3.1.57 \(\int \frac {1}{(d+e x^n)^2 (a+c x^{2 n})^3} \, dx\) [57]

Optimal. Leaf size=701 \[ \frac {c x \left (c d^2-a e^2-2 c d e x^n\right )}{4 a \left (c d^2+a e^2\right )^2 n \left (a+c x^{2 n}\right )^2}+\frac {c e^2 x \left (3 c d^2-a e^2-4 c d e x^n\right )}{2 a \left (c d^2+a e^2\right )^3 n \left (a+c x^{2 n}\right )}-\frac {c x \left (\left (c d^2-a e^2\right ) (1-4 n)-2 c d e (1-3 n) x^n\right )}{8 a^2 \left (c d^2+a e^2\right )^2 n^2 \left (a+c x^{2 n}\right )}+\frac {c e^4 \left (5 c d^2-a e^2\right ) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^4}+\frac {c \left (c d^2-a e^2\right ) (1-4 n) (1-2 n) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{8 a^3 \left (c d^2+a e^2\right )^2 n^2}-\frac {c e^2 \left (3 c d^2-a e^2\right ) (1-2 n) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 \left (c d^2+a e^2\right )^3 n}+\frac {6 c e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{\left (c d^2+a e^2\right )^4}-\frac {6 c^2 d e^5 x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^4 (1+n)}-\frac {c^2 d e (1-3 n) (1-n) x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{4 a^3 \left (c d^2+a e^2\right )^2 n^2 (1+n)}+\frac {2 c^2 d e^3 (1-n) x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2 \left (c d^2+a e^2\right )^3 n (1+n)}+\frac {e^6 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2+a e^2\right )^3} \]

[Out]

1/4*c*x*(c*d^2-a*e^2-2*c*d*e*x^n)/a/(a*e^2+c*d^2)^2/n/(a+c*x^(2*n))^2+1/2*c*e^2*x*(3*c*d^2-a*e^2-4*c*d*e*x^n)/
a/(a*e^2+c*d^2)^3/n/(a+c*x^(2*n))-1/8*c*x*((-a*e^2+c*d^2)*(1-4*n)-2*c*d*e*(1-3*n)*x^n)/a^2/(a*e^2+c*d^2)^2/n^2
/(a+c*x^(2*n))+c*e^4*(-a*e^2+5*c*d^2)*x*hypergeom([1, 1/2/n],[1+1/2/n],-c*x^(2*n)/a)/a/(a*e^2+c*d^2)^4+1/8*c*(
-a*e^2+c*d^2)*(1-4*n)*(1-2*n)*x*hypergeom([1, 1/2/n],[1+1/2/n],-c*x^(2*n)/a)/a^3/(a*e^2+c*d^2)^2/n^2-1/2*c*e^2
*(-a*e^2+3*c*d^2)*(1-2*n)*x*hypergeom([1, 1/2/n],[1+1/2/n],-c*x^(2*n)/a)/a^2/(a*e^2+c*d^2)^3/n+6*c*e^6*x*hyper
geom([1, 1/n],[1+1/n],-e*x^n/d)/(a*e^2+c*d^2)^4-6*c^2*d*e^5*x^(1+n)*hypergeom([1, 1/2*(1+n)/n],[3/2+1/2/n],-c*
x^(2*n)/a)/a/(a*e^2+c*d^2)^4/(1+n)-1/4*c^2*d*e*(1-3*n)*(1-n)*x^(1+n)*hypergeom([1, 1/2*(1+n)/n],[3/2+1/2/n],-c
*x^(2*n)/a)/a^3/(a*e^2+c*d^2)^2/n^2/(1+n)+2*c^2*d*e^3*(1-n)*x^(1+n)*hypergeom([1, 1/2*(1+n)/n],[3/2+1/2/n],-c*
x^(2*n)/a)/a^2/(a*e^2+c*d^2)^3/n/(1+n)+e^6*x*hypergeom([2, 1/n],[1+1/n],-e*x^n/d)/d^2/(a*e^2+c*d^2)^3

