Optimal. Leaf size=40 \[ -\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{x}} \]
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Rubi [A] time = 0.0129225, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 205} \[ -\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} (a+b x)} \, dx &=-\frac{2}{a \sqrt{x}}-\frac{b \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{a}\\ &=-\frac{2}{a \sqrt{x}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{a}\\ &=-\frac{2}{a \sqrt{x}}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0045101, size = 25, normalized size = 0.62 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{b x}{a}\right )}{a \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 32, normalized size = 0.8 \begin{align*} -2\,{\frac{b}{a\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }-2\,{\frac{1}{a\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63636, size = 207, normalized size = 5.18 \begin{align*} \left [\frac{x \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) - 2 \, \sqrt{x}}{a x}, \frac{2 \,{\left (x \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - \sqrt{x}\right )}}{a x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.38717, size = 102, normalized size = 2.55 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{a \sqrt{x}} & \text{for}\: b = 0 \\- \frac{2}{3 b x^{\frac{3}{2}}} & \text{for}\: a = 0 \\- \frac{2}{a \sqrt{x}} + \frac{i \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{i \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{3}{2}} \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19662, size = 42, normalized size = 1.05 \begin{align*} -\frac{2 \, b \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{2}{a \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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