Integrand size = 17, antiderivative size = 68 \[ \int x \sqrt {b x^2+c x^4} \, dx=\frac {\left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{8 c}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2038, 626, 634, 212} \[ \int x \sqrt {b x^2+c x^4} \, dx=\frac {\left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{8 c}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c^{3/2}} \]
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Rule 212
Rule 626
Rule 634
Rule 2038
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \sqrt {b x+c x^2} \, dx,x,x^2\right ) \\ & = \frac {\left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{8 c}-\frac {b^2 \text {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{16 c} \\ & = \frac {\left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{8 c}-\frac {b^2 \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c} \\ & = \frac {\left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{8 c}-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c^{3/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.43 \[ \int x \sqrt {b x^2+c x^4} \, dx=\frac {x \sqrt {b+c x^2} \left (\sqrt {c} x \sqrt {b+c x^2} \left (b+2 c x^2\right )+2 b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b}-\sqrt {b+c x^2}}\right )\right )}{8 c^{3/2} \sqrt {x^2 \left (b+c x^2\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {\left (2 c \,x^{2}+b \right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{8 c}-\frac {b^{2} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{8 c^{\frac {3}{2}} x \sqrt {c \,x^{2}+b}}\) | \(77\) |
default | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (2 x \left (c \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {c}-\sqrt {c}\, \sqrt {c \,x^{2}+b}\, b x -\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) b^{2}\right )}{8 x \sqrt {c \,x^{2}+b}\, c^{\frac {3}{2}}}\) | \(84\) |
pseudoelliptic | \(\frac {4 c^{\frac {3}{2}} x^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}+2 b \sqrt {c}\, \sqrt {x^{2} \left (c \,x^{2}+b \right )}-\ln \left (\frac {2 c \,x^{2}+2 \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {c}+b}{\sqrt {c}}\right ) b^{2}+\ln \left (2\right ) b^{2}}{16 c^{\frac {3}{2}}}\) | \(89\) |
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Time = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.06 \[ \int x \sqrt {b x^2+c x^4} \, dx=\left [\frac {b^{2} \sqrt {c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} {\left (2 \, c^{2} x^{2} + b c\right )}}{16 \, c^{2}}, \frac {b^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + \sqrt {c x^{4} + b x^{2}} {\left (2 \, c^{2} x^{2} + b c\right )}}{8 \, c^{2}}\right ] \]
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\[ \int x \sqrt {b x^2+c x^4} \, dx=\int x \sqrt {x^{2} \left (b + c x^{2}\right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07 \[ \int x \sqrt {b x^2+c x^4} \, dx=\frac {1}{4} \, \sqrt {c x^{4} + b x^{2}} x^{2} - \frac {b^{2} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{16 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{4} + b x^{2}} b}{8 \, c} \]
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Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int x \sqrt {b x^2+c x^4} \, dx=\frac {1}{8} \, \sqrt {c x^{2} + b} {\left (2 \, x^{2} \mathrm {sgn}\left (x\right ) + \frac {b \mathrm {sgn}\left (x\right )}{c}\right )} x + \frac {b^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, c^{\frac {3}{2}}} - \frac {b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, c^{\frac {3}{2}}} \]
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Time = 13.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94 \[ \int x \sqrt {b x^2+c x^4} \, dx=\frac {\left (\frac {b}{4\,c}+\frac {x^2}{2}\right )\,\sqrt {c\,x^4+b\,x^2}}{2}-\frac {b^2\,\ln \left (\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}+\sqrt {c\,x^4+b\,x^2}\right )}{16\,c^{3/2}} \]
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