Optimal. Leaf size=85 \[ \frac{2}{7} a^2 x^{7/2} (a B+3 A b)+\frac{2}{3} a^3 A x^{3/2}+\frac{2}{15} b^2 x^{15/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{19} b^3 B x^{19/2} \]
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Rubi [A] time = 0.0412365, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{2}{7} a^2 x^{7/2} (a B+3 A b)+\frac{2}{3} a^3 A x^{3/2}+\frac{2}{15} b^2 x^{15/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{19} b^3 B x^{19/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \sqrt{x} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx &=\int \left (a^3 A \sqrt{x}+a^2 (3 A b+a B) x^{5/2}+3 a b (A b+a B) x^{9/2}+b^2 (A b+3 a B) x^{13/2}+b^3 B x^{17/2}\right ) \, dx\\ &=\frac{2}{3} a^3 A x^{3/2}+\frac{2}{7} a^2 (3 A b+a B) x^{7/2}+\frac{6}{11} a b (A b+a B) x^{11/2}+\frac{2}{15} b^2 (A b+3 a B) x^{15/2}+\frac{2}{19} b^3 B x^{19/2}\\ \end{align*}
Mathematica [A] time = 0.03822, size = 71, normalized size = 0.84 \[ \frac{2 x^{3/2} \left (3135 a^2 x^2 (a B+3 A b)+7315 a^3 A+1463 b^2 x^6 (3 a B+A b)+5985 a b x^4 (a B+A b)+1155 b^3 B x^8\right )}{21945} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 80, normalized size = 0.9 \begin{align*}{\frac{2310\,{b}^{3}B{x}^{8}+2926\,{x}^{6}{b}^{3}A+8778\,{x}^{6}a{b}^{2}B+11970\,{x}^{4}a{b}^{2}A+11970\,{x}^{4}{a}^{2}bB+18810\,{x}^{2}A{a}^{2}b+6270\,{x}^{2}B{a}^{3}+14630\,{a}^{3}A}{21945}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08603, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{19} \, B b^{3} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{15}{2}} + \frac{6}{11} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{11}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} + \frac{2}{7} \,{\left (B a^{3} + 3 \, A a^{2} 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 = 0.854758, size = 192, normalized size = 2.26 \begin{align*} \frac{2}{21945} \,{\left (1155 \, B b^{3} x^{9} + 1463 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + 5985 \,{\left (B a^{2} b + A a b^{2}\right )} x^{5} + 7315 \, A a^{3} x + 3135 \,{\left (B a^{3} + 3 \, A a^{2} 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 = 3.65904, size = 95, normalized size = 1.12 \begin{align*} \frac{2 A a^{3} x^{\frac{3}{2}}}{3} + \frac{2 B b^{3} x^{\frac{19}{2}}}{19} + \frac{2 x^{\frac{15}{2}} \left (A b^{3} + 3 B a b^{2}\right )}{15} + \frac{2 x^{\frac{11}{2}} \left (3 A a b^{2} + 3 B a^{2} b\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (3 A a^{2} b + B a^{3}\right )}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13797, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{19} \, B b^{3} x^{\frac{19}{2}} + \frac{2}{5} \, B a b^{2} x^{\frac{15}{2}} + \frac{2}{15} \, A b^{3} x^{\frac{15}{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}{7} \, B a^{3} x^{\frac{7}{2}} + \frac{6}{7} \, A a^{2} b x^{\frac{7}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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