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

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Rubi [A]  time = 0.17, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.26, 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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fricas [F]  time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a e^{2} x^{2 \, n} + 2 \, a d e x^{n} + a d^{2} + {\left (c e^{2} x^{2 \, n} + 2 \, c d e x^{n} + c d^{2}\right )} x^{2 \, n}}, x\right ) \]

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

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)

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maple [F]  time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \,x^{n}+d \right )^{2} \left (c \,x^{2 n}+a \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {e^{2} x}{c d^{4} n + a d^{2} e^{2} n + {\left (c d^{3} e n + a d e^{3} n\right )} x^{n}} + {\left (c d^{2} e^{2} {\left (3 \, n - 1\right )} + a e^{4} {\left (n - 1\right )}\right )} \int \frac {1}{c^{2} d^{6} n + 2 \, a c d^{4} e^{2} n + a^{2} d^{2} e^{4} n + {\left (c^{2} d^{5} e n + 2 \, a c d^{3} e^{3} n + a^{2} d e^{5} n\right )} x^{n}}\,{d x} - \int \frac {2 \, c^{2} d e x^{n} - c^{2} d^{2} + a c e^{2}}{a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} + {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{2 \, n}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

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]

Exception raised: HeuristicGCDFailed

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