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

Optimal. Leaf size=45 \[ -\frac{2 \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt{a+c x^n}} \]

[Out]

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

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Rubi [A]  time = 0.07873, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 52, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.019, Rules used = {1816} \[ -\frac{2 \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt{a+c x^n}} \]

Antiderivative was successfully verified.

[In]

Int[(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 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{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}}\\ \end{align*}

Mathematica [A]  time = 0.222079, size = 45, normalized size = 1. \[ \frac{2 c f x^{n/2}-2 a \left (g+2 h x^{n/4}\right )}{a n \sqrt{a+c x^n}} \]

Antiderivative was successfully verified.

[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])

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-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(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)

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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 )} x^{\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(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)

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Fricas [A]  time = 1.3892, size = 107, normalized size = 2.38 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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(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 )} x^{\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(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)*x^(1/4*n - 1)/(c*x^n + a)^(3/2), x)