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

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

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Rubi [A]  time = 0.53, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6741, 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[(-(a*h*x^(-1 + n/2)) + c*f*x^(-1 + n) + c*g*x^(-1 + 2*n) + c*h*x^(-1 + (5*n)/2))/(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]

Rule 6741

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

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Mathematica [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 0.76, size = 137, normalized size = 1.83 \[ -\frac {2 \, \sqrt {c x^{4} x^{2 \, n - 4} + b x^{2} x^{n - 2} + a} {\left ({\left (2 \, c^{2} f - b c g\right )} x^{2} x^{n - 2} + {\left (b^{2} - 4 \, a c\right )} h x x^{\frac {1}{2} \, n - 1} + b c f - 2 \, a c g\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{4} x^{2 \, n - 4} + {\left (b^{3} - 4 \, a b c\right )} n x^{2} x^{n - 2} + {\left (a b^{2} - 4 \, a^{2} c\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*h*x^(-1+1/2*n)+c*f*x^(-1+n)+c*g*x^(-1+2*n)+c*h*x^(-1+5/2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x, algori
thm="fricas")

[Out]

-2*sqrt(c*x^4*x^(2*n - 4) + b*x^2*x^(n - 2) + a)*((2*c^2*f - b*c*g)*x^2*x^(n - 2) + (b^2 - 4*a*c)*h*x*x^(1/2*n
- 1) + b*c*f - 2*a*c*g)/((b^2*c - 4*a*c^2)*n*x^4*x^(2*n - 4) + (b^3 - 4*a*b*c)*n*x^2*x^(n - 2) + (a*b^2 - 4*a
^2*c)*n)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {c h x^{\frac {5}{2} \, n - 1} + c g x^{2 \, n - 1} + c f x^{n - 1} - a h x^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*h*x^(-1+1/2*n)+c*f*x^(-1+n)+c*g*x^(-1+2*n)+c*h*x^(-1+5/2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x, algori
thm="giac")

[Out]

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

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {-a h \,x^{\frac {n}{2}-1}+c f \,x^{n -1}+c g \,x^{2 n -1}+c h \,x^{\frac {5 n}{2}-1}}{\left (b \,x^{n}+c \,x^{2 n}+a \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {c h x^{\frac {5}{2} \, n - 1} + c g x^{2 \, n - 1} + c f x^{n - 1} - a h x^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*h*x^(-1+1/2*n)+c*f*x^(-1+n)+c*g*x^(-1+2*n)+c*h*x^(-1+5/2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x, algori
thm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {c\,g\,x^{2\,n-1}-a\,h\,x^{\frac {n}{2}-1}+c\,h\,x^{\frac {5\,n}{2}-1}+c\,f\,x^{n-1}}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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