3.2717 \(\int x^{-1-n-n p} \left (a+b x^n\right )^p \, dx\)

Optimal. Leaf size=32 \[ -\frac{x^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a n (p+1)} \]

[Out]

-((a + b*x^n)^(1 + p)/(a*n*(1 + p)*x^(n*(1 + p))))

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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]

-((a + b*x^n)^(1 + p)/(a*n*(1 + p)*x^(n*(1 + p))))

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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]

-x**(-n*(p + 1))*(a + b*x**n)**(p + 1)/(a*n*(p + 1))

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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]

-((a + b*x^n)^(1 + p)/(a*n*(1 + p)*x^(n*(1 + p))))

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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]

int(x^(-n*p-n-1)*(a+b*x^n)^p,x)

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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]

integrate((b*x^n + a)^p*x^(-n*p - n - 1), x)

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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]

-(b*x*x^(-n*p - n - 1)*x^n + a*x*x^(-n*p - n - 1))*(b*x^n + a)^p/(a*n*p + a*n)

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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]

Timed 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]

integrate((b*x^n + a)^p*x^(-n*p - n - 1), x)