Optimal. Leaf size=111 \[ \frac{4}{9} a^2 b x^{9/2} (2 a B+3 A b)+\frac{2}{7} a^3 x^{7/2} (a B+4 A b)+\frac{2}{5} a^4 A x^{5/2}+\frac{2}{13} b^3 x^{13/2} (4 a B+A b)+\frac{4}{11} a b^2 x^{11/2} (3 a B+2 A b)+\frac{2}{15} b^4 B x^{15/2} \]
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Rubi [A] time = 0.0522332, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 76} \[ \frac{4}{9} a^2 b x^{9/2} (2 a B+3 A b)+\frac{2}{7} a^3 x^{7/2} (a B+4 A b)+\frac{2}{5} a^4 A x^{5/2}+\frac{2}{13} b^3 x^{13/2} (4 a B+A b)+\frac{4}{11} a b^2 x^{11/2} (3 a B+2 A b)+\frac{2}{15} b^4 B x^{15/2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int x^{3/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int x^{3/2} (a+b x)^4 (A+B x) \, dx\\ &=\int \left (a^4 A x^{3/2}+a^3 (4 A b+a B) x^{5/2}+2 a^2 b (3 A b+2 a B) x^{7/2}+2 a b^2 (2 A b+3 a B) x^{9/2}+b^3 (A b+4 a B) x^{11/2}+b^4 B x^{13/2}\right ) \, dx\\ &=\frac{2}{5} a^4 A x^{5/2}+\frac{2}{7} a^3 (4 A b+a B) x^{7/2}+\frac{4}{9} a^2 b (3 A b+2 a B) x^{9/2}+\frac{4}{11} a b^2 (2 A b+3 a B) x^{11/2}+\frac{2}{13} b^3 (A b+4 a B) x^{13/2}+\frac{2}{15} b^4 B x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.0598668, size = 81, normalized size = 0.73 \[ \frac{2 \left (\frac{x^{5/2} \left (10010 a^2 b^2 x^2+8580 a^3 b x+3003 a^4+5460 a b^3 x^3+1155 b^4 x^4\right ) (3 A b-a B)}{3003}+B x^{5/2} (a+b x)^5\right )}{15 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 100, normalized size = 0.9 \begin{align*}{\frac{6006\,{b}^{4}B{x}^{5}+6930\,A{b}^{4}{x}^{4}+27720\,B{x}^{4}a{b}^{3}+32760\,aA{b}^{3}{x}^{3}+49140\,B{x}^{3}{a}^{2}{b}^{2}+60060\,{a}^{2}A{b}^{2}{x}^{2}+40040\,B{x}^{2}{a}^{3}b+51480\,{a}^{3}Abx+12870\,{a}^{4}Bx+18018\,A{a}^{4}}{45045}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02669, size = 134, normalized size = 1.21 \begin{align*} \frac{2}{15} \, B b^{4} x^{\frac{15}{2}} + \frac{2}{5} \, A a^{4} x^{\frac{5}{2}} + \frac{2}{13} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac{13}{2}} + \frac{4}{11} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac{11}{2}} + \frac{4}{9} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56019, size = 254, normalized size = 2.29 \begin{align*} \frac{2}{45045} \,{\left (3003 \, B b^{4} x^{7} + 9009 \, A a^{4} x^{2} + 3465 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{6} + 8190 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{5} + 10010 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{4} + 6435 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.4561, size = 148, normalized size = 1.33 \begin{align*} \frac{2 A a^{4} x^{\frac{5}{2}}}{5} + \frac{8 A a^{3} b x^{\frac{7}{2}}}{7} + \frac{4 A a^{2} b^{2} x^{\frac{9}{2}}}{3} + \frac{8 A a b^{3} x^{\frac{11}{2}}}{11} + \frac{2 A b^{4} x^{\frac{13}{2}}}{13} + \frac{2 B a^{4} x^{\frac{7}{2}}}{7} + \frac{8 B a^{3} b x^{\frac{9}{2}}}{9} + \frac{12 B a^{2} b^{2} x^{\frac{11}{2}}}{11} + \frac{8 B a b^{3} x^{\frac{13}{2}}}{13} + \frac{2 B b^{4} x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15417, size = 136, normalized size = 1.23 \begin{align*} \frac{2}{15} \, B b^{4} x^{\frac{15}{2}} + \frac{8}{13} \, B a b^{3} x^{\frac{13}{2}} + \frac{2}{13} \, A b^{4} x^{\frac{13}{2}} + \frac{12}{11} \, B a^{2} b^{2} x^{\frac{11}{2}} + \frac{8}{11} \, A a b^{3} x^{\frac{11}{2}} + \frac{8}{9} \, B a^{3} b x^{\frac{9}{2}} + \frac{4}{3} \, A a^{2} b^{2} x^{\frac{9}{2}} + \frac{2}{7} \, B a^{4} x^{\frac{7}{2}} + \frac{8}{7} \, A a^{3} b x^{\frac{7}{2}} + \frac{2}{5} \, A a^{4} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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