Integrand size = 20, antiderivative size = 72 \[ \int \frac {1}{x^3 \sqrt {a+b x^2+c x^4}} \, dx=-\frac {\sqrt {a+b x^2+c x^4}}{2 a x^2}+\frac {b \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{4 a^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 72, 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, 744, 738, 212} \[ \int \frac {1}{x^3 \sqrt {a+b x^2+c x^4}} \, dx=\frac {b \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{4 a^{3/2}}-\frac {\sqrt {a+b x^2+c x^4}}{2 a x^2} \]
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Rule 212
Rule 738
Rule 744
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {a+b x^2+c x^4}}{2 a x^2}-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{4 a} \\ & = -\frac {\sqrt {a+b x^2+c x^4}}{2 a x^2}+\frac {b \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 )}{2 a} \\ & = -\frac {\sqrt {a+b x^2+c x^4}}{2 a x^2}+\frac {b \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{4 a^{3/2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \sqrt {a+b x^2+c x^4}} \, dx=-\frac {\sqrt {a+b x^2+c x^4}}{2 a x^2}-\frac {b \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{2 a^{3/2}} \]
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Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\) | \(63\) |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\) | \(63\) |
elliptic | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\) | \(63\) |
pseudoelliptic | \(\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) b \,x^{2}-2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 a^{\frac {3}{2}} x^{2}}\) | \(67\) |
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none
Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.49 \[ \int \frac {1}{x^3 \sqrt {a+b x^2+c x^4}} \, dx=\left [\frac {\sqrt {a} b x^{2} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, \sqrt {c x^{4} + b x^{2} + a} a}{8 \, a^{2} x^{2}}, -\frac {\sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2} + a} a}{4 \, a^{2} x^{2}}\right ] \]
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\[ \int \frac {1}{x^3 \sqrt {a+b x^2+c x^4}} \, dx=\int \frac {1}{x^{3} \sqrt {a + b x^{2} + c x^{4}}}\, dx \]
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Exception generated. \[ \int \frac {1}{x^3 \sqrt {a+b x^2+c x^4}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.58 \[ \int \frac {1}{x^3 \sqrt {a+b x^2+c x^4}} \, dx=-\frac {b \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a} + \frac {{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} b + 2 \, a \sqrt {c}}{2 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )} a} \]
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Time = 13.33 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^3 \sqrt {a+b x^2+c x^4}} \, dx=\frac {b\,\mathrm {atanh}\left (\frac {\frac {b\,x^2}{2}+a}{\sqrt {a}\,\sqrt {c\,x^4+b\,x^2+a}}\right )}{4\,a^{3/2}}-\frac {\sqrt {c\,x^4+b\,x^2+a}}{2\,a\,x^2} \]
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