\(\int \frac {(a+c x^{2 n})^p}{(d+e x^n)^2} \, dx\) [64]

Optimal result

Integrand size = 21, antiderivative size = 261 \[ \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\frac {e^2 x^{1+2 n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} \left (2+\frac {1}{n}\right ),-p,2,\frac {1}{2} \left (4+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (1+2 n)}+\frac {x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2 n},-p,2,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2}-\frac {2 e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+n}{2 n},-p,2,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (1+n)} \]

[Out]

e^2*x^(1+2*n)*(a+c*x^(2*n))^p*AppellF1(1+1/2/n,2,-p,2+1/2/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a)/d^4/(1+2*n)/((1+c*x^
(2*n)/a)^p)+x*(a+c*x^(2*n))^p*AppellF1(1/2/n,2,-p,1+1/2/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a)/d^2/((1+c*x^(2*n)/a)^p
)-2*e*x^(1+n)*(a+c*x^(2*n))^p*AppellF1(1/2*(1+n)/n,2,-p,3/2+1/2/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a)/d^3/(1+n)/((1+
c*x^(2*n)/a)^p)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1452, 441, 440, 525, 524} \[ \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\frac {x \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2 n},-p,2,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2}+\frac {e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} \left (2+\frac {1}{n}\right ),-p,2,\frac {1}{2} \left (4+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (2 n+1)}-\frac {2 e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {n+1}{2 n},-p,2,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (n+1)} \]

[In]

Int[(a + c*x^(2*n))^p/(d + e*x^n)^2,x]

[Out]

(e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*AppellF1[(2 + n^(-1))/2, -p, 2, (4 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2
*n))/d^2])/(d^4*(1 + 2*n)*(1 + (c*x^(2*n))/a)^p) + (x*(a + c*x^(2*n))^p*AppellF1[1/(2*n), -p, 2, (2 + n^(-1))/
2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^2*(1 + (c*x^(2*n))/a)^p) - (2*e*x^(1 + n)*(a + c*x^(2*n))^p*Appell
F1[(1 + n)/(2*n), -p, 2, (3 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^3*(1 + n)*(1 + (c*x^(2*n))/a
)^p)

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 1452

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[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[q, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 \left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^2}-\frac {2 d e x^n \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2}+\frac {e^2 x^{2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2}\right ) \, dx \\ & = d^2 \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^2} \, dx-(2 d e) \int \frac {x^n \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx+e^2 \int \frac {x^{2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx \\ & = \left (d^2 \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {c x^{2 n}}{a}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^2} \, dx-\left (2 d e \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^n \left (1+\frac {c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx+\left (e^2 \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^{2 n} \left (1+\frac {c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx \\ & = \frac {e^2 x^{1+2 n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1}{2} \left (2+\frac {1}{n}\right );-p,2;\frac {1}{2} \left (4+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (1+2 n)}+\frac {x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1}{2 n};-p,2;\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^2}-\frac {2 e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1+n}{2 n};-p,2;\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^3 (1+n)} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int \frac {\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]

[In]

Integrate[(a + c*x^(2*n))^p/(d + e*x^n)^2,x]

[Out]

Integrate[(a + c*x^(2*n))^p/(d + e*x^n)^2, x]

Maple [F]

\[\int \frac {\left (a +c \,x^{2 n}\right )^{p}}{\left (d +e \,x^{n}\right )^{2}}d x\]

[In]

int((a+c*x^(2*n))^p/(d+e*x^n)^2,x)

[Out]

int((a+c*x^(2*n))^p/(d+e*x^n)^2,x)

Fricas [F]

\[ \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]

[In]

integrate((a+c*x^(2*n))^p/(d+e*x^n)^2,x, algorithm="fricas")

[Out]

integral((c*x^(2*n) + a)^p/(e^2*x^(2*n) + 2*d*e*x^n + d^2), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((a+c*x**(2*n))**p/(d+e*x**n)**2,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]

[In]

integrate((a+c*x^(2*n))^p/(d+e*x^n)^2,x, algorithm="maxima")

[Out]

integrate((c*x^(2*n) + a)^p/(e*x^n + d)^2, x)

Giac [F]

\[ \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]

[In]

integrate((a+c*x^(2*n))^p/(d+e*x^n)^2,x, algorithm="giac")

[Out]

integrate((c*x^(2*n) + a)^p/(e*x^n + d)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int \frac {{\left (a+c\,x^{2\,n}\right )}^p}{{\left (d+e\,x^n\right )}^2} \,d x \]

[In]

int((a + c*x^(2*n))^p/(d + e*x^n)^2,x)

[Out]

int((a + c*x^(2*n))^p/(d + e*x^n)^2, x)