Integrand size = 22, antiderivative size = 72 \[ \int \frac {1}{c^4 x^4 \left (\frac {a}{x^2}+b x^n\right )^{3/2}} \, dx=\frac {2}{a c^4 (2+n) x \sqrt {\frac {a}{x^2}+b x^n}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{a^{3/2} c^4 (2+n)} \]
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Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2055, 2054, 212} \[ \int \frac {1}{c^4 x^4 \left (\frac {a}{x^2}+b x^n\right )^{3/2}} \, dx=\frac {2}{a c^4 (n+2) x \sqrt {\frac {a}{x^2}+b x^n}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{a^{3/2} c^4 (n+2)} \]
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Rule 12
Rule 212
Rule 2054
Rule 2055
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{x^4 \left (\frac {a}{x^2}+b x^n\right )^{3/2}} \, dx}{c^4} \\ & = \frac {2}{a c^4 (2+n) x \sqrt {\frac {a}{x^2}+b x^n}}+\frac {\int \frac {1}{x^2 \sqrt {\frac {a}{x^2}+b x^n}} \, dx}{a c^4} \\ & = \frac {2}{a c^4 (2+n) x \sqrt {\frac {a}{x^2}+b x^n}}-\frac {2 \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {1}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{a c^4 (2+n)} \\ & = \frac {2}{a c^4 (2+n) x \sqrt {\frac {a}{x^2}+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{a^{3/2} c^4 (2+n)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.03 \[ \int \frac {1}{c^4 x^4 \left (\frac {a}{x^2}+b x^n\right )^{3/2}} \, dx=\frac {2 \left (\sqrt {a}-\sqrt {a+b x^{2+n}} \text {arctanh}\left (\frac {\sqrt {a+b x^{2+n}}}{\sqrt {a}}\right )\right )}{a^{3/2} c^4 (2+n) x \sqrt {\frac {a}{x^2}+b x^n}} \]
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\[\int \frac {1}{c^{4} x^{4} \left (\frac {a}{x^{2}}+b \,x^{n}\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{c^4 x^4 \left (\frac {a}{x^2}+b x^n\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{c^4 x^4 \left (\frac {a}{x^2}+b x^n\right )^{3/2}} \, dx=\frac {\int \frac {1}{a x^{2} \sqrt {\frac {a}{x^{2}} + b x^{n}} + b x^{4} x^{n} \sqrt {\frac {a}{x^{2}} + b x^{n}}}\, dx}{c^{4}} \]
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\[ \int \frac {1}{c^4 x^4 \left (\frac {a}{x^2}+b x^n\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{n} + \frac {a}{x^{2}}\right )}^{\frac {3}{2}} c^{4} x^{4}} \,d x } \]
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\[ \int \frac {1}{c^4 x^4 \left (\frac {a}{x^2}+b x^n\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{n} + \frac {a}{x^{2}}\right )}^{\frac {3}{2}} c^{4} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {1}{c^4 x^4 \left (\frac {a}{x^2}+b x^n\right )^{3/2}} \, dx=\int \frac {1}{c^4\,x^4\,{\left (b\,x^n+\frac {a}{x^2}\right )}^{3/2}} \,d x \]
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