3.6.85 \(\int \frac {-a h x^{-1+\frac {n}{4}}+b f x^{-1+\frac {n}{2}}+b g x^{-1+n}+b h x^{-1+\frac {5 n}{4}}}{(a+b x^n)^{3/2}} \, dx\) [585]

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^(1/4*n)-b*f*x^(1/2*n))/a/n/(a+b*x^n)^(1/2)

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Rubi [A]
time = 0.31, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 = {6873, 1830} \begin {gather*} -\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 {gather*}

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 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 6873

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

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*}

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Mathematica [A]
time = 0.67, size = 45, normalized size = 1.00 \begin {gather*} \frac {2 b f x^{n/2}-2 a \left (g+2 h x^{n/4}\right )}{a n \sqrt {a+b x^n}} \end {gather*}

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 = 0.08, size = 0, normalized size = 0.00 \[\int \frac {-a h \,x^{-1+\frac {n}{4}}+b f \,x^{-1+\frac {n}{2}}+b g \,x^{-1+n}+b h \,x^{-1+\frac {5 n}{4}}}{\left (a +b \,x^{n}\right )^{\frac {3}{2}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.38, size = 66, normalized size = 1.47 \begin {gather*} \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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 8010 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {b\,f\,x^{\frac {n}{2}-1}-a\,h\,x^{\frac {n}{4}-1}+b\,h\,x^{\frac {5\,n}{4}-1}+b\,g\,x^{n-1}}{{\left (a+b\,x^n\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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