Integrand size = 26, antiderivative size = 112 \[ \int \frac {1}{x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} \sqrt {e}}+\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{\sqrt {b} \sqrt {e}} \]
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Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1981, 1980, 400, 214} \[ \int \frac {1}{x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{\sqrt {b} \sqrt {e}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} \sqrt {e}} \]
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Rule 214
Rule 400
Rule 1980
Rule 1981
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {e (a+b x)}{c+d x}}} \, dx,x,x^2\right ) \\ & = ((b c-a d) e) \text {Subst}\left (\int \frac {1}{\left (-a e+c x^2\right ) \left (b e-d x^2\right )} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right ) \\ & = c \text {Subst}\left (\int \frac {1}{-a e+c x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )+d \text {Subst}\left (\int \frac {1}{b e-d x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right ) \\ & = -\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} \sqrt {e}}+\frac {\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{\sqrt {b} \sqrt {e}} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\sqrt {a+b x^2} \left (-\sqrt {b} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )+\sqrt {a} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )\right )}{\sqrt {a} \sqrt {b} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(84)=168\).
Time = 0.09 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.60
method | result | size |
default | \(-\frac {\left (b \,x^{2}+a \right ) \left (c \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) \sqrt {b d}-\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, d \right )}{2 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b d}\, \sqrt {a c}}\) | \(179\) |
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Time = 0.52 (sec) , antiderivative size = 881, normalized size of antiderivative = 7.87 \[ \int \frac {1}{x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\left [\frac {1}{4} \, \sqrt {\frac {d}{b e}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + a b c d + {\left (3 \, b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {d}{b e}}\right ) + \frac {1}{4} \, \sqrt {\frac {c}{a e}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (a b c d + a^{2} d^{2}\right )} x^{4} + 2 \, a^{2} c^{2} + {\left (a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {c}{a e}}}{x^{4}}\right ), -\frac {1}{2} \, \sqrt {-\frac {d}{b e}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {d}{b e}}}{2 \, {\left (b d x^{2} + a d\right )}}\right ) + \frac {1}{4} \, \sqrt {\frac {c}{a e}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (a b c d + a^{2} d^{2}\right )} x^{4} + 2 \, a^{2} c^{2} + {\left (a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {c}{a e}}}{x^{4}}\right ), \frac {1}{2} \, \sqrt {-\frac {c}{a e}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {c}{a e}}}{2 \, {\left (b c x^{2} + a c\right )}}\right ) + \frac {1}{4} \, \sqrt {\frac {d}{b e}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + a b c d + {\left (3 \, b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {d}{b e}}\right ), \frac {1}{2} \, \sqrt {-\frac {c}{a e}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {c}{a e}}}{2 \, {\left (b c x^{2} + a c\right )}}\right ) - \frac {1}{2} \, \sqrt {-\frac {d}{b e}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {d}{b e}}}{2 \, {\left (b d x^{2} + a d\right )}}\right )\right ] \]
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Timed out. \[ \int \frac {1}{x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {1}{x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\int \frac {1}{x\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}} \,d x \]
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