Optimal. Leaf size=29 \[ \frac{\left (-a+b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]
[Out]
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Rubi [A] time = 0.0692443, antiderivative size = 29, 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 \[ \frac{\left (-a+b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + n)*(b + 2*c*x^n)*(-a + b*x^n + c*x^(2*n))^p,x]
[Out]
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Rubi in Sympy [A] time = 13.2978, size = 20, normalized size = 0.69 \[ \frac{\left (- a + b x^{n} + c x^{2 n}\right )^{p + 1}}{n \left (p + 1\right )} \]
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))**p,x)
[Out]
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Mathematica [A] time = 0.0875493, size = 28, normalized size = 0.97 \[ \frac{\left (-a+b x^n+c x^{2 n}\right )^{p+1}}{n p+n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + n)*(b + 2*c*x^n)*(-a + b*x^n + c*x^(2*n))^p,x]
[Out]
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Maple [A] time = 0.094, size = 45, normalized size = 1.6 \[ -{\frac{ \left ( -c \left ({x}^{n} \right ) ^{2}-b{x}^{n}+a \right ) \left ( -a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) ^{p}}{n \left ( 1+p \right ) }} \]
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))^p,x)
[Out]
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Maxima [A] time = 1.04987, size = 58, normalized size = 2. \[ \frac{{\left (c x^{2 \, n} + b x^{n} - a\right )}{\left (c x^{2 \, n} + b x^{n} - a\right )}^{p}}{n{\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^n + b)*(c*x^(2*n) + b*x^n - a)^p*x^(n - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.307078, size = 57, normalized size = 1.97 \[ \frac{{\left (c x^{2 \, n} + b x^{n} - a\right )}{\left (c x^{2 \, n} + b x^{n} - a\right )}^{p}}{n p + n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^n + b)*(c*x^(2*n) + b*x^n - a)^p*x^(n - 1),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))**p,x)
[Out]
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GIAC/XCAS [A] time = 0.286956, size = 39, normalized size = 1.34 \[ \frac{{\left (c x^{2 \, n} + b x^{n} - a\right )}^{p + 1}}{n{\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^n + b)*(c*x^(2*n) + b*x^n - a)^p*x^(n - 1),x, algorithm="giac")
[Out]