Integrand size = 19, antiderivative size = 58 \[ \int \frac {x^3}{\sqrt {b x^2+c x^4}} \, dx=\frac {\sqrt {b x^2+c x^4}}{2 c}-\frac {b \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2 c^{3/2}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2043, 654, 634, 212} \[ \int \frac {x^3}{\sqrt {b x^2+c x^4}} \, dx=\frac {\sqrt {b x^2+c x^4}}{2 c}-\frac {b \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2 c^{3/2}} \]
[In]
[Out]
Rule 212
Rule 634
Rule 654
Rule 2043
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{\sqrt {b x+c x^2}} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {b x^2+c x^4}}{2 c}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{4 c} \\ & = \frac {\sqrt {b x^2+c x^4}}{2 c}-\frac {b \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{2 c} \\ & = \frac {\sqrt {b x^2+c x^4}}{2 c}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2 c^{3/2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.43 \[ \int \frac {x^3}{\sqrt {b x^2+c x^4}} \, dx=\frac {x \left (\sqrt {c} x \left (b+c x^2\right )+2 b \sqrt {b+c x^2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b}-\sqrt {b+c x^2}}\right )\right )}{2 c^{3/2} \sqrt {x^2 \left (b+c x^2\right )}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {c}-b \ln \left (\frac {2 c \,x^{2}+2 \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {c}+b}{\sqrt {c}}\right )+b \ln \left (2\right )}{4 c^{\frac {3}{2}}}\) | \(63\) |
default | \(-\frac {x \sqrt {c \,x^{2}+b}\, \left (-x \sqrt {c \,x^{2}+b}\, c^{\frac {3}{2}}+b \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) c \right )}{2 \sqrt {c \,x^{4}+b \,x^{2}}\, c^{\frac {5}{2}}}\) | \(64\) |
risch | \(\frac {x^{2} \left (c \,x^{2}+b \right )}{2 c \sqrt {x^{2} \left (c \,x^{2}+b \right )}}-\frac {b \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) x \sqrt {c \,x^{2}+b}}{2 c^{\frac {3}{2}} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(75\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.97 \[ \int \frac {x^3}{\sqrt {b x^2+c x^4}} \, dx=\left [\frac {b \sqrt {c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} c}{4 \, c^{2}}, \frac {b \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + \sqrt {c x^{4} + b x^{2}} c}{2 \, c^{2}}\right ] \]
[In]
[Out]
\[ \int \frac {x^3}{\sqrt {b x^2+c x^4}} \, dx=\int \frac {x^{3}}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{\sqrt {b x^2+c x^4}} \, dx=-\frac {b \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{4 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{4} + b x^{2}}}{2 \, c} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02 \[ \int \frac {x^3}{\sqrt {b x^2+c x^4}} \, dx=-\frac {b \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{4 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + b} x}{2 \, c \mathrm {sgn}\left (x\right )} + \frac {b \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{2 \, c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
[In]
[Out]
Time = 13.94 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{\sqrt {b x^2+c x^4}} \, dx=\frac {\sqrt {c\,x^4+b\,x^2}}{2\,c}-\frac {b\,\ln \left (\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}+\sqrt {c\,x^4+b\,x^2}\right )}{4\,c^{3/2}} \]
[In]
[Out]