Integrand size = 26, antiderivative size = 292 \[ \int \left (d+e x^n\right ) \sqrt {a+b x^n+c x^{2 n}} \, dx=\frac {e x^{1+n} \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (1+\frac {1}{n},-\frac {1}{2},-\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{(1+n) \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}}+\frac {d x \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (\frac {1}{n},-\frac {1}{2},-\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \]
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Time = 0.21 (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} \[ \int \left (d+e x^n\right ) \sqrt {a+b x^n+c x^{2 n}} \, dx=\frac {d x \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (\frac {1}{n},-\frac {1}{2},-\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}}+\frac {e x^{n+1} \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (1+\frac {1}{n},-\frac {1}{2},-\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{(n+1) \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rule 440
Rule 524
Rule 1362
Rule 1399
Rule 1446
Rubi steps \begin{align*} \text {integral}& = \int \left (d \sqrt {a+b x^n+c x^{2 n}}+e x^n \sqrt {a+b x^n+c x^{2 n}}\right ) \, dx \\ & = d \int \sqrt {a+b x^n+c x^{2 n}} \, dx+e \int x^n \sqrt {a+b x^n+c x^{2 n}} \, dx \\ & = \frac {\left (d \sqrt {a+b x^n+c x^{2 n}}\right ) \int \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 {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 {\left (e \sqrt {a+b x^n+c x^{2 n}}\right ) \int 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 {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 {e x^{1+n} \sqrt {a+b x^n+c x^{2 n}} 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 {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 {d x \sqrt {a+b x^n+c x^{2 n}} 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 {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*}
Time = 1.22 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.45 \[ \int \left (d+e x^n\right ) \sqrt {a+b x^n+c x^{2 n}} \, dx=\frac {x \left (-n \left (-4 a c e (1+n)+b^2 e (2+n)-2 b c d (1+2 n)\right ) x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+2 (1+n) \left (\left (a+x^n \left (b+c x^n\right )\right ) \left (b e n+2 c \left (d+2 d n+e (1+n) x^n\right )\right )+a n (-b e+2 c (d+2 d n)) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )}{4 (1+n)^2 (c+2 c n) \sqrt {a+x^n \left (b+c x^n\right )}} \]
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\[\int \left (d +e \,x^{n}\right ) \sqrt {a +b \,x^{n}+c \,x^{2 n}}d x\]
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Exception generated. \[ \int \left (d+e x^n\right ) \sqrt {a+b x^n+c x^{2 n}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (d+e x^n\right ) \sqrt {a+b x^n+c x^{2 n}} \, dx=\int \left (d + e x^{n}\right ) \sqrt {a + b x^{n} + c x^{2 n}}\, dx \]
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\[ \int \left (d+e x^n\right ) \sqrt {a+b x^n+c x^{2 n}} \, dx=\int { \sqrt {c x^{2 \, n} + b x^{n} + a} {\left (e x^{n} + d\right )} \,d x } \]
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\[ \int \left (d+e x^n\right ) \sqrt {a+b x^n+c x^{2 n}} \, dx=\int { \sqrt {c x^{2 \, n} + b x^{n} + a} {\left (e x^{n} + d\right )} \,d x } \]
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Timed out. \[ \int \left (d+e x^n\right ) \sqrt {a+b x^n+c x^{2 n}} \, dx=\int \left (d+e\,x^n\right )\,\sqrt {a+b\,x^n+c\,x^{2\,n}} \,d x \]
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