Optimal. Leaf size=40 \[ -\frac{2 (A b-a B)}{3 b^2 (a+b x)^{3/2}}-\frac{2 B}{b^2 \sqrt{a+b x}} \]
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Rubi [A] time = 0.0131098, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ -\frac{2 (A b-a B)}{3 b^2 (a+b x)^{3/2}}-\frac{2 B}{b^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{A+B x}{(a+b x)^{5/2}} \, dx &=\int \left (\frac{A b-a B}{b (a+b x)^{5/2}}+\frac{B}{b (a+b x)^{3/2}}\right ) \, dx\\ &=-\frac{2 (A b-a B)}{3 b^2 (a+b x)^{3/2}}-\frac{2 B}{b^2 \sqrt{a+b x}}\\ \end{align*}
Mathematica [A] time = 0.0181173, size = 29, normalized size = 0.72 \[ -\frac{2 (2 a B+A b+3 b B x)}{3 b^2 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 26, normalized size = 0.7 \begin{align*} -{\frac{6\,bBx+2\,Ab+4\,Ba}{3\,{b}^{2}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12953, size = 38, normalized size = 0.95 \begin{align*} -\frac{2 \,{\left (3 \,{\left (b x + a\right )} B - B a + A b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37413, size = 103, normalized size = 2.58 \begin{align*} -\frac{2 \,{\left (3 \, B b x + 2 \, B a + A b\right )} \sqrt{b x + a}}{3 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.22473, size = 124, normalized size = 3.1 \begin{align*} \begin{cases} - \frac{2 A b}{3 a b^{2} \sqrt{a + b x} + 3 b^{3} x \sqrt{a + b x}} - \frac{4 B a}{3 a b^{2} \sqrt{a + b x} + 3 b^{3} x \sqrt{a + b x}} - \frac{6 B b x}{3 a b^{2} \sqrt{a + b x} + 3 b^{3} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{A x + \frac{B x^{2}}{2}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16561, size = 38, normalized size = 0.95 \begin{align*} -\frac{2 \,{\left (3 \,{\left (b x + a\right )} B - B a + A b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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