\(\int \frac {(c+d x^n)^{-1-\frac {1}{n}}}{a+b x^n} \, dx\) [333]

Optimal result

Integrand size = 25, antiderivative size = 95 \[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=-\frac {d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)}+\frac {b x \left (c+d x^n\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a (b c-a d)} \]

[Out]

-d*x/c/(-a*d+b*c)/((c+d*x^n)^(1/n))+b*x*hypergeom([1, 1/n],[1+1/n],-(-a*d+b*c)*x^n/a/(c+d*x^n))/a/(-a*d+b*c)/(
(c+d*x^n)^(1/n))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {390, 387} \[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=\frac {b x \left (c+d x^n\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a (b c-a d)}-\frac {d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)} \]

[In]

Int[(c + d*x^n)^(-1 - n^(-1))/(a + b*x^n),x]

[Out]

-((d*x)/(c*(b*c - a*d)*(c + d*x^n)^n^(-1))) + (b*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -(((b*c - a*d)*x^n
)/(a*(c + d*x^n)))])/(a*(b*c - a*d)*(c + d*x^n)^n^(-1))

Rule 387

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*(x/(c^(p + 1)*(c + d*x^
n)^(1/n)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(a*(c + d*x^n)))], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && ILtQ[p, 0]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)}+\frac {b \int \frac {\left (c+d x^n\right )^{-1/n}}{a+b x^n} \, dx}{b c-a d} \\ & = -\frac {d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)}+\frac {b x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a (b c-a d)} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 6.53 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.61 \[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=\frac {x \left (c+d x^n\right )^{-\frac {1+n}{n}} \left (\frac {a \left (c+d x^n\right )}{c \left (a+b x^n\right )}+\frac {b x^n \Phi \left (\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )},1,1+\frac {1}{n}\right )}{a}+\frac {b (-b c+a d) n x^{2 n} \operatorname {Hypergeometric2F1}\left (2,2+\frac {1}{n},3+\frac {1}{n},\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )}\right )}{a^2 (1+2 n) \left (c+d x^n\right )}\right )}{a} \]

[In]

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

[Out]

(x*((a*(c + d*x^n))/(c*(a + b*x^n)) + (b*x^n*HurwitzLerchPhi[((-(b*c) + a*d)*x^n)/(a*(c + d*x^n)), 1, 1 + n^(-
1)])/a + (b*(-(b*c) + a*d)*n*x^(2*n)*Hypergeometric2F1[2, 2 + n^(-1), 3 + n^(-1), ((-(b*c) + a*d)*x^n)/(a*(c +
d*x^n))])/(a^2*(1 + 2*n)*(c + d*x^n))))/(a*(c + d*x^n)^((1 + n)/n))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(1/((b*x^n + a)*(d*x^n + c)^((n + 1)/n)), x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/((a + b*x^n)*(c + d*x^n)^(1/n + 1)),x)

[Out]

int(1/((a + b*x^n)*(c + d*x^n)^(1/n + 1)), x)