Optimal. Leaf size=49 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{A \sqrt{a+b x}}{a x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0218117, 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, 208} \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{A \sqrt{a+b x}}{a x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{x^2 \sqrt{a+b x}} \, dx &=-\frac{A \sqrt{a+b x}}{a x}+\frac{\left (-\frac{A b}{2}+a B\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{a}\\ &=-\frac{A \sqrt{a+b x}}{a x}+\frac{\left (2 \left (-\frac{A b}{2}+a B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{a b}\\ &=-\frac{A \sqrt{a+b x}}{a x}+\frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.025058, size = 49, normalized size = 1. \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{A \sqrt{a+b x}}{a x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 42, normalized size = 0.9 \begin{align*}{(Ab-2\,Ba){\it Artanh} \left ({\sqrt{bx+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}}-{\frac{A}{ax}\sqrt{bx+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.71236, size = 270, normalized size = 5.51 \begin{align*} \left [-\frac{{\left (2 \, B a - A b\right )} \sqrt{a} x \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \, \sqrt{b x + a} A a}{2 \, a^{2} x}, \frac{{\left (2 \, B a - A b\right )} \sqrt{-a} x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - \sqrt{b x + a} A a}{a^{2} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 17.0983, size = 82, normalized size = 1.67 \begin{align*} - \frac{A \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a \sqrt{x}} + \frac{A b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{3}{2}}} + \frac{2 B \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + b x}} \right )}}{a \sqrt{- \frac{1}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13294, size = 78, normalized size = 1.59 \begin{align*} -\frac{\frac{\sqrt{b x + a} A b}{a x} - \frac{{\left (2 \, B a b - A b^{2}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]