________________________________________________________________________________________

Rubi [A]
time = 0.48, antiderivative size = 701, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1451, 251, 1445, 1432, 371} \begin {gather*} -\frac {c^2 d e (1-3 n) (1-n) x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{4 a^3 n^2 (n+1) \left (a e^2+c d^2\right )^2}+\frac {c (1-4 n) (1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{8 a^3 n^2 \left (a e^2+c d^2\right )^2}+\frac {2 c^2 d e^3 (1-n) x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2 n (n+1) \left (a e^2+c d^2\right )^3}-\frac {c e^2 (1-2 n) x \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^3}-\frac {c x \left ((1-4 n) \left (c d^2-a e^2\right )-2 c d e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )}-\frac {6 c^2 d e^5 x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^4}+\frac {c e^2 x \left (-a e^2+3 c d^2-4 c d e x^n\right )}{2 a n \left (a e^2+c d^2\right )^3 \left (a+c x^{2 n}\right )}+\frac {c x \left (-a e^2+c d^2-2 c d e x^n\right )}{4 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )^2}+\frac {6 c e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{\left (a e^2+c d^2\right )^4}+\frac {e^6 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )^3}+\frac {c e^4 x \left (5 c d^2-a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x^n)^2*(a + c*x^(2*n))^3),x]

[Out]

(c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^n))/(4*a*(c*d^2 + a*e^2)^2*n*(a + c*x^(2*n))^2) + (c*e^2*x*(3*c*d^2 - a*e^2 -
4*c*d*e*x^n))/(2*a*(c*d^2 + a*e^2)^3*n*(a + c*x^(2*n))) - (c*x*((c*d^2 - a*e^2)*(1 - 4*n) - 2*c*d*e*(1 - 3*n)*
x^n))/(8*a^2*(c*d^2 + a*e^2)^2*n^2*(a + c*x^(2*n))) + (c*e^4*(5*c*d^2 - a*e^2)*x*Hypergeometric2F1[1, 1/(2*n),
(2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^4) + (c*(c*d^2 - a*e^2)*(1 - 4*n)*(1 - 2*n)*x*Hypergeom
etric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*(c*d^2 + a*e^2)^2*n^2) - (c*e^2*(3*c*d^2 - a*e^
2)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)^3*n) +
(6*c*e^6*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(c*d^2 + a*e^2)^4 - (6*c^2*d*e^5*x^(1 + n)*
Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^4*(1 + n)) - (c^2*d*
e*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(4*a^3*(c
*d^2 + a*e^2)^2*n^2*(1 + n)) + (2*c^2*d*e^3*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))
/2, -((c*x^(2*n))/a)])/(a^2*(c*d^2 + a*e^2)^3*n*(1 + n)) + (e^6*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((
e*x^n)/d)])/(d^2*(c*d^2 + a*e^2)^3)

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1432

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Dist[d, Int[1/(a + c*x^(2*n)), x], x] + D
ist[e, Int[x^n/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &
& (PosQ[a*c] ||  !IntegerQ[n])

Rule 1445

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^n)*((a + c*x^(2*n
))^(p + 1)/(2*a*n*(p + 1))), x] + Dist[1/(2*a*n*(p + 1)), Int[(d*(2*n*p + 2*n + 1) + e*(2*n*p + 3*n + 1)*x^n)*
(a + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && ILtQ[p, -1]

Rule 1451

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)
^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && ((
IntegersQ[p, q] &&  !IntegerQ[n]) || IGtQ[p, 0] || (IGtQ[q, 0] &&  !IntegerQ[n]))

