Integrand size = 26, antiderivative size = 292 \[ \int \frac {d+e x^n}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\frac {e x^{1+n} \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{(1+n) \sqrt {a+b x^n+c x^{2 n}}}+\frac {d x \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {a+b x^n+c x^{2 n}}} \]
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Time = 0.20 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1446, 1362, 440, 1399, 524} \[ \int \frac {d+e x^n}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\frac {d x \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {a+b x^n+c x^{2 n}}}+\frac {e x^{n+1} \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{(n+1) \sqrt {a+b x^n+c x^{2 n}}} \]
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Rule 440
Rule 524
Rule 1362
Rule 1399
Rule 1446
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d}{\sqrt {a+b x^n+c x^{2 n}}}+\frac {e x^n}{\sqrt {a+b x^n+c x^{2 n}}}\right ) \, dx \\ & = d \int \frac {1}{\sqrt {a+b x^n+c x^{2 n}}} \, dx+e \int \frac {x^n}{\sqrt {a+b x^n+c x^{2 n}}} \, dx \\ & = \frac {\left (d \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}}}\right ) \int \frac {1}{\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}}}} \, dx}{\sqrt {a+b x^n+c x^{2 n}}}+\frac {\left (e \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}}}\right ) \int \frac {x^n}{\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}}}} \, dx}{\sqrt {a+b x^n+c x^{2 n}}} \\ & = \frac {e x^{1+n} \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}}} F_1\left (1+\frac {1}{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 )}{(1+n) \sqrt {a+b x^n+c x^{2 n}}}+\frac {d 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}}} F_1\left (\frac {1}{n};\frac {1}{2},\frac {1}{2};1+\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 )}{\sqrt {a+b x^n+c x^{2 n}}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.84 \[ \int \frac {d+e x^n}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\frac {x \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \left (e x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+d (1+n) \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )}{(1+n) \sqrt {a+x^n \left (b+c x^n\right )}} \]
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\[\int \frac {d +e \,x^{n}}{\sqrt {a +b \,x^{n}+c \,x^{2 n}}}d x\]
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Exception generated. \[ \int \frac {d+e x^n}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {d+e x^n}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\int \frac {d + e x^{n}}{\sqrt {a + b x^{n} + c x^{2 n}}}\, dx \]
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\[ \int \frac {d+e x^n}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\int { \frac {e x^{n} + d}{\sqrt {c x^{2 \, n} + b x^{n} + a}} \,d x } \]
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\[ \int \frac {d+e x^n}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\int { \frac {e x^{n} + d}{\sqrt {c x^{2 \, n} + b x^{n} + a}} \,d x } \]
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Timed out. \[ \int \frac {d+e x^n}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\int \frac {d+e\,x^n}{\sqrt {a+b\,x^n+c\,x^{2\,n}}} \,d x \]
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