Integrand size = 22, antiderivative size = 149 \[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x^2} \, dx=-\frac {\sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (-\frac {1}{n},-\frac {1}{2},-\frac {1}{2},-\frac {1-n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \]
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Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1399, 524} \[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x^2} \, dx=-\frac {\sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (-\frac {1}{n},-\frac {1}{2},-\frac {1}{2},-\frac {1-n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rule 524
Rule 1399
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b x^n+c x^{2 n}} \int \frac {\sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}}{x^2} \, dx}{\sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \\ & = -\frac {\sqrt {a+b x^n+c x^{2 n}} F_1\left (-\frac {1}{n};-\frac {1}{2},-\frac {1}{2};-\frac {1-n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(365\) vs. \(2(149)=298\).
Time = 0.48 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.45 \[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x^2} \, dx=\frac {2 (-1+n) \left (a+x^n \left (b+c x^n\right )\right )-2 a (-1+n) n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (-\frac {1}{n},\frac {1}{2},\frac {1}{2},\frac {-1+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+b n x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {-1+n}{n},\frac {1}{2},\frac {1}{2},2-\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{2 (-1+n)^2 x \sqrt {a+x^n \left (b+c x^n\right )}} \]
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\[\int \frac {\sqrt {a +b \,x^{n}+c \,x^{2 n}}}{x^{2}}d x\]
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Exception generated. \[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x^2} \, dx=\int \frac {\sqrt {a + b x^{n} + c x^{2 n}}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x^2} \, dx=\int { \frac {\sqrt {c x^{2 \, n} + b x^{n} + a}}{x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x^2} \, dx=\int { \frac {\sqrt {c x^{2 \, n} + b x^{n} + a}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x^n+c x^{2 n}}}{x^2} \, dx=\int \frac {\sqrt {a+b\,x^n+c\,x^{2\,n}}}{x^2} \,d x \]
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