Optimal. Leaf size=38 \[ -\frac{x^{-2 n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{b n (p+1)} \]
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Rubi [A] time = 0.0668931, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ -\frac{x^{-2 n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - n*(1 + 2*p))*(b*x^n + c*x^(2*n))^p,x]
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Rubi in Sympy [A] time = 9.56211, size = 31, normalized size = 0.82 \[ - \frac{x^{- 2 n \left (p + 1\right )} \left (b x^{n} + c x^{2 n}\right )^{p + 1}}{b n \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-n*(1+2*p))*(b*x**n+c*x**(2*n))**p,x)
[Out]
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Mathematica [A] time = 0.0855039, size = 43, normalized size = 1.13 \[ -\frac{x^{-n (2 p+1)} \left (b+c x^n\right ) \left (x^n \left (b+c x^n\right )\right )^p}{b n (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - n*(1 + 2*p))*(b*x^n + c*x^(2*n))^p,x]
[Out]
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Maple [F] time = 0.112, size = 0, normalized size = 0. \[ \int{x}^{-1-n \left ( 1+2\,p \right ) } \left ( b{x}^{n}+c{x}^{2\,n} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-n*(1+2*p))*(b*x^n+c*x^(2*n))^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2 \, n} + b x^{n}\right )}^{p} x^{-n{\left (2 \, p + 1\right )} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^(2*n) + b*x^n)^p*x^(-n*(2*p + 1) - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.292187, size = 80, normalized size = 2.11 \[ -\frac{{\left (c x x^{-2 \, n p - n - 1} x^{n} + b x x^{-2 \, n p - n - 1}\right )}{\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{b n p + b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^(2*n) + b*x^n)^p*x^(-n*(2*p + 1) - 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*(1+2*p))*(b*x**n+c*x**(2*n))**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2 \, n} + b x^{n}\right )}^{p} x^{-n{\left (2 \, p + 1\right )} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^(2*n) + b*x^n)^p*x^(-n*(2*p + 1) - 1),x, algorithm="giac")
[Out]