3.506 \(\int x^{-1-n (-1+p)} \left (b x^n+c x^{2 n}\right )^p \, dx\)

Optimal. Leaf size=37 \[ \frac{x^{-n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{c n (p+1)} \]

[Out]

(b*x^n + c*x^(2*n))^(1 + p)/(c*n*(1 + p)*x^(n*(1 + p)))

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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]

(b*x^n + c*x^(2*n))^(1 + p)/(c*n*(1 + p)*x^(n*(1 + p)))

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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]

x**(-n*(p + 1))*(b*x**n + c*x**(2*n))**(p + 1)/(c*n*(p + 1))

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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]

((b + c*x^n)*(x^n*(b + c*x^n))^p)/(c*n*(1 + p)*x^(n*p))

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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]

int(x^(-1-n*(-1+p))*(b*x^n+c*x^(2*n))^p,x)

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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]

(c*x^n + b)*e^(-n*p*log(x) + p*log(c*x^n + b) + p*log(x^n))/(c*n*(p + 1))

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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]

(c*x*x^(-n*p + n - 1)*x^n + b*x*x^(-n*p + n - 1))*(c*x^(2*n) + b*x^n)^p/((c*n*p
+ c*n)*x^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**(-1-n*(-1+p))*(b*x**n+c*x**(2*n))**p,x)

[Out]

Timed 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]

integrate((c*x^(2*n) + b*x^n)^p*x^(-n*(p - 1) - 1), x)