Integrand size = 63, antiderivative size = 95 \[ \int \frac {(d x)^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 x^{1-\frac {n}{2}} (d x)^{\frac {1}{2} (-2+n)} \left (c (b f-2 a g)+\left (b^2-4 a c\right ) h x^{n/2}+c (2 c f-b g) x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \]
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Time = 0.14 (sec) , antiderivative size = 95, 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 = {1768, 1767} \[ \int \frac {(d x)^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 x^{1-\frac {n}{2}} (d x)^{\frac {n-2}{2}} \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt {a+b x^n+c x^{2 n}}} \]
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Rule 1767
Rule 1768
Rubi steps \begin{align*} \text {integral}& = \left (x^{1-\frac {n}{2}} (d x)^{-1+\frac {n}{2}}\right ) \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 x^{1-\frac {n}{2}} (d x)^{\frac {1}{2} (-2+n)} \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*}
Leaf count is larger than twice the leaf count of optimal. \(242\) vs. \(2(95)=190\).
Time = 3.46 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.55 \[ \int \frac {(d x)^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {x^{-n/2} (d x)^{n/2} \left (-2 a b^2 h x^{n/2}-2 a b c \left (f-g x^n\right )+4 a c \left (-c f x^n+a \left (g+2 h x^{n/2}\right )\right )+b \sqrt {c} (b f-2 a g) \sqrt {a+x^n \left (b+c x^n\right )} \text {arctanh}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+x^n \left (b+c x^n\right )}}\right )+b \sqrt {c} (b f-2 a g) \sqrt {a+x^n \left (b+c x^n\right )} \log \left (b+2 c x^n-2 \sqrt {c} \sqrt {a+x^n \left (b+c x^n\right )}\right )\right )}{a \left (-b^2+4 a c\right ) d n \sqrt {a+x^n \left (b+c x^n\right )}} \]
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\[\int \frac {\left (d x \right )^{-1+\frac {n}{2}} \left (-a h +c f \,x^{\frac {n}{2}}+c g \,x^{\frac {3 n}{2}}+c h \,x^{2 n}\right )}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {3}{2}}}d x\]
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none
Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.39 \[ \int \frac {(d x)^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \, {\left ({\left (b^{2} - 4 \, a c\right )} d^{\frac {1}{2} \, n - 1} h x^{\frac {1}{2} \, n} + {\left (2 \, c^{2} f - b c g\right )} d^{\frac {1}{2} \, n - 1} x^{n} + {\left (b c f - 2 \, a c g\right )} d^{\frac {1}{2} \, n - 1}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{2 \, n} + {\left (b^{3} - 4 \, a b c\right )} n x^{n} + {\left (a b^{2} - 4 \, a^{2} c\right )} n} \]
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Timed out. \[ \int \frac {(d x)^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(d x)^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {{\left (c h x^{2 \, n} + c g x^{\frac {3}{2} \, n} + c f x^{\frac {1}{2} \, n} - a h\right )} \left (d x\right )^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(d x)^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {{\left (c h x^{2 \, n} + c g x^{\frac {3}{2} \, n} + c f x^{\frac {1}{2} \, n} - a h\right )} \left (d x\right )^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d x)^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int \frac {{\left (d\,x\right )}^{\frac {n}{2}-1}\,\left (c\,f\,x^{n/2}-a\,h+c\,g\,x^{\frac {3\,n}{2}}+c\,h\,x^{2\,n}\right )}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}} \,d x \]
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