3.585 \(\int \frac{-a h x^{-1+\frac{n}{4}}+b f x^{-1+\frac{n}{2}}+b g x^{-1+n}+b h x^{-1+\frac{5 n}{4}}}{(a+b x^n)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.394729, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 58, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {6741, 1816} \[ -\frac{2 \left (a g+2 a h x^{n/4}-b f x^{n/2}\right )}{a n \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[(-(a*h*x^(-1 + n/4)) + b*f*x^(-1 + n/2) + b*g*x^(-1 + n) + b*h*x^(-1 + (5*n)/4))/(a + b*x^n)^(3/2),x]

[Out]

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

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{-a h x^{-1+\frac{n}{4}}+b f x^{-1+\frac{n}{2}}+b g x^{-1+n}+b h x^{-1+\frac{5 n}{4}}}{\left (a+b x^n\right )^{3/2}} \, dx &=\int \frac{x^{-1+\frac{n}{4}} \left (-a h+b f x^{n/4}+b g x^{3 n/4}+b h x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx\\ &=-\frac{2 \left (a g+2 a h x^{n/4}-b f x^{n/2}\right )}{a n \sqrt{a+b x^n}}\\ \end{align*}

Mathematica [A]  time = 0.21546, size = 45, normalized size = 1. \[ \frac{2 b f x^{n/2}-2 a \left (g+2 h x^{n/4}\right )}{a n \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-(a*h*x^(-1 + n/4)) + b*f*x^(-1 + n/2) + b*g*x^(-1 + n) + b*h*x^(-1 + (5*n)/4))/(a + b*x^n)^(3/2),x
]

[Out]

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

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Maple [F]  time = 0.487, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -ah{x}^{-1+{\frac{n}{4}}}+bf{x}^{-1+{\frac{n}{2}}}+bg{x}^{-1+n}+bh{x}^{-1+{\frac{5\,n}{4}}} \right ) \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*h*x^(-1+1/4*n)+b*f*x^(-1+1/2*n)+b*g*x^(-1+n)+b*h*x^(-1+5/4*n))/(a+b*x^n)^(3/2),x)

[Out]

int((-a*h*x^(-1+1/4*n)+b*f*x^(-1+1/2*n)+b*g*x^(-1+n)+b*h*x^(-1+5/4*n))/(a+b*x^n)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b h x^{\frac{5}{4} \, n - 1} + b g x^{n - 1} + b f x^{\frac{1}{2} \, n - 1} - a h x^{\frac{1}{4} \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*h*x^(-1+1/4*n)+b*f*x^(-1+1/2*n)+b*g*x^(-1+n)+b*h*x^(-1+5/4*n))/(a+b*x^n)^(3/2),x, algorithm="max
ima")

[Out]

integrate((b*h*x^(5/4*n - 1) + b*g*x^(n - 1) + b*f*x^(1/2*n - 1) - a*h*x^(1/4*n - 1))/(b*x^n + a)^(3/2), x)

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Fricas [A]  time = 1.895, size = 153, normalized size = 3.4 \begin{align*} \frac{2 \, \sqrt{b x^{4} x^{n - 4} + a}{\left (b f x^{2} x^{\frac{1}{2} \, n - 2} - 2 \, a h x x^{\frac{1}{4} \, n - 1} - a g\right )}}{a b n x^{4} x^{n - 4} + a^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*h*x^(-1+1/4*n)+b*f*x^(-1+1/2*n)+b*g*x^(-1+n)+b*h*x^(-1+5/4*n))/(a+b*x^n)^(3/2),x, algorithm="fri
cas")

[Out]

2*sqrt(b*x^4*x^(n - 4) + a)*(b*f*x^2*x^(1/2*n - 2) - 2*a*h*x*x^(1/4*n - 1) - a*g)/(a*b*n*x^4*x^(n - 4) + 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((-a*h*x**(-1+1/4*n)+b*f*x**(-1+1/2*n)+b*g*x**(-1+n)+b*h*x**(-1+5/4*n))/(a+b*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{b h x^{\frac{5}{4} \, n - 1} + b g x^{n - 1} + b f x^{\frac{1}{2} \, n - 1} - a h x^{\frac{1}{4} \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*h*x^(-1+1/4*n)+b*f*x^(-1+1/2*n)+b*g*x^(-1+n)+b*h*x^(-1+5/4*n))/(a+b*x^n)^(3/2),x, algorithm="gia
c")

[Out]

integrate((b*h*x^(5/4*n - 1) + b*g*x^(n - 1) + b*f*x^(1/2*n - 1) - a*h*x^(1/4*n - 1))/(b*x^n + a)^(3/2), x)