Optimal. Leaf size=32 \[ -\frac{x^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a n (p+1)} \]
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Rubi [A] time = 0.0336558, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{x^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a n (p+1)} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - n - n*p)*(a + b*x^n)^p,x]
[Out]
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Rubi in Sympy [A] time = 3.93618, size = 24, normalized size = 0.75 \[ - \frac{x^{- n \left (p + 1\right )} \left (a + b x^{n}\right )^{p + 1}}{a n \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-n*p-n-1)*(a+b*x**n)**p,x)
[Out]
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Mathematica [A] time = 0.0811157, size = 32, normalized size = 1. \[ -\frac{x^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a n (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - n - n*p)*(a + b*x^n)^p,x]
[Out]
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Maple [F] time = 0.108, size = 0, normalized size = 0. \[ \int{x}^{-np-n-1} \left ( a+b{x}^{n} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-n*p-n-1)*(a+b*x^n)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{p} x^{-n p - n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^p*x^(-n*p - n - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239435, size = 72, normalized size = 2.25 \[ -\frac{{\left (b x x^{-n p - n - 1} x^{n} + a x x^{-n p - n - 1}\right )}{\left (b x^{n} + a\right )}^{p}}{a n p + a n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^p*x^(-n*p - 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**(-n*p-n-1)*(a+b*x**n)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{p} x^{-n p - n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^p*x^(-n*p - n - 1),x, algorithm="giac")
[Out]