Optimal. Leaf size=25 \[ \frac{1}{7 n \left (a-b x^n-c x^{2 n}\right )^7} \]
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Rubi [A] time = 0.0764887, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{1}{7 n \left (a-b x^n-c x^{2 n}\right )^7} \]
Antiderivative was successfully verified.
[In] Int[(x^(-1 + n)*(b + 2*c*x^n))/(-a + b*x^n + c*x^(2*n))^8,x]
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Rubi in Sympy [A] time = 13.2993, size = 19, normalized size = 0.76 \[ \frac{1}{7 n \left (a - b x^{n} - c x^{2 n}\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+n)*(b+2*c*x**n)/(-a+b*x**n+c*x**(2*n))**8,x)
[Out]
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Mathematica [A] time = 0.0910688, size = 25, normalized size = 1. \[ -\frac{1}{7 n \left (-a+b x^n+c x^{2 n}\right )^7} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(-a + b*x^n + c*x^(2*n))^8,x]
[Out]
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Maple [A] time = 0.096, size = 24, normalized size = 1. \[{\frac{1}{7\,n \left ( -c \left ({x}^{n} \right ) ^{2}-b{x}^{n}+a \right ) ^{7}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+n)*(b+2*c*x^n)/(-a+b*x^n+c*x^(2*n))^8,x)
[Out]
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Maxima [A] time = 1.22849, size = 566, normalized size = 22.64 \[ -\frac{1}{7 \,{\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} - a^{7} n + 7 \,{\left (3 \, b^{2} c^{5} n - a c^{6} n\right )} x^{12 \, n} + 7 \,{\left (5 \, b^{3} c^{4} n - 6 \, a b c^{5} n\right )} x^{11 \, n} + 7 \,{\left (5 \, b^{4} c^{3} n - 15 \, a b^{2} c^{4} n + 3 \, a^{2} c^{5} n\right )} x^{10 \, n} + 7 \,{\left (3 \, b^{5} c^{2} n - 20 \, a b^{3} c^{3} n + 15 \, a^{2} b c^{4} n\right )} x^{9 \, n} + 7 \,{\left (b^{6} c n - 15 \, a b^{4} c^{2} n + 30 \, a^{2} b^{2} c^{3} n - 5 \, a^{3} c^{4} n\right )} x^{8 \, n} +{\left (b^{7} n - 42 \, a b^{5} c n + 210 \, a^{2} b^{3} c^{2} n - 140 \, a^{3} b c^{3} n\right )} x^{7 \, n} - 7 \,{\left (a b^{6} n - 15 \, a^{2} b^{4} c n + 30 \, a^{3} b^{2} c^{2} n - 5 \, a^{4} c^{3} n\right )} x^{6 \, n} + 7 \,{\left (3 \, a^{2} b^{5} n - 20 \, a^{3} b^{3} c n + 15 \, a^{4} b c^{2} n\right )} x^{5 \, n} - 7 \,{\left (5 \, a^{3} b^{4} n - 15 \, a^{4} b^{2} c n + 3 \, a^{5} c^{2} n\right )} x^{4 \, n} + 7 \,{\left (5 \, a^{4} b^{3} n - 6 \, a^{5} b c n\right )} x^{3 \, n} - 7 \,{\left (3 \, a^{5} b^{2} n - a^{6} c n\right )} x^{2 \, n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^n + b)*x^(n - 1)/(c*x^(2*n) + b*x^n - a)^8,x, algorithm="maxima")
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Fricas [A] time = 0.348273, size = 536, normalized size = 21.44 \[ -\frac{1}{7 \,{\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} - a^{7} n + 7 \,{\left (3 \, b^{2} c^{5} - a c^{6}\right )} n x^{12 \, n} + 7 \,{\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} n x^{11 \, n} + 7 \,{\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} n x^{10 \, n} + 7 \,{\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} n x^{9 \, n} + 7 \,{\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} n x^{8 \, n} +{\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} n x^{7 \, n} - 7 \,{\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} n x^{6 \, n} + 7 \,{\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} n x^{5 \, n} - 7 \,{\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} n x^{4 \, n} + 7 \,{\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} n x^{3 \, n} - 7 \,{\left (3 \, a^{5} b^{2} - a^{6} c\right )} n x^{2 \, n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^n + b)*x^(n - 1)/(c*x^(2*n) + b*x^n - a)^8,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(x**(-1+n)*(b+2*c*x**n)/(-a+b*x**n+c*x**(2*n))**8,x)
[Out]
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GIAC/XCAS [A] time = 0.281261, size = 31, normalized size = 1.24 \[ -\frac{1}{7 \,{\left (c x^{2 \, n} + b x^{n} - a\right )}^{7} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^n + b)*x^(n - 1)/(c*x^(2*n) + b*x^n - a)^8,x, algorithm="giac")
[Out]