Optimal. Leaf size=45 \[ -\frac{2 \left (a g+2 a h x^{n/4}-b f x^{n/2}\right )}{a n \sqrt{a+b x^n}} \]
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Rubi [A] time = 0.655136, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 58, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034 \[ -\frac{2 \left (a g+2 a h x^{n/4}-b f x^{n/2}\right )}{a n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[(-(a*h*x^(-1 + n/4)) + b*f*x^(-1 + n/2) + b*g*x^(-1 + n) + b*h*x^(-1 + (5*n)/4))/(a + b*x^n)^(3/2),x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-a*h*x**(-1+1/4*n)+b*f*x**(-1+1/2*n)+b*g*x**(-1+n)+b*h*x**(-1+5/4*n))/(a+b*x**n)**(3/2),x)
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Mathematica [A] time = 0.139775, size = 45, normalized size = 1. \[ -\frac{2 \left (a g+2 a h x^{n/4}-b f x^{n/2}\right )}{a n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[(-(a*h*x^(-1 + n/4)) + b*f*x^(-1 + n/2) + b*g*x^(-1 + n) + b*h*x^(-1 + (5*n)/4))/(a + b*x^n)^(3/2),x]
[Out]
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Maple [F] time = 0.125, size = 0, normalized size = 0. \[ \int{1 \left ( -ah{x}^{-1+{\frac{n}{4}}}+bf{x}^{-1+{\frac{n}{2}}}+bg{x}^{-1+n}+bh{x}^{-1+{\frac{5\,n}{4}}} \right ) \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-a*h*x^(-1+1/4*n)+b*f*x^(-1+1/2*n)+b*g*x^(-1+n)+b*h*x^(-1+5/4*n))/(a+b*x^n)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b h x^{\frac{5}{4} \, n - 1} + b g x^{n - 1} + b f x^{\frac{1}{2} \, n - 1} - a h x^{\frac{1}{4} \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*h*x^(5/4*n - 1) + b*g*x^(n - 1) + b*f*x^(1/2*n - 1) - a*h*x^(1/4*n - 1))/(b*x^n + a)^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.232558, size = 89, normalized size = 1.98 \[ \frac{2 \, \sqrt{b x^{4} x^{n - 4} + a}{\left (b f x^{2} x^{\frac{1}{2} \, n - 2} - 2 \, a h x x^{\frac{1}{4} \, n - 1} - a g\right )}}{a b n x^{4} x^{n - 4} + a^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*h*x^(5/4*n - 1) + b*g*x^(n - 1) + b*f*x^(1/2*n - 1) - a*h*x^(1/4*n - 1))/(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((-a*h*x**(-1+1/4*n)+b*f*x**(-1+1/2*n)+b*g*x**(-1+n)+b*h*x**(-1+5/4*n))/(a+b*x**n)**(3/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b h x^{\frac{5}{4} \, n - 1} + b g x^{n - 1} + b f x^{\frac{1}{2} \, n - 1} - a h x^{\frac{1}{4} \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*h*x^(5/4*n - 1) + b*g*x^(n - 1) + b*f*x^(1/2*n - 1) - a*h*x^(1/4*n - 1))/(b*x^n + a)^(3/2),x, algorithm="giac")
[Out]