Integrand size = 63, antiderivative size = 75 \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \left (c (b f-2 a g)+\left (b^2-4 a c\right ) h x^{n/2}+c (2 c f-b g) x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \]
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Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6873, 1767} \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt {a+b x^n+c x^{2 n}}} \]
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Rule 1767
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx \\ & = -\frac {2 \left (c (b f-2 a g)+\left (b^2-4 a c\right ) h x^{n/2}+c (2 c f-b g) x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \\ \end{align*}
Time = 2.15 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.12 \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \left (b c f-2 a c g+b^2 h x^{n/2}-4 a c h x^{n/2}+2 c^2 f x^n-b c g x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \]
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\[\int \frac {-a h \,x^{-1+\frac {n}{2}}+c f \,x^{-1+n}+c g \,x^{-1+2 n}+c h \,x^{-1+\frac {5 n}{2}}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {3}{2}}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.83 \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c x^{4} x^{2 \, n - 4} + b x^{2} x^{n - 2} + a} {\left ({\left (2 \, c^{2} f - b c g\right )} x^{2} x^{n - 2} + {\left (b^{2} - 4 \, a c\right )} h x x^{\frac {1}{2} \, n - 1} + b c f - 2 \, a c g\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{4} x^{2 \, n - 4} + {\left (b^{3} - 4 \, a b c\right )} n x^{2} x^{n - 2} + {\left (a b^{2} - 4 \, a^{2} c\right )} n} \]
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Timed out. \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {c h x^{\frac {5}{2} \, n - 1} + c g x^{2 \, n - 1} + c f x^{n - 1} - a h x^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {c h x^{\frac {5}{2} \, n - 1} + c g x^{2 \, n - 1} + c f x^{n - 1} - a h x^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int \frac {c\,g\,x^{2\,n-1}-a\,h\,x^{\frac {n}{2}-1}+c\,h\,x^{\frac {5\,n}{2}-1}+c\,f\,x^{n-1}}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}} \,d x \]
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