Optimal. Leaf size=60 \[ \frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {612, 620, 206} \begin {gather*} \frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 612
Rule 620
Rubi steps
\begin {align*} \int \sqrt {b x+c x^2} \, dx &=\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{8 c}\\ &=\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{4 c}\\ &=\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 74, normalized size = 1.23 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} (b+2 c x)-\frac {b^{3/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}\right )}{4 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.17, size = 70, normalized size = 1.17 \begin {gather*} \frac {b^2 \log \left (-2 c^{3/2} \sqrt {b x+c x^2}+b c+2 c^2 x\right )}{8 c^{3/2}}+\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 121, normalized size = 2.02 \begin {gather*} \left [\frac {b^{2} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (2 \, c^{2} x + b c\right )} \sqrt {c x^{2} + b x}}{8 \, c^{2}}, \frac {b^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (2 \, c^{2} x + b c\right )} \sqrt {c x^{2} + b x}}{4 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 61, normalized size = 1.02 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} + b x} {\left (2 \, x + \frac {b}{c}\right )} + \frac {b^{2} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 56, normalized size = 0.93 \begin {gather*} -\frac {b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}+\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.02, size = 63, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} + b x} x - \frac {b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + b x} b}{4 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 55, normalized size = 0.92 \begin {gather*} \sqrt {c\,x^2+b\,x}\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )-\frac {b^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{8\,c^{3/2}} \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 \sqrt {b x + c x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________