Optimal. Leaf size=159 \[ \frac{2}{3} a^2 b^3 x^{15/2} (4 a B+3 A b)+\frac{10}{13} a^3 b^2 x^{13/2} (3 a B+4 A b)+\frac{6}{11} a^4 b x^{11/2} (2 a B+5 A b)+\frac{2}{9} a^5 x^{9/2} (a B+6 A b)+\frac{2}{7} a^6 A x^{7/2}+\frac{2}{19} b^5 x^{19/2} (6 a B+A b)+\frac{6}{17} a b^4 x^{17/2} (5 a B+2 A b)+\frac{2}{21} b^6 B x^{21/2} \]
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Rubi [A] time = 0.0795575, antiderivative size = 159, 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} \[ \frac{2}{3} a^2 b^3 x^{15/2} (4 a B+3 A b)+\frac{10}{13} a^3 b^2 x^{13/2} (3 a B+4 A b)+\frac{6}{11} a^4 b x^{11/2} (2 a B+5 A b)+\frac{2}{9} a^5 x^{9/2} (a B+6 A b)+\frac{2}{7} a^6 A x^{7/2}+\frac{2}{19} b^5 x^{19/2} (6 a B+A b)+\frac{6}{17} a b^4 x^{17/2} (5 a B+2 A b)+\frac{2}{21} b^6 B x^{21/2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int x^{5/2} (a+b x)^6 (A+B x) \, dx\\ &=\int \left (a^6 A x^{5/2}+a^5 (6 A b+a B) x^{7/2}+3 a^4 b (5 A b+2 a B) x^{9/2}+5 a^3 b^2 (4 A b+3 a B) x^{11/2}+5 a^2 b^3 (3 A b+4 a B) x^{13/2}+3 a b^4 (2 A b+5 a B) x^{15/2}+b^5 (A b+6 a B) x^{17/2}+b^6 B x^{19/2}\right ) \, dx\\ &=\frac{2}{7} a^6 A x^{7/2}+\frac{2}{9} a^5 (6 A b+a B) x^{9/2}+\frac{6}{11} a^4 b (5 A b+2 a B) x^{11/2}+\frac{10}{13} a^3 b^2 (4 A b+3 a B) x^{13/2}+\frac{2}{3} a^2 b^3 (3 A b+4 a B) x^{15/2}+\frac{6}{17} a b^4 (2 A b+5 a B) x^{17/2}+\frac{2}{19} b^5 (A b+6 a B) x^{19/2}+\frac{2}{21} b^6 B x^{21/2}\\ \end{align*}
Mathematica [A] time = 0.0880796, size = 128, normalized size = 0.81 \[ \frac{2 \left (B x^{7/2} (a+b x)^7-\frac{7}{2} \left (2 a^2 b^4 x^{15/2}+\frac{40}{13} a^3 b^3 x^{13/2}+\frac{30}{11} a^4 b^2 x^{11/2}+\frac{4}{3} a^5 b x^{9/2}+\frac{2}{7} a^6 x^{7/2}+\frac{12}{17} a b^5 x^{17/2}+\frac{2}{19} b^6 x^{19/2}\right ) (a B-3 A b)\right )}{21 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 148, normalized size = 0.9 \begin{align*}{\frac{277134\,B{b}^{6}{x}^{7}+306306\,A{b}^{6}{x}^{6}+1837836\,B{x}^{6}a{b}^{5}+2054052\,aA{b}^{5}{x}^{5}+5135130\,B{x}^{5}{a}^{2}{b}^{4}+5819814\,{a}^{2}A{b}^{4}{x}^{4}+7759752\,B{x}^{4}{a}^{3}{b}^{3}+8953560\,{a}^{3}A{b}^{3}{x}^{3}+6715170\,B{x}^{3}{a}^{4}{b}^{2}+7936110\,{a}^{4}A{b}^{2}{x}^{2}+3174444\,B{x}^{2}{a}^{5}b+3879876\,{a}^{5}Abx+646646\,B{a}^{6}x+831402\,A{a}^{6}}{2909907}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0118, size = 198, normalized size = 1.25 \begin{align*} \frac{2}{21} \, B b^{6} x^{\frac{21}{2}} + \frac{2}{7} \, A a^{6} x^{\frac{7}{2}} + \frac{2}{19} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{19}{2}} + \frac{6}{17} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac{17}{2}} + \frac{2}{3} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac{15}{2}} + \frac{10}{13} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac{13}{2}} + \frac{6}{11} \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B a^{6} + 6 \, A a^{5} 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 = 1.56087, size = 385, normalized size = 2.42 \begin{align*} \frac{2}{2909907} \,{\left (138567 \, B b^{6} x^{10} + 415701 \, A a^{6} x^{3} + 153153 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{9} + 513513 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{8} + 969969 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{7} + 1119195 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{6} + 793611 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{5} + 323323 \,{\left (B a^{6} + 6 \, A a^{5} 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 = 18.4307, size = 214, normalized size = 1.35 \begin{align*} \frac{2 A a^{6} x^{\frac{7}{2}}}{7} + \frac{4 A a^{5} b x^{\frac{9}{2}}}{3} + \frac{30 A a^{4} b^{2} x^{\frac{11}{2}}}{11} + \frac{40 A a^{3} b^{3} x^{\frac{13}{2}}}{13} + 2 A a^{2} b^{4} x^{\frac{15}{2}} + \frac{12 A a b^{5} x^{\frac{17}{2}}}{17} + \frac{2 A b^{6} x^{\frac{19}{2}}}{19} + \frac{2 B a^{6} x^{\frac{9}{2}}}{9} + \frac{12 B a^{5} b x^{\frac{11}{2}}}{11} + \frac{30 B a^{4} b^{2} x^{\frac{13}{2}}}{13} + \frac{8 B a^{3} b^{3} x^{\frac{15}{2}}}{3} + \frac{30 B a^{2} b^{4} x^{\frac{17}{2}}}{17} + \frac{12 B a b^{5} x^{\frac{19}{2}}}{19} + \frac{2 B b^{6} x^{\frac{21}{2}}}{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13989, size = 201, normalized size = 1.26 \begin{align*} \frac{2}{21} \, B b^{6} x^{\frac{21}{2}} + \frac{12}{19} \, B a b^{5} x^{\frac{19}{2}} + \frac{2}{19} \, A b^{6} x^{\frac{19}{2}} + \frac{30}{17} \, B a^{2} b^{4} x^{\frac{17}{2}} + \frac{12}{17} \, A a b^{5} x^{\frac{17}{2}} + \frac{8}{3} \, B a^{3} b^{3} x^{\frac{15}{2}} + 2 \, A a^{2} b^{4} x^{\frac{15}{2}} + \frac{30}{13} \, B a^{4} b^{2} x^{\frac{13}{2}} + \frac{40}{13} \, A a^{3} b^{3} x^{\frac{13}{2}} + \frac{12}{11} \, B a^{5} b x^{\frac{11}{2}} + \frac{30}{11} \, A a^{4} b^{2} x^{\frac{11}{2}} + \frac{2}{9} \, B a^{6} x^{\frac{9}{2}} + \frac{4}{3} \, A a^{5} b x^{\frac{9}{2}} + \frac{2}{7} \, A a^{6} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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