Integrand size = 52, antiderivative size = 60 \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=-\frac {2 g}{n \sqrt {a+c x^n}}-\frac {2 x^{n/4} \left (2 a h-c f x^{n/4}\right )}{a n \sqrt {a+c x^n}} \] Output:
-2*g/n/(a+c*x^n)^(1/2)-2*x^(1/4*n)*(2*a*h-c*f*x^(1/4*n))/a/n/(a+c*x^n)^(1/ 2)
Time = 0.94 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.75 \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\frac {2 c f x^{n/2}-2 a \left (g+2 h x^{n/4}\right )}{a n \sqrt {a+c x^n}} \] Input:
Integrate[(x^(-1 + n/4)*(-(a*h) + c*f*x^(n/4) + c*g*x^((3*n)/4) + c*h*x^n) )/(a + c*x^n)^(3/2),x]
Output:
(2*c*f*x^(n/2) - 2*a*(g + 2*h*x^(n/4)))/(a*n*Sqrt[a + c*x^n])
Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.75, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2356}
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.
\(\displaystyle \int \frac {x^{\frac {n}{4}-1} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2356 |
\(\displaystyle -\frac {2 \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt {a+c x^n}}\) |
Input:
Int[(x^(-1 + n/4)*(-(a*h) + c*f*x^(n/4) + c*g*x^((3*n)/4) + c*h*x^n))/(a + c*x^n)^(3/2),x]
Output:
(-2*(a*g + 2*a*h*x^(n/4) - c*f*x^(n/2)))/(a*n*Sqrt[a + c*x^n])
Int[((x_)^(m_.)*((e_) + (h_.)*(x_)^(n_.) + (f_.)*(x_)^(q_.) + (g_.)*(x_)^(r _.)))/((a_) + (c_.)*(x_)^(n_.))^(3/2), x_Symbol] :> Simp[-(2*a*g + 4*a*h*x^ (n/4) - 2*c*f*x^(n/2))/(a*c*n*Sqrt[a + c*x^n]), x] /; FreeQ[{a, c, e, f, g, h, m, n}, x] && EqQ[q, n/4] && EqQ[r, 3*(n/4)] && EqQ[4*m - n + 4, 0] && E qQ[c*e + a*h, 0]
\[\int \frac {x^{-1+\frac {n}{4}} \left (-a h +c f \,x^{\frac {n}{4}}+c g \,x^{\frac {3 n}{4}}+c h \,x^{n}\right )}{\left (a +c \,x^{n}\right )^{\frac {3}{2}}}d x\]
Input:
int(x^(-1+1/4*n)*(-a*h+c*f*x^(1/4*n)+c*g*x^(3/4*n)+c*h*x^n)/(a+c*x^n)^(3/2 ),x)
Output:
int(x^(-1+1/4*n)*(-a*h+c*f*x^(1/4*n)+c*g*x^(3/4*n)+c*h*x^n)/(a+c*x^n)^(3/2 ),x)
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.80 \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\frac {2 \, {\left (c f x^{\frac {1}{2} \, n} - 2 \, a h x^{\frac {1}{4} \, n} - a g\right )} \sqrt {c x^{n} + a}}{a c n x^{n} + a^{2} n} \] Input:
integrate(x^(-1+1/4*n)*(-a*h+c*f*x^(1/4*n)+c*g*x^(3/4*n)+c*h*x^n)/(a+c*x^n )^(3/2),x, algorithm="fricas")
Output:
2*(c*f*x^(1/2*n) - 2*a*h*x^(1/4*n) - a*g)*sqrt(c*x^n + a)/(a*c*n*x^n + a^2 *n)
Timed out. \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x**(-1+1/4*n)*(-a*h+c*f*x**(1/4*n)+c*g*x**(3/4*n)+c*h*x**n)/(a+c *x**n)**(3/2),x)
Output:
Timed out
