\(\int (a+b x^n)^{-1-\frac {1}{n}} \, dx\) [2708]

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^{-1/n}}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {197} \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^{-1/n}}{a} \]

[In]

Int[(a + b*x^n)^(-1 - n^(-1)),x]

[Out]

x/(a*(a + b*x^n)^n^(-1))

Rule 197

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

Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b x^n\right )^{-1/n}}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^{-1/n}}{a} \]

[In]

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

[Out]

x/(a*(a + b*x^n)^n^(-1))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(18)=36\).

Time = 3.91 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.61

method	result	size
parallelrisch	\(\frac {x \,x^{n} \left (a +b \,x^{n}\right )^{-\frac {1+n}{n}} b +x \left (a +b \,x^{n}\right )^{-\frac {1+n}{n}} a}{a}\)	\(47\)
norman	\(x \,{\mathrm e}^{\left (-1-\frac {1}{n}\right ) \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}+\frac {b x \,{\mathrm e}^{n \ln \left (x \right )} {\mathrm e}^{\left (-1-\frac {1}{n}\right ) \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}}{a}\)	\(53\)

[In]

int((a+b*x^n)^(-1-1/n),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {b x x^{n} + a x}{{\left (b x^{n} + a\right )}^{\frac {n + 1}{n}} a} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (12) = 24\).

Time = 0.69 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.33 \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {a^{\frac {1}{n}} a^{-1 - \frac {1}{n}} b^{- \frac {1}{n}} \left (\frac {a x^{- n}}{b} + 1\right )^{- \frac {1}{n}} \Gamma \left (\frac {1}{n}\right )}{n \Gamma \left (1 + \frac {1}{n}\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{-\frac {1}{n} - 1} \,d x } \]

[In]

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

[Out]

integrate((b*x^n + a)^(-1/n - 1), x)

Giac [F]

\[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{-\frac {1}{n} - 1} \,d x } \]

[In]

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

[Out]

integrate((b*x^n + a)^(-1/n - 1), x)

Mupad [B] (verification not implemented)

Time = 5.90 (sec) , antiderivative size = 75, normalized size of antiderivative = 4.17 \[ \int \left (a+b x^n\right )^{-1-\frac {1}{n}} \, dx=\frac {b\,x^{n+1}\,\left (\frac {a}{b\,x^n}-{\left (\frac {a}{b\,x^n}+1\right )}^{\frac {n+1}{n}}+1\right )}{a\,n\,\left (\frac {n+1}{n}-1\right )\,{\left (a+b\,x^n\right )}^{\frac {n+1}{n}}} \]

[In]

int(1/(a + b*x^n)^(1/n + 1),x)

[Out]

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