Rubi steps

\begin {align*} \int \frac {1}{\left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^3} \, dx &=\int \left (\frac {e^6}{\left (c d^2+a e^2\right )^3 \left (d+e x^n\right )^2}+\frac {6 c d e^6}{\left (c d^2+a e^2\right )^4 \left (d+e x^n\right )}-\frac {c \left (-c d^2+a e^2+2 c d e x^n\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^{2 n}\right )^3}-\frac {c e^2 \left (-3 c d^2+a e^2+4 c d e x^n\right )}{\left (c d^2+a e^2\right )^3 \left (a+c x^{2 n}\right )^2}-\frac {c e^4 \left (-5 c d^2+a e^2+6 c d e x^n\right )}{\left (c d^2+a e^2\right )^4 \left (a+c x^{2 n}\right )}\right ) \, dx\\ &=-\frac {\left (c e^4\right ) \int \frac {-5 c d^2+a e^2+6 c d e x^n}{a+c x^{2 n}} \, dx}{\left (c d^2+a e^2\right )^4}+\frac {\left (6 c d e^6\right ) \int \frac {1}{d+e x^n} \, dx}{\left (c d^2+a e^2\right )^4}-\frac {\left (c e^2\right ) \int \frac {-3 c d^2+a e^2+4 c d e x^n}{\left (a+c x^{2 n}\right )^2} \, dx}{\left (c d^2+a e^2\right )^3}+\frac {e^6 \int \frac {1}{\left (d+e x^n\right )^2} \, dx}{\left (c d^2+a e^2\right )^3}-\frac {c \int \frac {-c d^2+a e^2+2 c d e x^n}{\left (a+c x^{2 n}\right )^3} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac {c x \left (c d^2-a e^2-2 c d e x^n\right )}{4 a \left (c d^2+a e^2\right )^2 n \left (a+c x^{2 n}\right )^2}+\frac {c e^2 x \left (3 c d^2-a e^2-4 c d e x^n\right )}{2 a \left (c d^2+a e^2\right )^3 n \left (a+c x^{2 n}\right )}+\frac {6 c e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{\left (c d^2+a e^2\right )^4}+\frac {e^6 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2+a e^2\right )^3}-\frac {\left (6 c^2 d e^5\right ) \int \frac {x^n}{a+c x^{2 n}} \, dx}{\left (c d^2+a e^2\right )^4}+\frac {\left (c e^4 \left (5 c d^2-a e^2\right )\right ) \int \frac {1}{a+c x^{2 n}} \, dx}{\left (c d^2+a e^2\right )^4}+\frac {\left (c e^2\right ) \int \frac {\left (-3 c d^2+a e^2\right ) (1-2 n)+4 c d e (1-n) x^n}{a+c x^{2 n}} \, dx}{2 a \left (c d^2+a e^2\right )^3 n}+\frac {c \int \frac {\left (-c d^2+a e^2\right ) (1-4 n)+2 c d e (1-3 n) x^n}{\left (a+c x^{2 n}\right )^2} \, dx}{4 a \left (c d^2+a e^2\right )^2 n}\\ &=\frac {c x \left (c d^2-a e^2-2 c d e x^n\right )}{4 a \left (c d^2+a e^2\right )^2 n \left (a+c x^{2 n}\right )^2}+\frac {c e^2 x \left (3 c d^2-a e^2-4 c d e x^n\right )}{2 a \left (c d^2+a e^2\right )^3 n \left (a+c x^{2 n}\right )}-\frac {c x \left (\left (c d^2-a e^2\right ) (1-4 n)-2 c d e (1-3 n) x^n\right )}{8 a^2 \left (c d^2+a e^2\right )^2 n^2 \left (a+c x^{2 n}\right )}+\frac {c e^4 \left (5 c d^2-a e^2\right ) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^4}+\frac {6 c e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{\left (c d^2+a e^2\right )^4}-\frac {6 c^2 d e^5 x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^4 (1+n)}+\frac {e^6 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2+a e^2\right )^3}-\frac {c \int \frac {\left (-c d^2+a e^2\right ) (1-4 n) (1-2 n)+2 c d e (1-3 n) (1-n) x^n}{a+c x^{2 n}} \, dx}{8 a^2 \left (c d^2+a e^2\right )^2 n^2}-\frac {\left (c e^2 \left (3 c d^2-a e^2\right ) (1-2 n)\right ) \int \frac {1}{a+c x^{2 n}} \, dx}{2 a \left (c d^2+a e^2\right )^3 n}+\frac {\left (2 c^2 d e^3 (1-n)\right ) \int \frac {x^n}{a+c x^{2 n}} \, dx}{a \left (c d^2+a e^2\right )^3 n}\\ &=\frac {c x \left (c d^2-a e^2-2 c d e x^n\right )}{4 a \left (c d^2+a e^2\right )^2 n \left (a+c x^{2 n}\right )^2}+\frac {c e^2 x \left (3 c d^2-a e^2-4 c d e x^n\right )}{2 a \left (c d^2+a e^2\right )^3 n \left (a+c x^{2 n}\right )}-\frac {c x \left (\left (c d^2-a e^2\right ) (1-4 n)-2 c d e (1-3 n) x^n\right )}{8 a^2 \left (c d^2+a e^2\right )^2 n^2 \left (a+c x^{2 n}\right )}+\frac {c e^4 \left (5 c d^2-a e^2\right ) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^4}-\frac {c e^2 \left (3 c d^2-a e^2\right ) (1-2 n) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 \left (c d^2+a e^2\right )^3 n}+\frac {6 c e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{\left (c d^2+a e^2\right )^4}-\frac {6 c^2 d e^5 x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^4 (1+n)}+\frac {2 c^2 d e^3 (1-n) x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2 \left (c d^2+a e^2\right )^3 n (1+n)}+\frac {e^6 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2+a e^2\right )^3}+\frac {\left (c \left (c d^2-a e^2\right ) (1-4 n) (1-2 n)\right ) \int \frac {1}{a+c x^{2 n}} \, dx}{8 a^2 \left (c d^2+a e^2\right )^2 n^2}-\frac {\left (c^2 d e (1-3 n) (1-n)\right ) \int \frac {x^n}{a+c x^{2 n}} \, dx}{4 a^2 \left (c d^2+a e^2\right )^2 n^2}\\ &=\frac {c x \left (c d^2-a e^2-2 c d e x^n\right )}{4 a \left (c d^2+a e^2\right )^2 n \left (a+c x^{2 n}\right )^2}+\frac {c e^2 x \left (3 c d^2-a e^2-4 c d e x^n\right )}{2 a \left (c d^2+a e^2\right )^3 n \left (a+c x^{2 n}\right )}-\frac {c x \left (\left (c d^2-a e^2\right ) (1-4 n)-2 c d e (1-3 n) x^n\right )}{8 a^2 \left (c d^2+a e^2\right )^2 n^2 \left (a+c x^{2 n}\right )}+\frac {c e^4 \left (5 c d^2-a e^2\right ) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^4}+\frac {c \left (c d^2-a e^2\right ) (1-4 n) (1-2 n) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{8 a^3 \left (c d^2+a e^2\right )^2 n^2}-\frac {c e^2 \left (3 c d^2-a e^2\right ) (1-2 n) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 \left (c d^2+a e^2\right )^3 n}+\frac {6 c e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{\left (c d^2+a e^2\right )^4}-\frac {6 c^2 d e^5 x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^4 (1+n)}-\frac {c^2 d e (1-3 n) (1-n) x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{4 a^3 \left (c d^2+a e^2\right )^2 n^2 (1+n)}+\frac {2 c^2 d e^3 (1-n) x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2 \left (c d^2+a e^2\right )^3 n (1+n)}+\frac {e^6 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2+a e^2\right )^3}\\ \end {align*}

