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

Optimal. Leaf size=193 \[ \frac{b^2 x (a d (1-3 n)-b (c-c n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n (b c-a d)^3}-\frac{d^2 x (b c (1-3 n)-a d (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 n (b c-a d)^3}+\frac{b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{d x (a d+b c)}{a c n (b c-a d)^2 \left (c+d x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.271006, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {414, 527, 522, 245} \[ \frac{b^2 x (a d (1-3 n)-b (c-c n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n (b c-a d)^3}-\frac{d^2 x (b c (1-3 n)-a d (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 n (b c-a d)^3}+\frac{b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{d x (a d+b c)}{a c n (b c-a d)^2 \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

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])

Rubi steps

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

Mathematica [A]  time = 0.208677, size = 147, normalized size = 0.76 \[ \frac{x \left (\frac{b^2 (a d (1-3 n)+b c (n-1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2}+\frac{b^2 (b c-a d)}{a \left (a+b x^n\right )}+\frac{d^2 (b c (3 n-1)-a d (n-1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2}+\frac{d^2 (b c-a d)}{c \left (c+d x^n\right )}\right )}{n (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*((b^2*(b*c - a*d))/(a*(a + b*x^n)) + (d^2*(b*c - a*d))/(c*(c + d*x^n)) + (b^2*(a*d*(1 - 3*n) + b*c*(-1 + n)
)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^2 + (d^2*(-(a*d*(-1 + n)) + b*c*(-1 + 3*n))*Hyperg
eometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/c^2))/((b*c - a*d)^3*n)

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Maple [F]  time = 0.734, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*b^2*d*(3*n - 1) - b^3*c*(n - 1))*integrate(-1/(a^2*b^3*c^3*n - 3*a^3*b^2*c^2*d*n + 3*a^4*b*c*d^2*n - a^5*d^
3*n + (a*b^4*c^3*n - 3*a^2*b^3*c^2*d*n + 3*a^3*b^2*c*d^2*n - a^4*b*d^3*n)*x^n), x) - (b*c*d^2*(3*n - 1) - a*d^
3*(n - 1))*integrate(-1/(b^3*c^5*n - 3*a*b^2*c^4*d*n + 3*a^2*b*c^3*d^2*n - a^3*c^2*d^3*n + (b^3*c^4*d*n - 3*a*
b^2*c^3*d^2*n + 3*a^2*b*c^2*d^3*n - a^3*c*d^4*n)*x^n), x) + ((b^2*c*d + a*b*d^2)*x*x^n + (b^2*c^2 + a^2*d^2)*x
)/(a^2*b^2*c^4*n - 2*a^3*b*c^3*d*n + a^4*c^2*d^2*n + (a*b^3*c^3*d*n - 2*a^2*b^2*c^2*d^2*n + a^3*b*c*d^3*n)*x^(
2*n) + (a*b^3*c^4*n - a^2*b^2*c^3*d*n - a^3*b*c^2*d^2*n + a^4*c*d^3*n)*x^n)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{2} d^{2} x^{4 \, n} + a^{2} c^{2} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} x^{3 \, n} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{2 \, n} + 2 \,{\left (a b c^{2} + a^{2} c d\right )} x^{n}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a + b*x**n)**2*(c + d*x**n)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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