c^{2}\right )} x^{8} + 4199 \,{\left (B b^{2} + 2 \, A b 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 = 17.42, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{4 A b c x^{\frac{15}{2}}}{15} + \frac{2 A c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B b^{2} x^{\frac{15}{2}}}{15} + \frac{4 B b c x^{\frac{17}{2}}}{17} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11315, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{4}{17} \, B b c x^{\frac{17}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{4}{15} \, A b c x^{\frac{15}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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