Integrand size = 61, antiderivative size = 75 \[ \int \frac {x^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \left (c (b f-2 a g)+\left (b^2-4 a c\right ) h x^{n/2}+c (2 c f-b g) x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \]
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Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {1767} \[ \int \frac {x^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt {a+b x^n+c x^{2 n}}} \]
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Rule 1767
Rubi steps \begin{align*} \text {integral}& = -\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 = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.12 \[ \int \frac {x^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \left (b c f-2 a c g+b^2 h x^{n/2}-4 a c h x^{n/2}+2 c^2 f x^n-b c g x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \]
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\[\int \frac {x^{-1+\frac {n}{2}} \left (-a h +c f \,x^{\frac {n}{2}}+c g \,x^{\frac {3 n}{2}}+c h \,x^{2 n}\right )}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {3}{2}}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.45 \[ \int \frac {x^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \, {\left (b c f - 2 \, a c g + {\left (b^{2} - 4 \, a c\right )} h x^{\frac {1}{2} \, n} + {\left (2 \, c^{2} f - b c g\right )} x^{n}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{2 \, n} + {\left (b^{3} - 4 \, a b c\right )} n x^{n} + {\left (a b^{2} - 4 \, a^{2} c\right )} n} \]
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Timed out. \[ \int \frac {x^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {{\left (c h x^{2 \, n} + c g x^{\frac {3}{2} \, n} + c f x^{\frac {1}{2} \, n} - a h\right )} x^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (71) = 142\).
Time = 1.00 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.49 \[ \int \frac {x^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \, {\left (\sqrt {x^{n}} {\left (\frac {{\left (2 \, b^{2} c^{2} f - 8 \, a c^{3} f - b^{3} c g + 4 \, a b c^{2} g\right )} \sqrt {x^{n}}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {b^{4} h - 8 \, a b^{2} c h + 16 \, a^{2} c^{2} h}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} + \frac {b^{3} c f - 4 \, a b c^{2} f - 2 \, a b^{2} c g + 8 \, a^{2} c^{2} g}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{\sqrt {c x^{2 \, n} + b x^{n} + a} n} \]
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Time = 8.88 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.07 \[ \int \frac {x^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2\,b^2\,h\,x^{n/2}-4\,a\,c\,g+2\,b\,c\,f+4\,c^2\,f\,x^n-8\,a\,c\,h\,x^{n/2}-2\,b\,c\,g\,x^n}{\left (b^2\,n-4\,a\,c\,n\right )\,\sqrt {a+b\,x^n+c\,x^{2\,n}}} \]
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