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Mathematica [A]
time = 1.55, size = 1241, normalized size = 1.77 \begin {gather*} \frac {x \left (\frac {8 c d^2 e^6+8 a e^8}{d^2 n+d e n x^n}+\frac {2 c \left (c d^2+a e^2\right )^2 \left (-a e^2+c d \left (d-2 e x^n\right )\right )}{a n \left (a+c x^{2 n}\right )^2}+\frac {c \left (c d^2+a e^2\right ) \left (a^2 e^4 (1-8 n)+c^2 d^3 \left (d (-1+4 n)-2 e (-1+3 n) x^n\right )+2 a c d e^2 \left (6 d n-e (-1+11 n) x^n\right )\right )}{a^2 n^2 \left (a+c x^{2 n}\right )}+\frac {8 c^4 d^6 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{a^3}+\frac {32 c^3 d^4 e^2 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{a^2}+\frac {48 c^2 d^2 e^4 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{a}-24 c e^6 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )+\frac {c^4 d^6 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{a^3 n^2}+\frac {c^3 d^4 e^2 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{a^2 n^2}-\frac {c^2 d^2 e^4 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{a n^2}-\frac {c e^6 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{n^2}-\frac {6 c^4 d^6 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{a^3 n}-\frac {18 c^3 d^4 e^2 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{a^2 n}-\frac {2 c^2 d^2 e^4 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{a n}+\frac {10 c e^6 \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{n}+\frac {8 e^6 \left (a e^2 (-1+n)+c d^2 (-1+7 n)\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 n}-\frac {6 c^4 d^5 e x^n \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^3 (1+n)}-\frac {28 c^3 d^3 e^3 x^n \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2 (1+n)}-\frac {70 c^2 d e^5 x^n \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a (1+n)}-\frac {2 c^4 d^5 e x^n \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^3 n^2 (1+n)}-\frac {4 c^3 d^3 e^3 x^n \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2 n^2 (1+n)}-\frac {2 c^2 d e^5 x^n \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a n^2 (1+n)}+\frac {8 c^4 d^5 e x^n \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^3 n (1+n)}+\frac {32 c^3 d^3 e^3 x^n \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2 n (1+n)}+\frac {24 c^2 d e^5 x^n \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a n (1+n)}\right )}{8 \left (c d^2+a e^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x^n)^2*(a + c*x^(2*n))^3),x]

