Integrand size = 63, antiderivative size = 75 \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \left (c (b f-2 a g)+\left (b^2-4 a c\right ) h x^{n/2}+c (2 c f-b g) x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \]
-2*(c*(-2*a*g+b*f)+(-4*a*c+b^2)*h*x^(1/2*n)+c*(-b*g+2*c*f)*x^n)/(-4*a*c+b^ 2)/n/(a+b*x^n+c*x^(2*n))^(1/2)
Time = 2.15 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.12 \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \left (b c f-2 a c g+b^2 h x^{n/2}-4 a c h x^{n/2}+2 c^2 f x^n-b c g x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \]
Integrate[(-(a*h*x^(-1 + n/2)) + c*f*x^(-1 + n) + c*g*x^(-1 + 2*n) + c*h*x ^(-1 + (5*n)/2))/(a + b*x^n + c*x^(2*n))^(3/2),x]
(-2*(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))/((b^2 - 4*a*c)*n*Sqrt[a + b*x^n + c*x^(2*n)])
Time = 0.38 (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 = {2029, 2289}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {-a h x^{\frac {n}{2}-1}+c f x^{n-1}+c g x^{2 n-1}+c h x^{\frac {5 n}{2}-1}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2029 |
\(\displaystyle \int \frac {x^{\frac {n}{2}-1} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}}dx\) |
\(\Big \downarrow \) 2289 |
\(\displaystyle -\frac {2 \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt {a+b x^n+c x^{2 n}}}\) |
Int[(-(a*h*x^(-1 + n/2)) + c*f*x^(-1 + n) + c*g*x^(-1 + 2*n) + c*h*x^(-1 + (5*n)/2))/(a + b*x^n + c*x^(2*n))^(3/2),x]
(-2*(c*(b*f - 2*a*g) + (b^2 - 4*a*c)*h*x^(n/2) + c*(2*c*f - b*g)*x^n))/((b ^2 - 4*a*c)*n*Sqrt[a + b*x^n + c*x^(2*n)])
3.1.10.3.1 Defintions of rubi rules used
Int[(Fx_.)*((d_.)*(x_)^(q_.) + (a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)* (x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r) + d*x^(q - r))^p*Fx, x] /; FreeQ[{a, b, c, d, r, s, t, q}, x] && IntegerQ[p ] && PosQ[s - r] && PosQ[t - r] && PosQ[q - r] && !(EqQ[p, 1] && EqQ[u, 1] )
Int[((x_)^(m_.)*((e_) + (f_.)*(x_)^(q_.) + (g_.)*(x_)^(r_.) + (h_.)*(x_)^(s _.)))/((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(3/2), x_Symbol] :> Sim p[-(2*c*(b*f - 2*a*g) + 2*h*(b^2 - 4*a*c)*x^(n/2) + 2*c*(2*c*f - b*g)*x^n)/ (c*n*(b^2 - 4*a*c)*Sqrt[a + b*x^n + c*x^(2*n)]), x] /; FreeQ[{a, b, c, e, f , g, h, m, n}, x] && EqQ[n2, 2*n] && EqQ[q, n/2] && EqQ[r, 3*(n/2)] && EqQ[ s, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*m - n + 2, 0] && EqQ[c*e + a*h, 0]
\[\int \frac {-a h \,x^{-1+\frac {n}{2}}+c f \,x^{-1+n}+c g \,x^{-1+2 n}+c h \,x^{-1+\frac {5 n}{2}}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {3}{2}}}d x\]
int((-a*h*x^(-1+1/2*n)+c*f*x^(-1+n)+c*g*x^(-1+2*n)+c*h*x^(-1+5/2*n))/(a+b* x^n+c*x^(2*n))^(3/2),x)
int((-a*h*x^(-1+1/2*n)+c*f*x^(-1+n)+c*g*x^(-1+2*n)+c*h*x^(-1+5/2*n))/(a+b* x^n+c*x^(2*n))^(3/2),x)
Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.83 \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c x^{4} x^{2 \, n - 4} + b x^{2} x^{n - 2} + a} {\left ({\left (2 \, c^{2} f - b c g\right )} x^{2} x^{n - 2} + {\left (b^{2} - 4 \, a c\right )} h x x^{\frac {1}{2} \, n - 1} + b c f - 2 \, a c g\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{4} x^{2 \, n - 4} + {\left (b^{3} - 4 \, a b c\right )} n x^{2} x^{n - 2} + {\left (a b^{2} - 4 \, a^{2} c\right )} n} \]
integrate((-a*h*x^(-1+1/2*n)+c*f*x^(-1+n)+c*g*x^(-1+2*n)+c*h*x^(-1+5/2*n)) /(a+b*x^n+c*x^(2*n))^(3/2),x, algorithm="fricas")
-2*sqrt(c*x^4*x^(2*n - 4) + b*x^2*x^(n - 2) + a)*((2*c^2*f - b*c*g)*x^2*x^ (n - 2) + (b^2 - 4*a*c)*h*x*x^(1/2*n - 1) + b*c*f - 2*a*c*g)/((b^2*c - 4*a *c^2)*n*x^4*x^(2*n - 4) + (b^3 - 4*a*b*c)*n*x^2*x^(n - 2) + (a*b^2 - 4*a^2 *c)*n)
Timed out. \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\text {Timed out} \]
integrate((-a*h*x**(-1+1/2*n)+c*f*x**(-1+n)+c*g*x**(-1+2*n)+c*h*x**(-1+5/2 *n))/(a+b*x**n+c*x**(2*n))**(3/2),x)
\[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {c h x^{\frac {5}{2} \, n - 1} + c g x^{2 \, n - 1} + c f x^{n - 1} - a h x^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \]
integrate((-a*h*x^(-1+1/2*n)+c*f*x^(-1+n)+c*g*x^(-1+2*n)+c*h*x^(-1+5/2*n)) /(a+b*x^n+c*x^(2*n))^(3/2),x, algorithm="maxima")
integrate((c*h*x^(5/2*n - 1) + c*g*x^(2*n - 1) + c*f*x^(n - 1) - a*h*x^(1/ 2*n - 1))/(c*x^(2*n) + b*x^n + a)^(3/2), x)
\[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {c h x^{\frac {5}{2} \, n - 1} + c g x^{2 \, n - 1} + c f x^{n - 1} - a h x^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \]
integrate((-a*h*x^(-1+1/2*n)+c*f*x^(-1+n)+c*g*x^(-1+2*n)+c*h*x^(-1+5/2*n)) /(a+b*x^n+c*x^(2*n))^(3/2),x, algorithm="giac")
integrate((c*h*x^(5/2*n - 1) + c*g*x^(2*n - 1) + c*f*x^(n - 1) - a*h*x^(1/ 2*n - 1))/(c*x^(2*n) + b*x^n + a)^(3/2), x)
Timed out. \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int \frac {c\,g\,x^{2\,n-1}-a\,h\,x^{\frac {n}{2}-1}+c\,h\,x^{\frac {5\,n}{2}-1}+c\,f\,x^{n-1}}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}} \,d x \]
int((c*g*x^(2*n - 1) - a*h*x^(n/2 - 1) + c*h*x^((5*n)/2 - 1) + c*f*x^(n - 1))/(a + b*x^n + c*x^(2*n))^(3/2),x)