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

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+1/p)/b/n/(1+p)

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Rubi [A]  time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1979, 2000} \[ \frac {p x^{(1-n) (p+1)} \left (a x^{-(1-n) p}+b x^{n-(1-n) p}\right )^{\frac {1}{p}+1}}{b n (p+1)} \]

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

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

\begin {align*} \int \left (x^{(-1+n) p} \left (a+b x^n\right )\right )^{\frac {1}{p}} \, dx &=\int \left (a x^{(-1+n) p}+b x^{n+(-1+n) p}\right )^{\frac {1}{p}} \, dx\\ &=\frac {p x^{(1-n) (1+p)} \left (a x^{-(1-n) p}+b x^{n-(1-n) p}\right )^{1+\frac {1}{p}}}{b n (1+p)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 47, normalized size = 0.82 \[ \frac {x^{1-n} \left (a+b x^n\right ) \left (x^{(n-1) p} \left (a+b x^n\right )\right )^{\frac {1}{p}}}{b n \left (\frac {1}{p}+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 47, normalized size = 0.82 \[ \frac {{\left (b p x x^{n} + a p x\right )} \left ({\left (b x^{n} + a\right )} x^{{\left (n - 1\right )} p}\right )^{\left (\frac {1}{p}\right )}}{{\left (b n p + b n\right )} x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^((-1+n)*p)*(a+b*x^n))^(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left ({\left (b x^{n} + a\right )} x^{{\left (n - 1\right )} p}\right )^{\left (\frac {1}{p}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.72, size = 0, normalized size = 0.00 \[ \int \left (\left (b \,x^{n}+a \right ) x^{\left (n -1\right ) p}\right )^{\frac {1}{p}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left ({\left (b x^{n} + a\right )} x^{{\left (n - 1\right )} p}\right )^{\left (\frac {1}{p}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (x^{p\,\left (n-1\right )}\,\left (a+b\,x^n\right )\right )}^{1/p} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (x^{p \left (n - 1\right )} \left (a + b x^{n}\right )\right )^{\frac {1}{p}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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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