Optimal. Leaf size=37 \[ \frac{x^{-n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{c n (p+1)} \]
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Rubi [A] time = 0.0673135, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{x^{-n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{c n (p+1)} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - n*(-1 + p))*(b*x^n + c*x^(2*n))^p,x]
[Out]
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Rubi in Sympy [A] time = 9.39798, size = 27, normalized size = 0.73 \[ \frac{x^{- n \left (p + 1\right )} \left (b x^{n} + c x^{2 n}\right )^{p + 1}}{c n \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-n*(-1+p))*(b*x**n+c*x**(2*n))**p,x)
[Out]
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Mathematica [A] time = 0.0490163, size = 38, normalized size = 1.03 \[ \frac{x^{-n p} \left (b+c x^n\right ) \left (x^n \left (b+c x^n\right )\right )^p}{c n (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - n*(-1 + p))*(b*x^n + c*x^(2*n))^p,x]
[Out]
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Maple [F] time = 0.109, size = 0, normalized size = 0. \[ \int{x}^{-1-n \left ( -1+p \right ) } \left ( b{x}^{n}+c{x}^{2\,n} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-n*(-1+p))*(b*x^n+c*x^(2*n))^p,x)
[Out]
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Maxima [A] time = 0.81335, size = 58, normalized size = 1.57 \[ \frac{{\left (c x^{n} + b\right )} e^{\left (-n p \log \left (x\right ) + p \log \left (c x^{n} + b\right ) + p \log \left (x^{n}\right )\right )}}{c n{\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^(2*n) + b*x^n)^p*x^(-n*(p - 1) - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.290735, size = 80, normalized size = 2.16 \[ \frac{{\left (c x x^{-n p + n - 1} x^{n} + b x x^{-n p + n - 1}\right )}{\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{{\left (c n p + c n\right )} x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^(2*n) + b*x^n)^p*x^(-n*(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+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. \[ \int{\left (c x^{2 \, n} + b x^{n}\right )}^{p} x^{-n{\left (p - 1\right )} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^(2*n) + b*x^n)^p*x^(-n*(p - 1) - 1),x, algorithm="giac")
[Out]