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