Optimal. Leaf size=63 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2} \]
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Rubi [A] time = 0.0282857, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int x^{7/2} (a+b x)^2 (A+B x) \, dx\\ &=\int \left (a^2 A x^{7/2}+a (2 A b+a B) x^{9/2}+b (A b+2 a B) x^{11/2}+b^2 B x^{13/2}\right ) \, dx\\ &=\frac{2}{9} a^2 A x^{9/2}+\frac{2}{11} a (2 A b+a B) x^{11/2}+\frac{2}{13} b (A b+2 a B) x^{13/2}+\frac{2}{15} b^2 B x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.0185611, size = 52, normalized size = 0.83 \[ \frac{2 x^{9/2} \left (65 a^2 (11 A+9 B x)+90 a b x (13 A+11 B x)+33 b^2 x^2 (15 A+13 B x)\right )}{6435} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 52, normalized size = 0.8 \begin{align*}{\frac{858\,{b}^{2}B{x}^{3}+990\,A{b}^{2}{x}^{2}+1980\,B{x}^{2}ab+2340\,aAbx+1170\,{a}^{2}Bx+1430\,A{a}^{2}}{6435}{x}^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06638, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B a^{2} + 2 \, 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 = 1.47502, size = 140, normalized size = 2.22 \begin{align*} \frac{2}{6435} \,{\left (429 \, B b^{2} x^{7} + 715 \, A a^{2} x^{4} + 495 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + 585 \,{\left (B a^{2} + 2 \, 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 = 9.58025, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a b x^{\frac{11}{2}}}{11} + \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{2 B a^{2} x^{\frac{11}{2}}}{11} + \frac{4 B a b x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14989, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{4}{13} \, B a b x^{\frac{13}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A a b x^{\frac{11}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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