Optimal. Leaf size=85 \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{7} a^3 A x^{7/2}+\frac{2}{13} b^2 x^{13/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{15} b^3 B x^{15/2} \]
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Rubi [A] time = 0.0376684, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {76} \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{7} a^3 A x^{7/2}+\frac{2}{13} b^2 x^{13/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{15} b^3 B x^{15/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int x^{5/2} (a+b x)^3 (A+B x) \, dx &=\int \left (a^3 A x^{5/2}+a^2 (3 A b+a B) x^{7/2}+3 a b (A b+a B) x^{9/2}+b^2 (A b+3 a B) x^{11/2}+b^3 B x^{13/2}\right ) \, dx\\ &=\frac{2}{7} a^3 A x^{7/2}+\frac{2}{9} a^2 (3 A b+a B) x^{9/2}+\frac{6}{11} a b (A b+a B) x^{11/2}+\frac{2}{13} b^2 (A b+3 a B) x^{13/2}+\frac{2}{15} b^3 B x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.024993, size = 71, normalized size = 0.84 \[ \frac{2 x^{7/2} \left (1365 a^2 b x (11 A+9 B x)+715 a^3 (9 A+7 B x)+945 a b^2 x^2 (13 A+11 B x)+231 b^3 x^3 (15 A+13 B x)\right )}{45045} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 76, normalized size = 0.9 \begin{align*}{\frac{6006\,B{b}^{3}{x}^{4}+6930\,A{b}^{3}{x}^{3}+20790\,B{x}^{3}a{b}^{2}+24570\,aA{b}^{2}{x}^{2}+24570\,B{x}^{2}{a}^{2}b+30030\,{a}^{2}Abx+10010\,{a}^{3}Bx+12870\,{a}^{3}A}{45045}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07894, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{15} \, B b^{3} x^{\frac{15}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} + \frac{2}{13} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{13}{2}} + \frac{6}{11} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35202, size = 196, normalized size = 2.31 \begin{align*} \frac{2}{45045} \,{\left (3003 \, B b^{3} x^{7} + 6435 \, A a^{3} x^{3} + 3465 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 12285 \,{\left (B a^{2} b + A a b^{2}\right )} x^{5} + 5005 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.05739, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{2 A a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 A a b^{2} x^{\frac{11}{2}}}{11} + \frac{2 A b^{3} x^{\frac{13}{2}}}{13} + \frac{2 B a^{3} x^{\frac{9}{2}}}{9} + \frac{6 B a^{2} b x^{\frac{11}{2}}}{11} + \frac{6 B a b^{2} x^{\frac{13}{2}}}{13} + \frac{2 B b^{3} x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24083, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{15} \, B b^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B a b^{2} x^{\frac{13}{2}} + \frac{2}{13} \, A b^{3} x^{\frac{13}{2}} + \frac{6}{11} \, B a^{2} b x^{\frac{11}{2}} + \frac{6}{11} \, A a b^{2} x^{\frac{11}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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