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

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 = 0.694641, 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 \[ \frac{b^2 x (a d (1-3 n)-b (c-c n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n (b c-a d)^3}+\frac{d^2 x (a d (1-n)-b (c-3 c n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 n (b c-a d)^3}+\frac{b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac{d x (a d+b c)}{a c n (b c-a d)^2 \left (c+d x^n\right )} \]

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*(a*d*(1 - n) -
b*(c - 3*c*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 in Sympy [A]  time = 105.36, size = 162, normalized size = 0.84 \[ \frac{d x}{c n \left (a + b x^{n}\right ) \left (c + d x^{n}\right ) \left (a d - b c\right )} - \frac{d^{2} x \left (- a d n + a d + 3 b c n - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c^{2} n \left (a d - b c\right )^{3}} + \frac{b x \left (a d + b c\right )}{a c n \left (a + b x^{n}\right ) \left (a d - b c\right )^{2}} - \frac{b^{2} x \left (- 3 a d n + a d + b c n - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} n \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x/(c*n*(a + b*x**n)*(c + d*x**n)*(a*d - b*c)) - d**2*x*(-a*d*n + a*d + 3*b*c*n
- b*c)*hyper((1, 1/n), (1 + 1/n,), -d*x**n/c)/(c**2*n*(a*d - b*c)**3) + b*x*(a*
d + b*c)/(a*c*n*(a + b*x**n)*(a*d - b*c)**2) - b**2*x*(-3*a*d*n + a*d + b*c*n -
b*c)*hyper((1, 1/n), (1 + 1/n,), -b*x**n/a)/(a**2*n*(a*d - b*c)**3)

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Mathematica [A]  time = 0.344232, 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))*Hypergeometric2F1[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 = 0.153, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \]

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. \[{\left (a b^{2} d{\left (3 \, n - 1\right )} - b^{3} c{\left (n - 1\right )}\right )} \int -\frac{1}{a^{2} b^{3} c^{3} n - 3 \, a^{3} b^{2} c^{2} d n + 3 \, a^{4} b c d^{2} n - a^{5} d^{3} n +{\left (a b^{4} c^{3} n - 3 \, a^{2} b^{3} c^{2} d n + 3 \, a^{3} b^{2} c d^{2} n - a^{4} b d^{3} n\right )} x^{n}}\,{d x} -{\left (b c d^{2}{\left (3 \, n - 1\right )} - a d^{3}{\left (n - 1\right )}\right )} \int -\frac{1}{b^{3} c^{5} n - 3 \, a b^{2} c^{4} d n + 3 \, a^{2} b c^{3} d^{2} n - a^{3} c^{2} d^{3} n +{\left (b^{3} c^{4} d n - 3 \, a b^{2} c^{3} d^{2} n + 3 \, a^{2} b c^{2} d^{3} n - a^{3} c d^{4} n\right )} x^{n}}\,{d x} + \frac{{\left (b^{2} c d + a b d^{2}\right )} x x^{n} +{\left (b^{2} c^{2} + a^{2} d^{2}\right )} x}{a^{2} b^{2} c^{4} n - 2 \, a^{3} b c^{3} d n + a^{4} c^{2} d^{2} n +{\left (a b^{3} c^{3} d n - 2 \, a^{2} b^{2} c^{2} d^{2} n + a^{3} b c d^{3} n\right )} x^{2 \, n} +{\left (a b^{3} c^{4} n - a^{2} b^{2} c^{3} d n - a^{3} b c^{2} d^{2} n + a^{4} c d^{3} n\right )} x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^2*(d*x^n + c)^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. \[{\rm integral}\left (\frac{1}{b^{2} d^{2} x^{4 \, n} + a^{2} c^{2} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} x^{3 \, n} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{2 \, n} + 2 \,{\left (a b c^{2} + a^{2} c d\right )} x^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^2*(d*x^n + c)^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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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