Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]
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Rubi [A] time = 0.0310831, antiderivative size = 61, 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} \[ -\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^2 \left (A+B x^3\right )}{x^{7/2}} \, dx &=\int \left (\frac{a^2 A}{x^{7/2}}+\frac{a (2 A b+a B)}{\sqrt{x}}+b (A b+2 a B) x^{5/2}+b^2 B x^{11/2}\right ) \, dx\\ &=-\frac{2 a^2 A}{5 x^{5/2}}+2 a (2 A b+a B) \sqrt{x}+\frac{2}{7} b (A b+2 a B) x^{7/2}+\frac{2}{13} b^2 B x^{13/2}\\ \end{align*}
Mathematica [A] time = 0.0161413, size = 57, normalized size = 0.93 \[ \frac{2 \left (-91 a^2 \left (A-5 B x^3\right )+130 a b x^3 \left (7 A+B x^3\right )+5 b^2 x^6 \left (13 A+7 B x^3\right )\right )}{455 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-70\,B{b}^{2}{x}^{9}-130\,A{b}^{2}{x}^{6}-260\,B{x}^{6}ab-1820\,aAb{x}^{3}-910\,B{x}^{3}{a}^{2}+182\,{a}^{2}A}{455}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943316, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{2}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{7}{2}} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} \sqrt{x} - \frac{2 \, A a^{2}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72558, size = 130, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (35 \, B b^{2} x^{9} + 65 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + 455 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} - 91 \, A a^{2}\right )}}{455 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.0103, size = 76, normalized size = 1.25 \begin{align*} - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} + 4 A a b \sqrt{x} + \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + 2 B a^{2} \sqrt{x} + \frac{4 B a b x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16006, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{7} \, B a b x^{\frac{7}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + 2 \, B a^{2} \sqrt{x} + 4 \, A a b \sqrt{x} - \frac{2 \, A a^{2}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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