Optimal. Leaf size=52 \[ \frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{\sqrt {c}}-\frac {2 A \sqrt {b x+c x^2}}{b x} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {792, 620, 206} \begin {gather*} \frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{\sqrt {c}}-\frac {2 A \sqrt {b x+c x^2}}{b x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 620
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x \sqrt {b x+c x^2}} \, dx &=-\frac {2 A \sqrt {b x+c x^2}}{b x}+B \int \frac {1}{\sqrt {b x+c x^2}} \, dx\\ &=-\frac {2 A \sqrt {b x+c x^2}}{b x}+(2 B) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )\\ &=-\frac {2 A \sqrt {b x+c x^2}}{b x}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{\sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 69, normalized size = 1.33 \begin {gather*} \frac {2 \sqrt {x (b+c x)} \left (\frac {\sqrt {b} B \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {c} \sqrt {\frac {c x}{b}+1}}-A\right )}{b x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.30, size = 58, normalized size = 1.12 \begin {gather*} -\frac {2 A \sqrt {b x+c x^2}}{b x}-\frac {B \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 116, normalized size = 2.23 \begin {gather*} \left [\frac {B b \sqrt {c} x \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, \sqrt {c x^{2} + b x} A c}{b c x}, -\frac {2 \, {\left (B b \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + \sqrt {c x^{2} + b x} A c\right )}}{b c x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.32, size = 59, normalized size = 1.13 \begin {gather*} -\frac {B \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{\sqrt {c}} + \frac {2 \, A}{\sqrt {c} x - \sqrt {c x^{2} + b x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 51, normalized size = 0.98 \begin {gather*} \frac {B \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}-\frac {2 \sqrt {c \,x^{2}+b x}\, A}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.73, size = 49, normalized size = 0.94 \begin {gather*} \frac {B \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{\sqrt {c}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.27, size = 50, normalized size = 0.96 \begin {gather*} \frac {B\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{\sqrt {c}}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{b\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{x \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________