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

Optimal. Leaf size=75 \[ -\frac{2 \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt{a+b x^n+c x^{2 n}}} \]

[Out]

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

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Rubi [A]  time = 0.10818, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 61, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.016, Rules used = {1753} \[ -\frac{2 \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt{a+b x^n+c x^{2 n}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1753

Int[((x_)^(m_.)*((e_) + (f_.)*(x_)^(q_.) + (g_.)*(x_)^(r_.) + (h_.)*(x_)^(s_.)))/((a_) + (b_.)*(x_)^(n_.) + (c
_.)*(x_)^(n2_.))^(3/2), x_Symbol] :> -Simp[(2*c*(b*f - 2*a*g) + 2*h*(b^2 - 4*a*c)*x^(n/2) + 2*c*(2*c*f - b*g)*
x^n)/(c*n*(b^2 - 4*a*c)*Sqrt[a + b*x^n + c*x^(2*n)]), x] /; FreeQ[{a, b, c, e, f, g, h, m, n}, x] && EqQ[n2, 2
*n] && EqQ[q, n/2] && EqQ[r, (3*n)/2] && EqQ[s, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*m - n + 2, 0] && EqQ[c*e
+ a*h, 0]

Rubi steps

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

Mathematica [F]  time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+{\frac{n}{2}}} \left ( -ah+cf{x}^{{\frac{n}{2}}}+cg{x}^{{\frac{3\,n}{2}}}+ch{x}^{2\,n} \right ) \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]  time = 1.50595, size = 240, normalized size = 3.2 \begin{align*} -\frac{2 \,{\left (b c f - 2 \, a c g +{\left (b^{2} - 4 \, a c\right )} h x^{\frac{1}{2} \, n} +{\left (2 \, c^{2} f - b c g\right )} x^{n}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{2 \, n} +{\left (b^{3} - 4 \, a b c\right )} n x^{n} +{\left (a b^{2} - 4 \, a^{2} c\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(b*c*f - 2*a*c*g + (b^2 - 4*a*c)*h*x^(1/2*n) + (2*c^2*f - b*c*g)*x^n)*sqrt(c*x^(2*n) + b*x^n + a)/((b^2*c -
4*a*c^2)*n*x^(2*n) + (b^3 - 4*a*b*c)*n*x^n + (a*b^2 - 4*a^2*c)*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+1/2*n)*(-a*h+c*f*x**(1/2*n)+c*g*x**(3/2*n)+c*h*x**(2*n))/(a+b*x**n+c*x**(2*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 h x^{2 \, n} + c g x^{\frac{3}{2} \, n} + c f x^{\frac{1}{2} \, n} - a h\right )} x^{\frac{1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b 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+1/2*n)*(-a*h+c*f*x^(1/2*n)+c*g*x^(3/2*n)+c*h*x^(2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x, algorithm="
giac")

[Out]

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