Integrand size = 25, antiderivative size = 75 \[ \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}+\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n} \]
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Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2053, 2054, 212} \[ \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n}-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n} \]
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Rule 212
Rule 2053
Rule 2054
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}+a \int \frac {x^{-1+\frac {j}{2}}}{\sqrt {a x^j+b x^n}} \, dx \\ & = -\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n} \\ & = -\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.39 \[ \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=-\frac {2 x^{-j/2} \left (a x^j+b x^n-\sqrt {a} \sqrt {b} x^{\frac {j+n}{2}} \sqrt {1+\frac {a x^{j-n}}{b}} \text {arcsinh}\left (\frac {\sqrt {a} x^{\frac {j-n}{2}}}{\sqrt {b}}\right )\right )}{(j-n) \sqrt {a x^j+b x^n}} \]
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\[\int x^{-1-\frac {j}{2}} \sqrt {a \,x^{j}+b \,x^{n}}d x\]
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Exception generated. \[ \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\int x^{- \frac {j}{2} - 1} \sqrt {a x^{j} + b x^{n}}\, dx \]
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\[ \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\int { \sqrt {a x^{j} + b x^{n}} x^{-\frac {1}{2} \, j - 1} \,d x } \]
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\[ \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\int { \sqrt {a x^{j} + b x^{n}} x^{-\frac {1}{2} \, j - 1} \,d x } \]
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Timed out. \[ \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\int \frac {\sqrt {a\,x^j+b\,x^n}}{x^{\frac {j}{2}+1}} \,d x \]
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