Optimal. Leaf size=55 \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{9} a A x^{9/2}+\frac{2}{13} x^{13/2} (A c+b B)+\frac{2}{15} B c x^{15/2} \]
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Rubi [A] time = 0.0260274, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{9} a A x^{9/2}+\frac{2}{13} x^{13/2} (A c+b B)+\frac{2}{15} B c x^{15/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int x^{7/2} (A+B x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a A x^{7/2}+(A b+a B) x^{9/2}+(b B+A c) x^{11/2}+B c x^{13/2}\right ) \, dx\\ &=\frac{2}{9} a A x^{9/2}+\frac{2}{11} (A b+a B) x^{11/2}+\frac{2}{13} (b B+A c) x^{13/2}+\frac{2}{15} B c x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.0511677, size = 47, normalized size = 0.85 \[ \frac{2 x^{9/2} \left (65 a (11 A+9 B x)+45 A x (13 b+11 c x)+33 B x^2 (15 b+13 c x)\right )}{6435} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 42, normalized size = 0.8 \begin{align*}{\frac{858\,Bc{x}^{3}+990\,Ac{x}^{2}+990\,Bb{x}^{2}+1170\,Abx+1170\,aBx+1430\,aA}{6435}{x}^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05717, size = 53, normalized size = 0.96 \begin{align*} \frac{2}{15} \, B c x^{\frac{15}{2}} + \frac{2}{13} \,{\left (B b + A c\right )} x^{\frac{13}{2}} + \frac{2}{9} \, A a x^{\frac{9}{2}} + \frac{2}{11} \,{\left (B a + A b\right )} x^{\frac{11}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29912, size = 119, normalized size = 2.16 \begin{align*} \frac{2}{6435} \,{\left (429 \, B c x^{7} + 495 \,{\left (B b + A c\right )} x^{6} + 715 \, A a x^{4} + 585 \,{\left (B a + A b\right )} x^{5}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.6265, size = 70, normalized size = 1.27 \begin{align*} \frac{2 A a x^{\frac{9}{2}}}{9} + \frac{2 A b x^{\frac{11}{2}}}{11} + \frac{2 A c x^{\frac{13}{2}}}{13} + \frac{2 B a x^{\frac{11}{2}}}{11} + \frac{2 B b x^{\frac{13}{2}}}{13} + \frac{2 B c x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18061, size = 58, normalized size = 1.05 \begin{align*} \frac{2}{15} \, B c x^{\frac{15}{2}} + \frac{2}{13} \, B b x^{\frac{13}{2}} + \frac{2}{13} \, A c x^{\frac{13}{2}} + \frac{2}{11} \, B a x^{\frac{11}{2}} + \frac{2}{11} \, A b x^{\frac{11}{2}} + \frac{2}{9} \, A a x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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