Optimal. Leaf size=37 \[ -\frac{2 (a B+A b)}{3 x^{3/2}}-\frac{2 a A}{5 x^{5/2}}-\frac{2 b B}{\sqrt{x}} \]
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Rubi [A] time = 0.0120989, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ -\frac{2 (a B+A b)}{3 x^{3/2}}-\frac{2 a A}{5 x^{5/2}}-\frac{2 b B}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x) (A+B x)}{x^{7/2}} \, dx &=\int \left (\frac{a A}{x^{7/2}}+\frac{A b+a B}{x^{5/2}}+\frac{b B}{x^{3/2}}\right ) \, dx\\ &=-\frac{2 a A}{5 x^{5/2}}-\frac{2 (A b+a B)}{3 x^{3/2}}-\frac{2 b B}{\sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0102009, size = 30, normalized size = 0.81 \[ -\frac{2 (a (3 A+5 B x)+5 b x (A+3 B x))}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 28, normalized size = 0.8 \begin{align*} -{\frac{30\,bB{x}^{2}+10\,Abx+10\,Bax+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 = 1.05272, size = 36, normalized size = 0.97 \begin{align*} -\frac{2 \,{\left (15 \, B b x^{2} + 3 \, A a + 5 \,{\left (B a + A b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39734, size = 73, normalized size = 1.97 \begin{align*} -\frac{2 \,{\left (15 \, B b x^{2} + 3 \, A a + 5 \,{\left (B a + A b\right )} x\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 = 1.94522, size = 46, normalized size = 1.24 \begin{align*} - \frac{2 A a}{5 x^{\frac{5}{2}}} - \frac{2 A b}{3 x^{\frac{3}{2}}} - \frac{2 B a}{3 x^{\frac{3}{2}}} - \frac{2 B b}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15904, size = 36, normalized size = 0.97 \begin{align*} -\frac{2 \,{\left (15 \, B b x^{2} + 5 \, B a x + 5 \, A b x + 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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