\(\int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{(a+b x^n+c x^{2 n})^{3/2}} \, dx\) [15]

Optimal result

Integrand size = 63, antiderivative size = 87 \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 h x^{n/2}}{n \sqrt {a+b x^n+c x^{2 n}}}-\frac {2 c \left (b f-2 a g+(2 c f-b g) x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \] Output:

-2*h*x^(1/2*n)/n/(a+b*x^n+c*x^(2*n))^(1/2)-2*c*(b*f-2*a*g+(-b*g+2*c*f)*x^n 
)/(-4*a*c+b^2)/n/(a+b*x^n+c*x^(2*n))^(1/2)
 

Mathematica [A] (verified)

Time = 2.94 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97 \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \left (b c f-2 a c g+b^2 h x^{n/2}-4 a c h x^{n/2}+2 c^2 f x^n-b c g x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \] Input:

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]
 

Output:

(-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)])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86, 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

\(\displaystyle -\frac {2 \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt {a+b x^n+c x^{2 n}}}\)

Input:

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]
 

Output:

(-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)])
 

Defintions of rubi rules used

rule 2029

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] 
)
 

rule 2289

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]
 

Maple [F]

\[\int \frac {-a h \,x^{-1+\frac {n}{2}}+c f \,x^{-1+n}+c g \,x^{-1+2 n}+c h \,x^{-1+\frac {5 n}{2}}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {3}{2}}}d x\]

Input:

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)
 

Output:

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)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.57 \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c x^{4} x^{2 \, n - 4} + b x^{2} x^{n - 2} + a} {\left ({\left (2 \, c^{2} f - b c g\right )} x^{2} x^{n - 2} + {\left (b^{2} - 4 \, a c\right )} h x x^{\frac {1}{2} \, n - 1} + b c f - 2 \, a c g\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{4} x^{2 \, n - 4} + {\left (b^{3} - 4 \, a b c\right )} n x^{2} x^{n - 2} + {\left (a b^{2} - 4 \, a^{2} c\right )} n} \] Input:

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")
 

Output:

-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)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\text {Timed out} \] Input:

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)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {c h x^{\frac {5}{2} \, n - 1} + c g x^{2 \, n - 1} + c f x^{n - 1} - a h x^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

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")
 

Output:

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)
 

Giac [F]

\[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {c h x^{\frac {5}{2} \, n - 1} + c g x^{2 \, n - 1} + c f x^{n - 1} - a h x^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

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")
 

Output:

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)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int \frac {c\,g\,x^{2\,n-1}-a\,h\,x^{\frac {n}{2}-1}+c\,h\,x^{\frac {5\,n}{2}-1}+c\,f\,x^{n-1}}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}} \,d x \] Input:

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)
 

Output:

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)
 

Reduce [F]

\[ \int \frac {-a h x^{-1+\frac {n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac {5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\text {too large to display} \] Input:

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)
 

Output:

( - 33948915000*x**((25*n)/2)*sqrt(x**(2*n)*c + x**n*b + a)*a*c**9*h + 848 
7228750*x**((25*n)/2)*sqrt(x**(2*n)*c + x**n*b + a)*b**2*c**8*h + 38798760 
000*x**((23*n)/2)*sqrt(x**(2*n)*c + x**n*b + a)*a*b*c**8*h - 9699690000*x* 
*((23*n)/2)*sqrt(x**(2*n)*c + x**n*b + a)*b**3*c**7*h - 164384220000*x**(( 
21*n)/2)*sqrt(x**(2*n)*c + x**n*b + a)*a**2*c**8*h + 181996815000*x**((21* 
n)/2)*sqrt(x**(2*n)*c + x**n*b + a)*a*b**2*c**7*h - 35225190000*x**((21*n) 
/2)*sqrt(x**(2*n)*c + x**n*b + a)*b**4*c**6*h + 140900760000*x**((19*n)/2) 
*sqrt(x**(2*n)*c + x**n*b + a)*a**2*b*c**7*h - 200990790000*x**((19*n)/2)* 
sqrt(x**(2*n)*c + x**n*b + a)*a*b**3*c**6*h + 41441400000*x**((19*n)/2)*sq 
rt(x**(2*n)*c + x**n*b + a)*b**5*c**5*h - 312330018000*x**((17*n)/2)*sqrt( 
x**(2*n)*c + x**n*b + a)*a**3*c**7*h + 626263918644*x**((17*n)/2)*sqrt(x** 
(2*n)*c + x**n*b + a)*a**2*b**2*c**6*h - 347580123072*x**((17*n)/2)*sqrt(x 
**(2*n)*c + x**n*b + a)*a*b**4*c**5*h + 52633692384*x**((17*n)/2)*sqrt(x** 
(2*n)*c + x**n*b + a)*b**6*c**4*h + 178474296000*x**((15*n)/2)*sqrt(x**(2* 
n)*c + x**n*b + a)*a**3*b*c**6*h - 477030591408*x**((15*n)/2)*sqrt(x**(2*n 
)*c + x**n*b + a)*a**2*b**3*c**5*h + 367222720704*x**((15*n)/2)*sqrt(x**(2 
*n)*c + x**n*b + a)*a*b**5*c**4*h - 64779929088*x**((15*n)/2)*sqrt(x**(2*n 
)*c + x**n*b + a)*b**7*c**3*h - 283936380000*x**((13*n)/2)*sqrt(x**(2*n)*c 
 + x**n*b + a)*a**4*c**6*h + 843283916760*x**((13*n)/2)*sqrt(x**(2*n)*c + 
x**n*b + a)*a**3*b**2*c**5*h - 806242031280*x**((13*n)/2)*sqrt(x**(2*n)...