Optimal. Leaf size=40 \[ \frac{2 (b c-a d)}{3 d^2 (c+d x)^{3/2}}-\frac{2 b}{d^2 \sqrt{c+d x}} \]
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Rubi [A] time = 0.0139255, 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} \[ \frac{2 (b c-a d)}{3 d^2 (c+d x)^{3/2}}-\frac{2 b}{d^2 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{(c+d x)^{5/2}} \, dx &=\int \left (\frac{-b c+a d}{d (c+d x)^{5/2}}+\frac{b}{d (c+d x)^{3/2}}\right ) \, dx\\ &=\frac{2 (b c-a d)}{3 d^2 (c+d x)^{3/2}}-\frac{2 b}{d^2 \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0275788, size = 29, normalized size = 0.72 \[ -\frac{2 (a d+2 b c+3 b d x)}{3 d^2 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 0.7 \begin{align*} -{\frac{6\,bdx+2\,ad+4\,bc}{3\,{d}^{2}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949559, size = 38, normalized size = 0.95 \begin{align*} -\frac{2 \,{\left (3 \,{\left (d x + c\right )} b - b c + a d\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00135, size = 103, normalized size = 2.58 \begin{align*} -\frac{2 \,{\left (3 \, b d x + 2 \, b c + a d\right )} \sqrt{d x + c}}{3 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.07312, size = 124, normalized size = 3.1 \begin{align*} \begin{cases} - \frac{2 a d}{3 c d^{2} \sqrt{c + d x} + 3 d^{3} x \sqrt{c + d x}} - \frac{4 b c}{3 c d^{2} \sqrt{c + d x} + 3 d^{3} x \sqrt{c + d x}} - \frac{6 b d x}{3 c d^{2} \sqrt{c + d x} + 3 d^{3} x \sqrt{c + d x}} & \text{for}\: d \neq 0 \\\frac{a x + \frac{b x^{2}}{2}}{c^{\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 = 1.05535, size = 38, normalized size = 0.95 \begin{align*} -\frac{2 \,{\left (3 \,{\left (d x + c\right )} b - b c + a d\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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