Optimal. Leaf size=37 \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{5 x^{5/2}}+\frac{2}{3} b B x^{3/2} \]
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Rubi [A] time = 0.0156172, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{5 x^{5/2}}+\frac{2}{3} b B x^{3/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (A+B x^2\right )}{x^{7/2}} \, dx &=\int \left (\frac{a A}{x^{7/2}}+\frac{A b+a B}{x^{3/2}}+b B \sqrt{x}\right ) \, dx\\ &=-\frac{2 a A}{5 x^{5/2}}-\frac{2 (A b+a B)}{\sqrt{x}}+\frac{2}{3} b B x^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0126628, size = 36, normalized size = 0.97 \[ \frac{10 b x^2 \left (B x^2-3 A\right )-6 a \left (A+5 B x^2\right )}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.9 \begin{align*} -{\frac{-10\,bB{x}^{4}+30\,A{x}^{2}b+30\,B{x}^{2}a+6\,Aa}{15}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03911, size = 39, normalized size = 1.05 \begin{align*} \frac{2}{3} \, B b x^{\frac{3}{2}} - \frac{2 \,{\left (5 \,{\left (B a + A b\right )} x^{2} + A a\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 = 0.858949, size = 74, normalized size = 2. \begin{align*} \frac{2 \,{\left (5 \, B b x^{4} - 15 \,{\left (B a + A b\right )} x^{2} - 3 \, A a\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.86614, size = 42, normalized size = 1.14 \begin{align*} - \frac{2 A a}{5 x^{\frac{5}{2}}} - \frac{2 A b}{\sqrt{x}} - \frac{2 B a}{\sqrt{x}} + \frac{2 B b x^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13884, size = 42, normalized size = 1.14 \begin{align*} \frac{2}{3} \, B b x^{\frac{3}{2}} - \frac{2 \,{\left (5 \, B a x^{2} + 5 \, A b x^{2} + A a\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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