Optimal. Leaf size=49 \[ -\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{2 A}{a \sqrt{x}} \]
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Rubi [A] time = 0.0247503, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {78, 63, 205} \[ -\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{2 A}{a \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{3/2} (a+b x)} \, dx &=-\frac{2 A}{a \sqrt{x}}+\frac{\left (2 \left (-\frac{A b}{2}+\frac{a B}{2}\right )\right ) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{a}\\ &=-\frac{2 A}{a \sqrt{x}}+\frac{\left (4 \left (-\frac{A b}{2}+\frac{a B}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{a}\\ &=-\frac{2 A}{a \sqrt{x}}-\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0260471, size = 49, normalized size = 1. \[ \frac{2 (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{2 A}{a \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 53, normalized size = 1.1 \begin{align*} -2\,{\frac{Ab}{a\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }+2\,{\frac{B}{\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }-2\,{\frac{A}{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 = 2.45293, size = 263, normalized size = 5.37 \begin{align*} \left [-\frac{2 \, A a b \sqrt{x} -{\left (B a - A b\right )} \sqrt{-a b} x \log \left (\frac{b x - a + 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right )}{a^{2} b x}, -\frac{2 \,{\left (A a b \sqrt{x} +{\left (B a - A b\right )} \sqrt{a b} x \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right )\right )}}{a^{2} b x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.9011, size = 216, normalized size = 4.41 \begin{align*} \begin{cases} \tilde{\infty } \left (- \frac{2 A}{3 x^{\frac{3}{2}}} - \frac{2 B}{\sqrt{x}}\right ) & \text{for}\: a = 0 \wedge b = 0 \\\frac{- \frac{2 A}{3 x^{\frac{3}{2}}} - \frac{2 B}{\sqrt{x}}}{b} & \text{for}\: a = 0 \\\frac{- \frac{2 A}{\sqrt{x}} + 2 B \sqrt{x}}{a} & \text{for}\: b = 0 \\- \frac{2 A}{a \sqrt{x}} + \frac{i A \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{i A \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{i B \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{\sqrt{a} b \sqrt{\frac{1}{b}}} + \frac{i B \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{\sqrt{a} b \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 = 1.25464, size = 53, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{2 \, A}{a \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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