Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{\sqrt{x}}{b (a+b x)} \]
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Rubi [A] time = 0.0133927, antiderivative size = 46, 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 = {47, 63, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{\sqrt{x}}{b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{(a+b x)^2} \, dx &=-\frac{\sqrt{x}}{b (a+b x)}+\frac{\int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 b}\\ &=-\frac{\sqrt{x}}{b (a+b x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{b}\\ &=-\frac{\sqrt{x}}{b (a+b x)}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.019689, size = 46, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{\sqrt{x}}{b (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 37, normalized size = 0.8 \begin{align*} -{\frac{1}{b \left ( bx+a \right ) }\sqrt{x}}+{\frac{1}{b}\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 = 1.68023, size = 277, normalized size = 6.02 \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 b^{3} x + a^{2} b^{2}\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 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 = 8.45385, size = 337, normalized size = 7.33 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{3}{2}}}{3 a^{2}} & \text{for}\: b = 0 \\- \frac{2}{b^{2} \sqrt{x}} & \text{for}\: a = 0 \\- \frac{2 i \sqrt{a} b \sqrt{x} \sqrt{\frac{1}{b}}}{2 i a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} + \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} + \frac{b x \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} - \frac{b x \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{3} 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 = 1.21819, size = 49, normalized size = 1.07 \begin{align*} \frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} - \frac{\sqrt{x}}{{\left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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