\[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\int { \frac {{\left (c g x^{\frac {3}{4} \, n} + c f x^{\frac {1}{4} \, n} + c h x^{n} - a h\right )} x^{\frac {1}{4} \, n - 1}}{{\left (c x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^(-1+1/4*n)*(-a*h+c*f*x^(1/4*n)+c*g*x^(3/4*n)+c*h*x^n)/(a+c*x^n )^(3/2),x, algorithm="maxima")
Output:
integrate((c*g*x^(3/4*n) + c*f*x^(1/4*n) + c*h*x^n - a*h)*x^(1/4*n - 1)/(c *x^n + a)^(3/2), x)
Time = 4.43 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.65 \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left (\frac {c f {\left (x^{n}\right )}^{\frac {1}{4}}}{a} - 2 \, h\right )} {\left (x^{n}\right )}^{\frac {1}{4}} - g\right )}}{\sqrt {c x^{n} + a} n} \] Input:
integrate(x^(-1+1/4*n)*(-a*h+c*f*x^(1/4*n)+c*g*x^(3/4*n)+c*h*x^n)/(a+c*x^n )^(3/2),x, algorithm="giac")
Output:
2*((c*f*(x^n)^(1/4)/a - 2*h)*(x^n)^(1/4) - g)/(sqrt(c*x^n + a)*n)
Time = 21.40 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.65 \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=-\frac {2\,\left (a\,g-c\,f\,x^{n/2}+2\,a\,h\,x^{n/4}\right )}{a\,n\,\sqrt {a+c\,x^n}} \] Input:
int((x^(n/4 - 1)*(c*h*x^n - a*h + c*f*x^(n/4) + c*g*x^((3*n)/4)))/(a + c*x ^n)^(3/2),x)
Output:
-(2*(a*g - c*f*x^(n/2) + 2*a*h*x^(n/4)))/(a*n*(a + c*x^n)^(1/2))
\[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\frac {2 x^{\frac {n}{2}} \sqrt {x^{n} c +a}\, c f -2 \sqrt {x^{n} c +a}\, a g +x^{n} \left (\int \frac {x^{\frac {5 n}{4}} \sqrt {x^{n} c +a}}{x^{2 n} c^{2} x +2 x^{n} a c x +a^{2} x}d x \right ) a \,c^{2} h n -x^{n} \left (\int \frac {x^{\frac {n}{4}} \sqrt {x^{n} c +a}}{x^{2 n} c^{2} x +2 x^{n} a c x +a^{2} x}d x \right ) a^{2} c h n +\left (\int \frac {x^{\frac {5 n}{4}} \sqrt {x^{n} c +a}}{x^{2 n} c^{2} x +2 x^{n} a c x +a^{2} x}d x \right ) a^{2} c h n -\left (\int \frac {x^{\frac {n}{4}} \sqrt {x^{n} c +a}}{x^{2 n} c^{2} x +2 x^{n} a c x +a^{2} x}d x \right ) a^{3} h n}{a n \left (x^{n} c +a \right )} \] Input:
int(x^(-1+1/4*n)*(-a*h+c*f*x^(1/4*n)+c*g*x^(3/4*n)+c*h*x^n)/(a+c*x^n)^(3/2 ),x)
Output:
(2*x**(n/2)*sqrt(x**n*c + a)*c*f - 2*sqrt(x**n*c + a)*a*g + x**n*int((x**( (5*n)/4)*sqrt(x**n*c + a))/(x**(2*n)*c**2*x + 2*x**n*a*c*x + a**2*x),x)*a* c**2*h*n - x**n*int((x**(n/4)*sqrt(x**n*c + a))/(x**(2*n)*c**2*x + 2*x**n* a*c*x + a**2*x),x)*a**2*c*h*n + int((x**((5*n)/4)*sqrt(x**n*c + a))/(x**(2 *n)*c**2*x + 2*x**n*a*c*x + a**2*x),x)*a**2*c*h*n - int((x**(n/4)*sqrt(x** n*c + a))/(x**(2*n)*c**2*x + 2*x**n*a*c*x + a**2*x),x)*a**3*h*n)/(a*n*(x** n*c + a))