Integrand size = 26, antiderivative size = 298 \[ \int \frac {d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx=\frac {e x^{1+n} \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {5}{2},\frac {5}{2},2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a^2 (1+n) \sqrt {a+b x^n+c x^{2 n}}}+\frac {d x \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {5}{2},\frac {5}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {a+b x^n+c x^{2 n}}} \]
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Time = 0.21 (sec) , antiderivative size = 298, 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} \[ \int \frac {d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx=\frac {d x \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {1}{n},\frac {5}{2},\frac {5}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {a+b x^n+c x^{2 n}}}+\frac {e x^{n+1} \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {5}{2},\frac {5}{2},2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a^2 (n+1) \sqrt {a+b x^n+c x^{2 n}}} \]
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Rule 440
Rule 524
Rule 1362
Rule 1399
Rule 1446
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d}{\left (a+b x^n+c x^{2 n}\right )^{5/2}}+\frac {e x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}}\right ) \, dx \\ & = d \int \frac {1}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx+e \int \frac {x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, 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}{\left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{5/2}} \, dx}{a^2 \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}{\left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{5/2}} \, dx}{a^2 \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 {5}{2},\frac {5}{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 )}{a^2 (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 {5}{2},\frac {5}{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 )}{a^2 \sqrt {a+b x^n+c x^{2 n}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(701\) vs. \(2(298)=596\).
Time = 5.36 (sec) , antiderivative size = 701, normalized size of antiderivative = 2.35 \[ \int \frac {d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx=\frac {x \left (-2 c \left (2 a b^2 e+4 a b c d (2-5 n)+8 a^2 c e (-1+2 n)+b^3 d (-2+3 n)\right ) x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \left (a+x^n \left (b+c x^n\right )\right ) \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+(1+n) \left (2 \left (b^3 d (-2+3 n) x^n \left (b+c x^n\right )^2+4 a^3 c \left (b e (-2+3 n)+c d (-2+8 n)+2 c e (-1+3 n) x^n\right )+2 a b \left (b+c x^n\right ) \left (-2 c^2 d (-2+5 n) x^{2 n}+b c x^n \left (d (5-11 n)+e x^n\right )+b^2 \left (d (-1+2 n)+e x^n\right )\right )+a^2 \left (-b^3 e (-2+n)+8 b c^2 e (-2+3 n) x^{2 n}-2 b^2 c \left (d (-5+14 n)-3 e (-1+n) x^n\right )+8 c^3 x^{2 n} \left (d (-1+3 n)+e (-1+2 n) x^n\right )\right )\right )+\left (2 a b^3 e (-2+n)-8 a^2 b c e (-2+3 n)+b^4 d \left (4-8 n+3 n^2\right )+16 a^2 c^2 d \left (1-4 n+3 n^2\right )-4 a b^2 c d \left (5-14 n+6 n^2\right )\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \left (a+x^n \left (b+c x^n\right )\right ) \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )}{3 a^2 \left (b^2-4 a c\right )^2 n^2 (1+n) \left (a+x^n \left (b+c x^n\right )\right )^{3/2}} \]
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\[\int \frac {d +e \,x^{n}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx=\int \frac {d + e x^{n}}{\left (a + b x^{n} + c x^{2 n}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx=\int { \frac {e x^{n} + d}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx=\int { \frac {e x^{n} + d}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx=\int \frac {d+e\,x^n}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{5/2}} \,d x \]
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