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

Optimal. Leaf size=95 \[ -\frac{2 x^{1-\frac{n}{2}} (d x)^{\frac{n-2}{2}} \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt{a+b x^n+c x^{2 n}}} \]

[Out]

(-2*x^(1 - n/2)*(d*x)^((-2 + n)/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 = 0.335067, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 63, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032 \[ -\frac{2 x^{1-\frac{n}{2}} (d x)^{\frac{n-2}{2}} \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt{a+b x^n+c x^{2 n}}} \]

Antiderivative was successfully verified.

[In]  Int[((d*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*x^(1 - n/2)*(d*x)^((-2 + n)/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 = 57.959, size = 85, normalized size = 0.89 \[ \frac{x^{- \frac{n}{2} + 1} \left (d x\right )^{\frac{n}{2} - 1} \left (2 c x^{n} \left (b g - 2 c f\right ) + 2 c \left (2 a g - b f\right ) - 2 h x^{\frac{n}{2}} \left (- 4 a c + b^{2}\right )\right )}{n \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{n} + c x^{2 n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*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]

x**(-n/2 + 1)*(d*x)**(n/2 - 1)*(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 = 0.814398, size = 100, normalized size = 1.05 \[ -\frac{2 x^{-n/2} (d x)^{n/2} \left (2 c \left (c f x^n-a \left (g+2 h x^{n/2}\right )\right )+b^2 h x^{n/2}+b c \left (f-g x^n\right )\right )}{d n \left (b^2-4 a c\right ) \sqrt{a+x^n \left (b+c x^n\right )}} \]

Antiderivative was successfully verified.

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

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{1 \left ( dx \right ) ^{-1+{\frac{n}{2}}} \left ( -ah+cf{x}^{{\frac{n}{2}}}+cg{x}^{{\frac{3\,n}{2}}}+ch{x}^{2\,n} \right ) \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

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)*(d*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)*(d*x)^(1/2*n - 1)/
(c*x^(2*n) + b*x^n + a)^(3/2), x)

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Fricas [A]  time = 0.294091, size = 178, normalized size = 1.87 \[ -\frac{2 \,{\left ({\left (b^{2} - 4 \, a c\right )} d^{\frac{1}{2} \, n - 1} h x^{\frac{1}{2} \, n} +{\left (2 \, c^{2} f - b c g\right )} d^{\frac{1}{2} \, n - 1} x^{n} +{\left (b c f - 2 \, a c g\right )} d^{\frac{1}{2} \, n - 1}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{2 \, n} +{\left (b^{3} - 4 \, a b c\right )} n x^{n} +{\left (a b^{2} - 4 \, a^{2} c\right )} n} \]

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

[Out]

-2*((b^2 - 4*a*c)*d^(1/2*n - 1)*h*x^(1/2*n) + (2*c^2*f - b*c*g)*d^(1/2*n - 1)*x^
n + (b*c*f - 2*a*c*g)*d^(1/2*n - 1))*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((d*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. \[ \int \frac{{\left (c h x^{2 \, n} + c g x^{\frac{3}{2} \, n} + c f x^{\frac{1}{2} \, n} - a h\right )} \left (d x\right )^{\frac{1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \]

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)*(d*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)*(d*x)^(1/2*n - 1)/
(c*x^(2*n) + b*x^n + a)^(3/2), x)