[Out]

(x*((8*c*d^2*e^6 + 8*a*e^8)/(d^2*n + d*e*n*x^n) + (2*c*(c*d^2 + a*e^2)^2*(-(a*e^2) + c*d*(d - 2*e*x^n)))/(a*n*
(a + c*x^(2*n))^2) + (c*(c*d^2 + a*e^2)*(a^2*e^4*(1 - 8*n) + c^2*d^3*(d*(-1 + 4*n) - 2*e*(-1 + 3*n)*x^n) + 2*a
*c*d*e^2*(6*d*n - e*(-1 + 11*n)*x^n)))/(a^2*n^2*(a + c*x^(2*n))) + (8*c^4*d^6*Hypergeometric2F1[1, 1/(2*n), 1
+ 1/(2*n), -((c*x^(2*n))/a)])/a^3 + (32*c^3*d^4*e^2*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a
)])/a^2 + (48*c^2*d^2*e^4*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/a - 24*c*e^6*Hypergeom
etric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)] + (c^4*d^6*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c
*x^(2*n))/a)])/(a^3*n^2) + (c^3*d^4*e^2*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a^2*n^2
) - (c^2*d^2*e^4*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a*n^2) - (c*e^6*Hypergeometric
2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/n^2 - (6*c^4*d^6*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -(
(c*x^(2*n))/a)])/(a^3*n) - (18*c^3*d^4*e^2*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a^2*
n) - (2*c^2*d^2*e^4*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a*n) + (10*c*e^6*Hypergeome
tric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/n + (8*e^6*(a*e^2*(-1 + n) + c*d^2*(-1 + 7*n))*Hypergeomet
ric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*n) - (6*c^4*d^5*e*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3
+ n^(-1))/2, -((c*x^(2*n))/a)])/(a^3*(1 + n)) - (28*c^3*d^3*e^3*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 +
n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*(1 + n)) - (70*c^2*d*e^5*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1
))/2, -((c*x^(2*n))/a)])/(a*(1 + n)) - (2*c^4*d^5*e*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -(
(c*x^(2*n))/a)])/(a^3*n^2*(1 + n)) - (4*c^3*d^3*e^3*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -(
(c*x^(2*n))/a)])/(a^2*n^2*(1 + n)) - (2*c^2*d*e^5*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c
*x^(2*n))/a)])/(a*n^2*(1 + n)) + (8*c^4*d^5*e*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(
2*n))/a)])/(a^3*n*(1 + n)) + (32*c^3*d^3*e^3*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2
*n))/a)])/(a^2*n*(1 + n)) + (24*c^2*d*e^5*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n)
)/a)])/(a*n*(1 + n))))/(8*(c*d^2 + a*e^2)^4)

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d +e \,x^{n}\right )^{2} \left (a +c \,x^{2 n}\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d+e*x^n)^2/(a+c*x^(2*n))^3,x)

[Out]

int(1/(d+e*x^n)^2/(a+c*x^(2*n))^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d+e*x^n)^2/(a+c*x^(2*n))^3,x, algorithm="maxima")

[Out]

