\(\int \frac {(d x)^{-1+\frac {n}{4}} (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n)}{(a+c x^n)^{3/2}} \, dx\) [13]

Optimal result

Integrand size = 54, antiderivative size = 65 \[ \int \frac {(d x)^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=-\frac {2 x^{1-\frac {n}{4}} (d x)^{\frac {1}{4} (-4+n)} \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt {a+c x^n}} \]

[Out]

-2*x^(1-1/4*n)*(d*x)^(-1+1/4*n)*(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 = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {1831, 1830} \[ \int \frac {(d x)^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=-\frac {2 x^{1-\frac {n}{4}} (d x)^{\frac {n-4}{4}} \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt {a+c x^n}} \]

[In]

Int[((d*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*x^(1 - n/4)*(d*x)^((-4 + n)/4)*(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]

Rule 1831

Int[(((d_)*(x_))^(m_.)*((e_) + (h_.)*(x_)^(n_.) + (f_.)*(x_)^(q_.) + (g_.)*(x_)^(r_.)))/((a_) + (c_.)*(x_)^(n_
.))^(3/2), x_Symbol] :> Dist[(d*x)^m/x^m, Int[x^m*((e + f*x^(n/4) + g*x^((3*n)/4) + h*x^n)/(a + c*x^n)^(3/2)),
x], x] /; FreeQ[{a, c, d, e, f, g, h, m, n}, x] && EqQ[4*m - n + 4, 0] && EqQ[q, n/4] && EqQ[r, 3*(n/4)] && E
qQ[c*e + a*h, 0]

Rubi steps \begin{align*} \text {integral}& = \left (x^{1-\frac {n}{4}} (d x)^{-1+\frac {n}{4}}\right ) \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^{1-\frac {n}{4}} (d x)^{\frac {1}{4} (-4+n)} \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 = 0.67 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {(d x)^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\frac {2 x^{-n/4} (d x)^{n/4} \left (c f x^{n/2}-a \left (g+2 h x^{n/4}\right )\right )}{a d n \sqrt {a+c x^n}} \]

[In]

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

Maple [F]

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

[In]

int((d*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((d*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 = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.06 \[ \int \frac {(d x)^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\frac {2 \, {\left (c d^{\frac {1}{4} \, n - 1} f x^{\frac {1}{2} \, n} - 2 \, a d^{\frac {1}{4} \, n - 1} h x^{\frac {1}{4} \, n} - a d^{\frac {1}{4} \, n - 1} g\right )} \sqrt {c x^{n} + a}}{a c n x^{n} + a^{2} n} \]

[In]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 117.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.46 \[ \int \frac {(d x)^{-1+\frac {n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx=\frac {2 \sqrt {c} d^{\frac {n}{4} - 1} f}{a n \sqrt {\frac {a x^{- n}}{c} + 1}} - \frac {2 d^{\frac {n}{4} - 1} g}{\sqrt {a} n \sqrt {1 + \frac {c x^{n}}{a}}} - \frac {d^{\frac {n}{4} - 1} h x^{\frac {n}{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (\frac {5}{4}\right )} + \frac {c d^{\frac {n}{4} - 1} h x^{\frac {5 n}{4}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{n} e^{i \pi }}{a}} \right )}}{a^{\frac {3}{2}} n \Gamma \left (\frac {9}{4}\right )} \]

[In]

integrate((d*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]

2*sqrt(c)*d**(n/4 - 1)*f/(a*n*sqrt(a/(c*x**n) + 1)) - 2*d**(n/4 - 1)*g/(sqrt(a)*n*sqrt(1 + c*x**n/a)) - d**(n/
4 - 1)*h*x**(n/4)*gamma(1/4)*hyper((1/4, 3/2), (5/4,), c*x**n*exp_polar(I*pi)/a)/(sqrt(a)*n*gamma(5/4)) + c*d*
*(n/4 - 1)*h*x**(5*n/4)*gamma(5/4)*hyper((5/4, 3/2), (9/4,), c*x**n*exp_polar(I*pi)/a)/(a**(3/2)*n*gamma(9/4))

Maxima [F]

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

[In]

integrate((d*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)*(d*x)^(1/4*n - 1)/(c*x^n + a)^(3/2), x)

Giac [F]

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

[In]

integrate((d*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]

integrate((c*g*x^(3/4*n) + c*f*x^(1/4*n) + c*h*x^n - a*h)*(d*x)^(1/4*n - 1)/(c*x^n + a)^(3/2), x)

Mupad [F(-1)]

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

[In]

int(((d*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]

int(((d*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)