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

Optimal. Leaf size=65 \[ -\frac{2 x^{1-\frac{n}{4}} (d x)^{\frac{n-4}{4}} \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt{a+c x^n}} \]

[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])

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Rubi [A]  time = 0.15557, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 54, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {1817, 1816} \[ -\frac{2 x^{1-\frac{n}{4}} (d x)^{\frac{n-4}{4}} \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt{a+c x^n}} \]

Antiderivative was successfully verified.

[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 1817

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]

Rule 1816

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*} \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 &=\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]  time = 0.156561, size = 64, normalized size = 0.98 \[ \frac{2 x^{-n/4} (d x)^{n/4} \left (c f x^{n/2}-a \left (g+2 h x^{n/4}\right )\right )}{a d n \sqrt{a+c x^n}} \]

Antiderivative was successfully verified.

[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])

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx \right ) ^{-1+{\frac{n}{4}}} \left ( -ah+cf{x}^{{\frac{n}{4}}}+cg{x}^{{\frac{3\,n}{4}}}+ch{x}^{n} \right ) \left ( a+c{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.44246, size = 163, normalized size = 2.51 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)