Integrand size = 20, antiderivative size = 88 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{8 a x^4}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 a^{3/2}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1128, 734, 738, 212} \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 a^{3/2}}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{8 a x^4} \]
[In]
[Out]
Rule 212
Rule 734
Rule 738
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^2\right ) \\ & = -\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{8 a x^4}-\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{16 a} \\ & = -\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{8 a x^4}+\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^2}{\sqrt {a+b x^2+c x^4}}\right )}{8 a} \\ & = -\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{8 a x^4}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 a^{3/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{8 a x^4}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{8 a^{3/2}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-\frac {\left (b \,x^{2}+2 a \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a \,x^{4}}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}\) | \(81\) |
pseudoelliptic | \(\frac {-4 \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) a c \,x^{4}+\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) b^{2} x^{4}-2 b \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2} \sqrt {a}-4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{\frac {3}{2}}}{16 a^{\frac {3}{2}} x^{4}}\) | \(132\) |
default | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}+\frac {b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a^{2} x^{2}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2}}+\frac {b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}-\frac {b c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{8 a^{2}}+\frac {c \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 a}-\frac {c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 \sqrt {a}}\) | \(193\) |
elliptic | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}+\frac {b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a^{2} x^{2}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2}}+\frac {b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}-\frac {b c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{8 a^{2}}+\frac {c \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 a}-\frac {c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 \sqrt {a}}\) | \(193\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=\left [-\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {a} x^{4} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (a b x^{2} + 2 \, a^{2}\right )}}{32 \, a^{2} x^{4}}, -\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (a b x^{2} + 2 \, a^{2}\right )}}{16 \, a^{2} x^{4}}\right ] \]
[In]
[Out]
\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=\int \frac {\sqrt {a + b x^{2} + c x^{4}}}{x^{5}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (74) = 148\).
Time = 0.30 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.74 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=-\frac {{\left (b^{2} - 4 \, a c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a} + \frac {{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} b^{2} + 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a c + 8 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a b \sqrt {c} + {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a b^{2} + 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{2} c}{8 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{2} a} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^5} \,d x \]
[In]
[Out]