Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]
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Rubi [A] time = 0.030532, 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} \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{3/2}} \, dx &=\int \left (\frac{a^2 A}{x^{3/2}}+a (2 A b+a B) \sqrt{x}+b (A b+2 a B) x^{5/2}+b^2 B x^{9/2}\right ) \, dx\\ &=-\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{3} a (2 A b+a B) x^{3/2}+\frac{2}{7} b (A b+2 a B) x^{7/2}+\frac{2}{11} b^2 B x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0203783, size = 60, normalized size = 0.98 \[ \frac{-154 a^2 \left (3 A-B x^2\right )+44 a b x^2 \left (7 A+3 B x^2\right )+6 b^2 x^4 \left (11 A+7 B x^2\right )}{231 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-42\,B{b}^{2}{x}^{6}-66\,A{b}^{2}{x}^{4}-132\,B{x}^{4}ab-308\,aAb{x}^{2}-154\,B{x}^{2}{a}^{2}+462\,{a}^{2}A}{231}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04592, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{2}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{7}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.886639, size = 130, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (21 \, B b^{2} x^{6} + 33 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 231 \, A a^{2} + 77 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{231 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.5104, size = 78, normalized size = 1.28 \begin{align*} - \frac{2 A a^{2}}{\sqrt{x}} + \frac{4 A a b x^{\frac{3}{2}}}{3} + \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + \frac{2 B a^{2} x^{\frac{3}{2}}}{3} + \frac{4 B a b x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16097, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{7} \, B a b x^{\frac{7}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + \frac{2}{3} \, B a^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A a b x^{\frac{3}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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