\(\int \frac {x^{-1+\frac {n}{4}} (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n)}{(a+c x^n)^{3/2}} \, dx\) [12]

Optimal result

Integrand size = 52, antiderivative size = 45 \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=-\frac {2 \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt {a+c x^n}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {1830} \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=-\frac {2 \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt {a+c x^n}} \]

[In]

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]

[Out]

(-2*(a*g + 2*a*h*x^(n/4) - c*f*x^(n/2)))/(a*n*Sqrt[a + c*x^n])

Rule 1830

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] && EqQ[c*e + a*h, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt {a+c x^n}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\frac {2 c f x^{n/2}-2 a \left (g+2 h x^{n/4}\right )}{a n \sqrt {a+c x^n}} \]

[In]

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]

[Out]

(2*c*f*x^(n/2) - 2*a*(g + 2*h*x^(n/4)))/(a*n*Sqrt[a + c*x^n])

Maple [F]

\[\int \frac {x^{-1+\frac {n}{4}} \left (-a h +c f \,x^{\frac {n}{4}}+c g \,x^{\frac {3 n}{4}}+c h \,x^{n}\right )}{\left (a +c \,x^{n}\right )^{\frac {3}{2}}}d x\]

[In]

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)

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.07 \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\frac {2 \, {\left (c f x^{\frac {1}{2} \, n} - 2 \, a h x^{\frac {1}{4} \, n} - a g\right )} \sqrt {c x^{n} + a}}{a c n x^{n} + a^{2} n} \]

[In]

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")

[Out]

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)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

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)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\int { \frac {{\left (c g x^{\frac {3}{4} \, n} + c f x^{\frac {1}{4} \, n} + c h x^{n} - a h\right )} x^{\frac {1}{4} \, n - 1}}{{\left (c x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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")

[Out]

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)

Giac [A] (verification not implemented)

none

Time = 5.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left (\frac {c f {\left (x^{n}\right )}^{\frac {1}{4}}}{a} - 2 \, h\right )} {\left (x^{n}\right )}^{\frac {1}{4}} - g\right )}}{\sqrt {c x^{n} + a} n} \]

[In]

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")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \frac {x^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=-\frac {2\,\left (a\,g-c\,f\,x^{n/2}+2\,a\,h\,x^{n/4}\right )}{a\,n\,\sqrt {a+c\,x^n}} \]

[In]

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)

[Out]

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