Integrand size = 25, antiderivative size = 94 \[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=\frac {x \left (c+d x^n\right )^{-1/n}}{a c}-\frac {b x \left (c+d x^n\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {a x^{-n} \left (c+d x^n\right )}{b c-a d}\right )}{a (b c-a d)} \] Output:
x/a/c/((c+d*x^n)^(1/n))-b*x*hypergeom([1, -1/n],[-(1-n)/n],-a*(c+d*x^n)/(- a*d+b*c)/(x^n))/a/(-a*d+b*c)/((c+d*x^n)^(1/n))
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 6.67 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.63 \[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=\frac {x \left (c+d x^n\right )^{-\frac {1+n}{n}} \left (\frac {a \left (c+d x^n\right )}{c \left (a+b x^n\right )}+\frac {b x^n \Phi \left (\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )},1,1+\frac {1}{n}\right )}{a}+\frac {b (-b c+a d) n x^{2 n} \operatorname {Hypergeometric2F1}\left (2,2+\frac {1}{n},3+\frac {1}{n},\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )}\right )}{a^2 (1+2 n) \left (c+d x^n\right )}\right )}{a} \] Input:
Integrate[(c + d*x^n)^(-1 - n^(-1))/(a + b*x^n),x]
Output:
(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)*H ypergeometric2F1[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))
Time = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {907, 904}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (c+d x^n\right )^{-\frac {1}{n}-1}}{a+b x^n} \, dx\) |
\(\Big \downarrow \) 907 |
\(\displaystyle \frac {b \int \frac {\left (d x^n+c\right )^{-1/n}}{b x^n+a}dx}{b c-a d}-\frac {d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)}\) |
\(\Big \downarrow \) 904 |
\(\displaystyle \frac {b x \left (c+d x^n\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a (b c-a d)}-\frac {d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)}\) |
Input:
Int[(c + d*x^n)^(-1 - n^(-1))/(a + b*x^n),x]
Output:
-((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))
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]
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] + Simp[(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] && EqQ[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] || ! LtQ[q, -1]) && NeQ[p, -1]
\[\int \frac {\left (c +d \,x^{n}\right )^{-1-\frac {1}{n}}}{a +b \,x^{n}}d x\]
Input:
int((c+d*x^n)^(-1-1/n)/(a+b*x^n),x)
Output:
int((c+d*x^n)^(-1-1/n)/(a+b*x^n),x)
\[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=\int { \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} - 1}}{b x^{n} + a} \,d x } \] Input:
integrate((c+d*x^n)^(-1-1/n)/(a+b*x^n),x, algorithm="fricas")
Output:
integral(1/((b*x^n + a)*(d*x^n + c)^((n + 1)/n)), x)
Exception generated. \[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((c+d*x**n)**(-1-1/n)/(a+b*x**n),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=\int { \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} - 1}}{b x^{n} + a} \,d x } \] Input:
integrate((c+d*x^n)^(-1-1/n)/(a+b*x^n),x, algorithm="maxima")
Output:
integrate((d*x^n + c)^(-1/n - 1)/(b*x^n + a), x)
\[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=\int { \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} - 1}}{b x^{n} + a} \,d x } \] Input:
integrate((c+d*x^n)^(-1-1/n)/(a+b*x^n),x, algorithm="giac")
Output:
integrate((d*x^n + c)^(-1/n - 1)/(b*x^n + a), x)
Timed out. \[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=\int \frac {1}{\left (a+b\,x^n\right )\,{\left (c+d\,x^n\right )}^{\frac {1}{n}+1}} \,d x \] Input:
int(1/((a + b*x^n)*(c + d*x^n)^(1/n + 1)),x)
Output:
int(1/((a + b*x^n)*(c + d*x^n)^(1/n + 1)), x)
\[ \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx=\int \frac {1}{x^{2 n} \left (x^{n} d +c \right )^{\frac {1}{n}} b d +x^{n} \left (x^{n} d +c \right )^{\frac {1}{n}} a d +x^{n} \left (x^{n} d +c \right )^{\frac {1}{n}} b c +\left (x^{n} d +c \right )^{\frac {1}{n}} a c}d x \] Input:
int((c+d*x^n)^(-1-1/n)/(a+b*x^n),x)
Output:
int(1/(x**(2*n)*(x**n*d + c)**(1/n)*b*d + x**n*(x**n*d + c)**(1/n)*a*d + x **n*(x**n*d + c)**(1/n)*b*c + (x**n*d + c)**(1/n)*a*c),x)