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

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 = 0.176618, 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 \[ -\frac{2 \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt{a+b x^n+c x^{2 n}}} \]

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)])

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Rubi in Sympy [A]  time = 29.3354, size = 70, normalized size = 0.93 \[ \frac{2 c x^{n} \left (b g - 2 c f\right ) + 2 c \left (2 a g - b f\right ) - 2 h x^{\frac{n}{2}} \left (- 4 a c + b^{2}\right )}{n \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{n} + c x^{2 n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

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

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Mathematica [A]  time = 0.167296, size = 84, normalized size = 1.12 \[ -\frac{2 \left (-2 a c g-4 a c h x^{n/2}+b^2 h x^{n/2}+b c f-b c g x^n+2 c^2 f x^n\right )}{n \left (b^2-4 a c\right ) \sqrt{a+b x^n+c x^{2 n}}} \]

Antiderivative was successfully verified.

[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]

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

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int{1{x}^{-1+{\frac{n}{2}}} \left ( -ah+cf{x}^{{\frac{n}{2}}}+cg{x}^{{\frac{3\,n}{2}}}+ch{x}^{2\,n} \right ) \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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, 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 = 0.292135, size = 147, normalized size = 1.96 \[ -\frac{2 \,{\left (b c f - 2 \, a c g +{\left (b^{2} - 4 \, a c\right )} h x^{\frac{1}{2} \, n} +{\left (2 \, c^{2} f - b c g\right )} x^{n}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{2 \, n} +{\left (b^{3} - 4 \, a b c\right )} n x^{n} +{\left (a b^{2} - 4 \, a^{2} c\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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, 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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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, 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)