Optimal. Leaf size=49 \[ \frac{2 B \sqrt{x}}{c}-\frac{2 (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} c^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.026395, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {781, 80, 63, 205} \[ \frac{2 B \sqrt{x}}{c}-\frac{2 (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} c^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 781
Rule 80
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{x} (A+B x)}{b x+c x^2} \, dx &=\int \frac{A+B x}{\sqrt{x} (b+c x)} \, dx\\ &=\frac{2 B \sqrt{x}}{c}+\frac{\left (2 \left (-\frac{b B}{2}+\frac{A c}{2}\right )\right ) \int \frac{1}{\sqrt{x} (b+c x)} \, dx}{c}\\ &=\frac{2 B \sqrt{x}}{c}+\frac{\left (4 \left (-\frac{b B}{2}+\frac{A c}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b+c x^2} \, dx,x,\sqrt{x}\right )}{c}\\ &=\frac{2 B \sqrt{x}}{c}-\frac{2 (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0283581, size = 49, normalized size = 1. \[ \frac{2 B \sqrt{x}}{c}-\frac{2 (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} c^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 53, normalized size = 1.1 \begin{align*} 2\,{\frac{B\sqrt{x}}{c}}+2\,{\frac{A}{\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \right ) }-2\,{\frac{bB}{c\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \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 = 2.11272, size = 250, normalized size = 5.1 \begin{align*} \left [\frac{2 \, B b c \sqrt{x} +{\left (B b - A c\right )} \sqrt{-b c} \log \left (\frac{c x - b - 2 \, \sqrt{-b c} \sqrt{x}}{c x + b}\right )}{b c^{2}}, \frac{2 \,{\left (B b c \sqrt{x} +{\left (B b - A c\right )} \sqrt{b c} \arctan \left (\frac{\sqrt{b c}}{c \sqrt{x}}\right )\right )}}{b c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.91469, size = 218, normalized size = 4.45 \begin{align*} \begin{cases} \tilde{\infty } \left (- \frac{2 A}{\sqrt{x}} + 2 B \sqrt{x}\right ) & \text{for}\: b = 0 \wedge c = 0 \\\frac{- \frac{2 A}{\sqrt{x}} + 2 B \sqrt{x}}{c} & \text{for}\: b = 0 \\\frac{2 A \sqrt{x} + \frac{2 B x^{\frac{3}{2}}}{3}}{b} & \text{for}\: c = 0 \\- \frac{i A \log{\left (- i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{\sqrt{b} c \sqrt{\frac{1}{c}}} + \frac{i A \log{\left (i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{\sqrt{b} c \sqrt{\frac{1}{c}}} + \frac{i B \sqrt{b} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{c^{2} \sqrt{\frac{1}{c}}} - \frac{i B \sqrt{b} \log{\left (i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{c^{2} \sqrt{\frac{1}{c}}} + \frac{2 B \sqrt{x}}{c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11409, size = 53, normalized size = 1.08 \begin{align*} \frac{2 \, B \sqrt{x}}{c} - \frac{2 \,{\left (B b - A c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]