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

Optimal. Leaf size=57 \[ \frac{p x^{(1-n) (p+1)} \left (a x^{-(1-n) p}+b x^{n-(1-n) p}\right )^{\frac{1}{p}+1}}{b n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0618204, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{p x^{(1-n) (p+1)} \left (a x^{-(1-n) p}+b x^{n p+n-p}\right )^{\frac{1}{p}+1}}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(x^((-1 + n)*p)*(a + b*x^n))^p^(-1),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (x^{p \left (n - 1\right )} \left (a + b x^{n}\right )\right )^{\frac{1}{p}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**((-1+n)*p)*(a+b*x**n))**(1/p),x)

[Out]

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

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Mathematica [A]  time = 0.0551251, size = 46, normalized size = 0.81 \[ \frac{p x^{1-n} \left (a+b x^n\right ) \left (x^{(n-1) p} \left (a+b x^n\right )\right )^{\frac{1}{p}}}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^((-1 + n)*p)*(a + b*x^n))^p^(-1),x]

[Out]

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

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \[ \int \sqrt [p]{{x}^{ \left ( -1+n \right ) p} \left ( a+b{x}^{n} \right ) }\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^((-1+n)*p)*(a+b*x^n))^(1/p),x)

[Out]

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

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Maxima [A]  time = 1.51668, size = 45, normalized size = 0.79 \[ \frac{{\left (b p x^{n} + a p\right )}{\left (b x^{n} + a\right )}^{\left (\frac{1}{p}\right )}}{b n{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n + a)*x^((n - 1)*p))^(1/p),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.264947, size = 63, normalized size = 1.11 \[ \frac{{\left (b p x x^{n} + a p x\right )} \left ({\left (b x^{n} + a\right )} x^{{\left (n - 1\right )} p}\right )^{\left (\frac{1}{p}\right )}}{{\left (b n p + b n\right )} x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n + a)*x^((n - 1)*p))^(1/p),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \left ({\left (b x^{n} + a\right )} x^{{\left (n - 1\right )} p}\right )^{\left (\frac{1}{p}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n + a)*x^((n - 1)*p))^(1/p),x, algorithm="giac")

[Out]

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