Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{3} b x^{3/2} (2 a B+A b)-\frac{2 a (a B+2 A b)}{\sqrt{x}}+\frac{2}{7} b^2 B x^{7/2} \]
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Rubi [A] time = 0.0287255, 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} \[ -\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{3} b x^{3/2} (2 a B+A b)-\frac{2 a (a B+2 A b)}{\sqrt{x}}+\frac{2}{7} b^2 B x^{7/2} \]
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^{7/2}} \, dx &=\int \left (\frac{a^2 A}{x^{7/2}}+\frac{a (2 A b+a B)}{x^{3/2}}+b (A b+2 a B) \sqrt{x}+b^2 B x^{5/2}\right ) \, dx\\ &=-\frac{2 a^2 A}{5 x^{5/2}}-\frac{2 a (2 A b+a B)}{\sqrt{x}}+\frac{2}{3} b (A b+2 a B) x^{3/2}+\frac{2}{7} b^2 B x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0155888, size = 57, normalized size = 0.93 \[ \frac{-42 a^2 \left (A+5 B x^2\right )+140 a b x^2 \left (B x^2-3 A\right )+10 b^2 x^4 \left (7 A+3 B x^2\right )}{105 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-30\,B{b}^{2}{x}^{6}-70\,A{b}^{2}{x}^{4}-140\,B{x}^{4}ab+420\,aAb{x}^{2}+210\,B{x}^{2}{a}^{2}+42\,{a}^{2}A}{105}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01632, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{2}{3} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (A a^{2} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.796934, size = 130, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (15 \, B b^{2} x^{6} + 35 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 21 \, A a^{2} - 105 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{105 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.14279, size = 76, normalized size = 1.25 \begin{align*} - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 A a b}{\sqrt{x}} + \frac{2 A b^{2} x^{\frac{3}{2}}}{3} - \frac{2 B a^{2}}{\sqrt{x}} + \frac{4 B a b x^{\frac{3}{2}}}{3} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12807, size = 74, normalized size = 1.21 \begin{align*} \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{4}{3} \, B a b x^{\frac{3}{2}} + \frac{2}{3} \, A b^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (5 \, B a^{2} x^{2} + 10 \, A a b x^{2} + A a^{2}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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