Optimal. Leaf size=63 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}+\frac{\sqrt{x} (A b-a B)}{a b (a+b x)} \]
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Rubi [A] time = 0.0251839, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {27, 78, 63, 205} \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}+\frac{\sqrt{x} (A b-a B)}{a b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{\sqrt{x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{A+B x}{\sqrt{x} (a+b x)^2} \, dx\\ &=\frac{(A b-a B) \sqrt{x}}{a b (a+b x)}+\frac{(A b+a B) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 a b}\\ &=\frac{(A b-a B) \sqrt{x}}{a b (a+b x)}+\frac{(A b+a B) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{a b}\\ &=\frac{(A b-a B) \sqrt{x}}{a b (a+b x)}+\frac{(A b+a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0360156, size = 63, normalized size = 1. \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}+\frac{\sqrt{x} (A b-a B)}{a b (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 69, normalized size = 1.1 \begin{align*}{\frac{Ab-aB}{ab \left ( bx+a \right ) }\sqrt{x}}+{\frac{A}{a}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{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 = 1.65496, size = 396, normalized size = 6.29 \begin{align*} \left [-\frac{{\left (B a^{2} + A a b +{\left (B a b + A b^{2}\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x - a - 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right ) + 2 \,{\left (B a^{2} b - A a b^{2}\right )} \sqrt{x}}{2 \,{\left (a^{2} b^{3} x + a^{3} b^{2}\right )}}, -\frac{{\left (B a^{2} + A a b +{\left (B a b + A b^{2}\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (B a^{2} b - A a b^{2}\right )} \sqrt{x}}{a^{2} b^{3} x + a^{3} b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.5024, size = 716, normalized size = 11.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10675, size = 81, normalized size = 1.29 \begin{align*} \frac{{\left (B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a b} - \frac{B a \sqrt{x} - A b \sqrt{x}}{{\left (b x + a\right )} a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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