Optimal. Leaf size=50 \[ \frac{2 (A b-a B)}{a b \sqrt{a+b x}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0150914, antiderivative size = 50, 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} \[ \frac{2 (A b-a B)}{a b \sqrt{a+b x}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{x (a+b x)^{3/2}} \, dx &=\frac{2 (A b-a B)}{a b \sqrt{a+b x}}+\frac{A \int \frac{1}{x \sqrt{a+b x}} \, dx}{a}\\ &=\frac{2 (A b-a B)}{a b \sqrt{a+b x}}+\frac{(2 A) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{a b}\\ &=\frac{2 (A b-a B)}{a b \sqrt{a+b x}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.034676, size = 50, normalized size = 1. \[ -\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2 (a B-A b)}{a b \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 46, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{b} \left ( -{\frac{Ab}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) }-{\frac{-Ab+Ba}{a\sqrt{bx+a}}} \right ) } \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 = 2.41383, size = 336, normalized size = 6.72 \begin{align*} \left [\frac{{\left (A b^{2} x + A a b\right )} \sqrt{a} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left (B a^{2} - A a b\right )} \sqrt{b x + a}}{a^{2} b^{2} x + a^{3} b}, \frac{2 \,{\left ({\left (A b^{2} x + A a b\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) -{\left (B a^{2} - A a b\right )} \sqrt{b x + a}\right )}}{a^{2} b^{2} x + a^{3} b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 16.5693, size = 49, normalized size = 0.98 \begin{align*} \frac{2 A \operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{a \sqrt{- a}} - \frac{2 \left (- A b + B a\right )}{a b \sqrt{a + b x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20671, size = 66, normalized size = 1.32 \begin{align*} \frac{2 \, A \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{2 \,{\left (B a - A b\right )}}{\sqrt{b x + a} a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]