Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{\sqrt{x}}{a (a+b x)} \]
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Rubi [A] time = 0.0128271, antiderivative size = 45, 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} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{\sqrt{x}}{a (a+b x)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} (a+b x)^2} \, dx &=\frac{\sqrt{x}}{a (a+b x)}+\frac{\int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 a}\\ &=\frac{\sqrt{x}}{a (a+b x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{\sqrt{x}}{a (a+b x)}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0181418, size = 45, normalized size = 1. \[ \frac{\sqrt{x}}{a^2+a b x}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 36, normalized size = 0.8 \begin{align*}{\frac{1}{a \left ( bx+a \right ) }\sqrt{x}}+{\frac{1}{a}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.6122, size = 274, normalized size = 6.09 \begin{align*} \left [\frac{2 \, a b \sqrt{x} - \sqrt{-a b}{\left (b x + a\right )} \log \left (\frac{b x - a - 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right )}{2 \,{\left (a^{2} b^{2} x + a^{3} b\right )}}, \frac{a b \sqrt{x} - \sqrt{a b}{\left (b x + a\right )} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right )}{a^{2} b^{2} x + a^{3} b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.8363, size = 328, normalized size = 7.29 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{3 b^{2} x^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{2 \sqrt{x}}{a^{2}} & \text{for}\: b = 0 \\\frac{2 i \sqrt{a} b \sqrt{x} \sqrt{\frac{1}{b}}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} + \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} + \frac{b x \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} - \frac{b x \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \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 = 1.2027, size = 47, normalized size = 1.04 \begin{align*} \frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} + \frac{\sqrt{x}}{{\left (b x + a\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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