3.46 \(\int \frac{1}{(d+e x^n)^2 (a+c x^{2 n})} \, dx\)

Optimal. Leaf size=205 \[ -\frac{2 c^2 d e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^2}+\frac{c x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^2}+\frac{2 c e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{\left (a e^2+c d^2\right )^2}+\frac{e^2 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )} \]

[Out]

(c*(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)^2) +
(2*c*e^2*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(c*d^2 + a*e^2)^2 - (2*c^2*d*e*x^(1 + n)*Hy
pergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^2*(1 + n)) + (e^2*x*Hy
pergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 + a*e^2))

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Rubi [A]  time = 0.173819, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {1425, 245, 1418, 364} \[ -\frac{2 c^2 d e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^2}+\frac{c x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^2}+\frac{2 c e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{\left (a e^2+c d^2\right )^2}+\frac{e^2 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(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)^2) +
(2*c*e^2*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(c*d^2 + a*e^2)^2 - (2*c^2*d*e*x^(1 + n)*Hy
pergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^2*(1 + n)) + (e^2*x*Hy
pergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 + a*e^2))

Rule 1425

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)^q/(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] && IntegerQ[q]

Rule 245

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 1418

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 364

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

Rubi steps

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

Mathematica [A]  time = 0.266714, size = 186, normalized size = 0.91 \[ \frac{x \left (e \left (-2 c^2 d^3 x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+a e (n+1) \left (a e^2+c d^2\right ) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )+2 a c d^2 e (n+1) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )\right )+c d^2 (n+1) \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )\right )}{a (n+1) \left (a d e^2+c d^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.141, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d+e{x}^{n} \right ) ^{2} \left ( a+c{x}^{2\,n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{e^{2} x}{c d^{4} n + a d^{2} e^{2} n +{\left (c d^{3} e n + a d e^{3} n\right )} x^{n}} +{\left (c d^{2} e^{2}{\left (3 \, n - 1\right )} + a e^{4}{\left (n - 1\right )}\right )} \int \frac{1}{c^{2} d^{6} n + 2 \, a c d^{4} e^{2} n + a^{2} d^{2} e^{4} n +{\left (c^{2} d^{5} e n + 2 \, a c d^{3} e^{3} n + a^{2} d e^{5} n\right )} x^{n}}\,{d x} - \int \frac{2 \, c^{2} d e x^{n} - c^{2} d^{2} + a c e^{2}}{a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{2 \, n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^2*x/(c*d^4*n + a*d^2*e^2*n + (c*d^3*e*n + a*d*e^3*n)*x^n) + (c*d^2*e^2*(3*n - 1) + a*e^4*(n - 1))*integrate(
1/(c^2*d^6*n + 2*a*c*d^4*e^2*n + a^2*d^2*e^4*n + (c^2*d^5*e*n + 2*a*c*d^3*e^3*n + a^2*d*e^5*n)*x^n), x) - inte
grate((2*c^2*d*e*x^n - c^2*d^2 + a*c*e^2)/(a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4 + (c^3*d^4 + 2*a*c^2*d^2*e^2
+ a^2*c*e^4)*x^(2*n)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a e^{2} x^{2 \, n} + 2 \, a d e x^{n} + a d^{2} +{\left (c e^{2} x^{2 \, n} + 2 \, c d e x^{n} + c d^{2}\right )} x^{2 \, n}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2 \, n} + a\right )}{\left (e x^{n} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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