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*(-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 = 0.11, antiderivative size = 75, normalized size of antiderivative = 1.00, 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*}

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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[(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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fricas [A]  time = 0.75, size = 109, normalized size = 1.45 \[ -\frac {2 \, {\left (b c f - 2 \, a c g + {\left (b^{2} - 4 \, a c\right )} h x^{\frac {1}{2} \, n} + {\left (2 \, c^{2} f - b c g\right )} x^{n}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{2 \, n} + {\left (b^{3} - 4 \, a b c\right )} n x^{n} + {\left (a b^{2} - 4 \, a^{2} c\right )} n} \]

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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giac [B]  time = 4.72, size = 187, normalized size = 2.49 \[ -\frac {2 \, {\left (\sqrt {x^{n}} {\left (\frac {{\left (2 \, b^{2} c^{2} f - 8 \, a c^{3} f - b^{3} c g + 4 \, a b c^{2} g\right )} \sqrt {x^{n}}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {b^{4} h - 8 \, a b^{2} c h + 16 \, a^{2} c^{2} h}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} + \frac {b^{3} c f - 4 \, a b c^{2} f - 2 \, a b^{2} c g + 8 \, a^{2} c^{2} g}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{\sqrt {c x^{2 \, n} + b x^{n} + a} n} \]

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]

-2*(sqrt(x^n)*((2*b^2*c^2*f - 8*a*c^3*f - b^3*c*g + 4*a*b*c^2*g)*sqrt(x^n)/(b^4 - 8*a*b^2*c + 16*a^2*c^2) + (b
^4*h - 8*a*b^2*c*h + 16*a^2*c^2*h)/(b^4 - 8*a*b^2*c + 16*a^2*c^2)) + (b^3*c*f - 4*a*b*c^2*f - 2*a*b^2*c*g + 8*
a^2*c^2*g)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))/(sqrt(c*x^(2*n) + b*x^n + a)*n)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (c f \,x^{\frac {n}{2}}+c g \,x^{\frac {3 n}{2}}+c h \,x^{2 n}-a h \right ) x^{\frac {n}{2}-1}}{\left (b \,x^{n}+c \,x^{2 n}+a \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^(1/2*n-1)*(-a*h+c*f*x^(1/2*n)+c*g*x^(3/2*n)+c*h*x^(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 \[ \int \frac {{\left (c h x^{2 \, n} + c g x^{\frac {3}{2} \, n} + c f x^{\frac {1}{2} \, n} - a h\right )} x^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}}\,{d x} \]

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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mupad [B]  time = 2.45, size = 80, normalized size = 1.07 \[ -\frac {2\,b^2\,h\,x^{n/2}-4\,a\,c\,g+2\,b\,c\,f+4\,c^2\,f\,x^n-8\,a\,c\,h\,x^{n/2}-2\,b\,c\,g\,x^n}{\left (b^2\,n-4\,a\,c\,n\right )\,\sqrt {a+b\,x^n+c\,x^{2\,n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b^2*h*x^(n/2) - 4*a*c*g + 2*b*c*f + 4*c^2*f*x^n - 8*a*c*h*x^(n/2) - 2*b*c*g*x^n)/((b^2*n - 4*a*c*n)*(a + b
*x^n + c*x^(2*n))^(1/2))

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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(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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