(c*d^2*(7*n - 1)*e^6 + a*(n - 1)*e^8)*integrate(1/(c^4*d^10*n + 4*a*c^3*d^8*n*e^2 + 6*a^2*c^2*d^6*n*e^4 + 4*a^
3*c*d^4*n*e^6 + a^4*d^2*n*e^8 + (c^4*d^9*n*e + 4*a*c^3*d^7*n*e^3 + 6*a^2*c^2*d^5*n*e^5 + 4*a^3*c*d^3*n*e^7 + a
^4*d*n*e^9)*x^n), x) - 1/8*(2*(c^4*d^4*(3*n - 1)*e^2 + a*c^3*d^2*(11*n - 1)*e^4 - 4*a^2*c^2*n*e^6)*x*x^(4*n) +
(c^4*d^5*(2*n - 1)*e + 2*a*c^3*d^3*(5*n - 1)*e^3 + a^2*c^2*d*(8*n - 1)*e^5)*x*x^(3*n) - (c^4*d^6*(4*n - 1) +
2*a*c^3*d^4*(n + 1)*e^2 - a^2*c^2*d^2*(34*n - 3)*e^4 + 16*a^3*c*n*e^6)*x*x^(2*n) + (a*c^3*d^5*(4*n - 1)*e + 2*
a^2*c^2*d^3*(7*n - 1)*e^3 + a^3*c*d*(10*n - 1)*e^5)*x*x^n - (a*c^3*d^6*(6*n - 1) + 12*a^2*c^2*d^4*n*e^2 - a^3*
c*d^2*(10*n - 1)*e^4 + 8*a^4*n*e^6)*x)/(a^4*c^3*d^8*n^2 + 3*a^5*c^2*d^6*n^2*e^2 + 3*a^6*c*d^4*n^2*e^4 + a^7*d^
2*n^2*e^6 + (a^2*c^5*d^7*n^2*e + 3*a^3*c^4*d^5*n^2*e^3 + 3*a^4*c^3*d^3*n^2*e^5 + a^5*c^2*d*n^2*e^7)*x^(5*n) +
(a^2*c^5*d^8*n^2 + 3*a^3*c^4*d^6*n^2*e^2 + 3*a^4*c^3*d^4*n^2*e^4 + a^5*c^2*d^2*n^2*e^6)*x^(4*n) + 2*(a^3*c^4*d
^7*n^2*e + 3*a^4*c^3*d^5*n^2*e^3 + 3*a^5*c^2*d^3*n^2*e^5 + a^6*c*d*n^2*e^7)*x^(3*n) + 2*(a^3*c^4*d^8*n^2 + 3*a
^4*c^3*d^6*n^2*e^2 + 3*a^5*c^2*d^4*n^2*e^4 + a^6*c*d^2*n^2*e^6)*x^(2*n) + (a^4*c^3*d^7*n^2*e + 3*a^5*c^2*d^5*n
^2*e^3 + 3*a^6*c*d^3*n^2*e^5 + a^7*d*n^2*e^7)*x^n) - integrate(-1/8*((8*n^2 - 6*n + 1)*c^4*d^6 + (32*n^2 - 18*
n + 1)*a*c^3*d^4*e^2 + (48*n^2 - 2*n - 1)*a^2*c^2*d^2*e^4 - (24*n^2 - 10*n + 1)*a^3*c*e^6 - 2*((3*n^2 - 4*n +
1)*c^4*d^5*e + 2*(7*n^2 - 8*n + 1)*a*c^3*d^3*e^3 + (35*n^2 - 12*n + 1)*a^2*c^2*d*e^5)*x^n)/(a^3*c^4*d^8*n^2 +
4*a^4*c^3*d^6*n^2*e^2 + 6*a^5*c^2*d^4*n^2*e^4 + 4*a^6*c*d^2*n^2*e^6 + a^7*n^2*e^8 + (a^2*c^5*d^8*n^2 + 4*a^3*c
^4*d^6*n^2*e^2 + 6*a^4*c^3*d^4*n^2*e^4 + 4*a^5*c^2*d^2*n^2*e^6 + a^6*c*n^2*e^8)*x^(2*n)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d+e*x^n)^2/(a+c*x^(2*n))^3,x, algorithm="fricas")

[Out]

integral(1/(2*a^3*d*x^n*e + a^3*d^2 + a^3*x^(2*n)*e^2 + (2*c^3*d*x^n*e + c^3*d^2 + c^3*x^(2*n)*e^2)*x^(6*n) +
3*(2*a*c^2*d*x^n*e + a*c^2*d^2 + a*c^2*x^(2*n)*e^2)*x^(4*n) + 3*(2*a^2*c*d*x^n*e + a^2*c*d^2 + a^2*c*x^(2*n)*e
^2)*x^(2*n)), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d+e*x**n)**2/(a+c*x**(2*n))**3,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d+e*x^n)^2/(a+c*x^(2*n))^3,x, algorithm="giac")

[Out]

integrate(1/((c*x^(2*n) + a)^3*(x^n*e + d)^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a+c\,x^{2\,n}\right )}^3\,{\left (d+e\,x^n\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^(2*n))^3*(d + e*x^n)^2),x)

[Out]

int(1/((a + c*x^(2*n))^3*(d + e*x^n)^2), x)

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