Integrand size = 21, antiderivative size = 167 \[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\frac {x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2 n},-p,1,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d}-\frac {e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+n}{2 n},-p,1,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (1+n)} \] Output:
x*(a+c*x^(2*n))^p*AppellF1(1/2/n,1,-p,1+1/2/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a )/d/((1+c*x^(2*n)/a)^p)-e*x^(1+n)*(a+c*x^(2*n))^p*AppellF1(1/2*(1+n)/n,1,- p,3/2+1/2/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a)/d^2/(1+n)/((1+c*x^(2*n)/a)^p)
\[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx \] Input:
Integrate[(a + c*x^(2*n))^p/(d + e*x^n),x]
Output:
Integrate[(a + c*x^(2*n))^p/(d + e*x^n), x]
Time = 0.35 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1768, 2009}
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.
\(\displaystyle \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx\) |
\(\Big \downarrow \) 1768 |
\(\displaystyle \int \left (\frac {d \left (a+c x^{2 n}\right )^p}{d^2-e^2 x^{2 n}}+\frac {e x^n \left (a+c x^{2 n}\right )^p}{e^2 x^{2 n}-d^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2 n},-p,1,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d}-\frac {e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {n+1}{2 n},-p,1,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1)}\) |
Input:
Int[(a + c*x^(2*n))^p/(d + e*x^n),x]
Output:
(x*(a + c*x^(2*n))^p*AppellF1[1/(2*n), -p, 1, (2 + n^(-1))/2, -((c*x^(2*n) )/a), (e^2*x^(2*n))/d^2])/(d*(1 + (c*x^(2*n))/a)^p) - (e*x^(1 + n)*(a + c* x^(2*n))^p*AppellF1[(1 + n)/(2*n), -p, 1, (3 + n^(-1))/2, -((c*x^(2*n))/a) , (e^2*x^(2*n))/d^2])/(d^2*(1 + n)*(1 + (c*x^(2*n))/a)^p)
Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^(2*n))^p, (d/(d^2 - e^2*x^(2*n)) - e*(x^n/ (d^2 - e^2*x^(2*n))))^(-q), x], x] /; FreeQ[{a, c, d, e, n, p}, x] && EqQ[n 2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ILtQ[q, 0]
\[\int \frac {\left (a +c \,x^{2 n}\right )^{p}}{d +e \,x^{n}}d x\]
Input:
int((a+c*x^(2*n))^p/(d+e*x^n),x)
Output:
int((a+c*x^(2*n))^p/(d+e*x^n),x)
\[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p}}{e x^{n} + d} \,d x } \] Input:
integrate((a+c*x^(2*n))^p/(d+e*x^n),x, algorithm="fricas")
Output:
integral((c*x^(2*n) + a)^p/(e*x^n + d), x)
Exception generated. \[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((a+c*x**(2*n))**p/(d+e*x**n),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p}}{e x^{n} + d} \,d x } \] Input:
integrate((a+c*x^(2*n))^p/(d+e*x^n),x, algorithm="maxima")
Output:
integrate((c*x^(2*n) + a)^p/(e*x^n + d), x)
\[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p}}{e x^{n} + d} \,d x } \] Input:
integrate((a+c*x^(2*n))^p/(d+e*x^n),x, algorithm="giac")
Output:
integrate((c*x^(2*n) + a)^p/(e*x^n + d), x)
Timed out. \[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int \frac {{\left (a+c\,x^{2\,n}\right )}^p}{d+e\,x^n} \,d x \] Input:
int((a + c*x^(2*n))^p/(d + e*x^n),x)
Output:
int((a + c*x^(2*n))^p/(d + e*x^n), x)
\[ \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx=\int \frac {\left (x^{2 n} c +a \right )^{p}}{x^{n} e +d}d x \] Input:
int((a+c*x^(2*n))^p/(d+e*x^n),x)
Output:
int((x**(2*n)*c + a)**p/(x**n*e + d),x)