Optimal. Leaf size=47 \[ \frac {\sqrt {b x+c x^2}}{c}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 634, 212}
\begin {gather*} \frac {\sqrt {b x+c x^2}}{c}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 654
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {b x+c x^2}} \, dx &=\frac {\sqrt {b x+c x^2}}{c}-\frac {b \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2 c}\\ &=\frac {\sqrt {b x+c x^2}}{c}-\frac {b \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c}\\ &=\frac {\sqrt {b x+c x^2}}{c}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 69, normalized size = 1.47 \begin {gather*} \frac {\sqrt {c} x (b+c x)+b \sqrt {x} \sqrt {b+c x} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{c^{3/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 47, normalized size = 1.00
method | result | size |
default | \(\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\) | \(47\) |
risch | \(\frac {x \left (c x +b \right )}{c \sqrt {x \left (c x +b \right )}}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 45, normalized size = 0.96 \begin {gather*} -\frac {b \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + b x}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.42, size = 98, normalized size = 2.09 \begin {gather*} \left [\frac {b \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, \sqrt {c x^{2} + b x} c}{2 \, c^{2}}, \frac {b \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + \sqrt {c x^{2} + b x} c}{c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.80, size = 52, normalized size = 1.11 \begin {gather*} \frac {b \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + b x}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 46, normalized size = 0.98 \begin {gather*} \frac {\sqrt {c\,x^2+b\,x}}{c}-\frac {b\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{2\,c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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