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

Optimal. Leaf size=142 \[ -\frac{b (b c (n+1)-a d (2 n+1)) \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^2 n x (b c-a d)^2}-\frac{d^2 \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{d x^n}{c}\right )}{c x (b c-a d)^2}+\frac{b}{a n x (b c-a d) \left (a+b x^n\right )} \]

[Out]

b/(a*(b*c - a*d)*n*x*(a + b*x^n)) - (b*(b*c*(1 + n) - a*d*(1 + 2*n))*Hypergeomet
ric2F1[1, -n^(-1), -((1 - n)/n), -((b*x^n)/a)])/(a^2*(b*c - a*d)^2*n*x) - (d^2*H
ypergeometric2F1[1, -n^(-1), -((1 - n)/n), -((d*x^n)/c)])/(c*(b*c - a*d)^2*x)

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Rubi [A]  time = 0.606682, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{b (b c (n+1)-a d (2 n+1)) \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^2 n x (b c-a d)^2}-\frac{d^2 \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{d x^n}{c}\right )}{c x (b c-a d)^2}+\frac{b}{a n x (b c-a d) \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

b/(a*(b*c - a*d)*n*x*(a + b*x^n)) - (b*(b*c*(1 + n) - a*d*(1 + 2*n))*Hypergeomet
ric2F1[1, -n^(-1), -((1 - n)/n), -((b*x^n)/a)])/(a^2*(b*c - a*d)^2*n*x) - (d^2*H
ypergeometric2F1[1, -n^(-1), -((1 - n)/n), -((d*x^n)/c)])/(c*(b*c - a*d)^2*x)

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Rubi in Sympy [A]  time = 154.167, size = 206, normalized size = 1.45 \[ - \frac{b}{a n x \left (a + b x^{n}\right ) \left (a d - b c\right )} + \frac{b d^{2} x^{n - 1} \left (n + 1\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n - 1}{n} \\ 2 - \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{a c n \left (- n + 1\right ) \left (a d - b c\right )^{2}} - \frac{d \left (a d n - b c \left (n + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{a c n x \left (a d - b c\right )^{2}} - \frac{b^{2} d x^{n - 1} \left (n + 1\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n - 1}{n} \\ 2 - \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} n \left (- n + 1\right ) \left (a d - b c\right )^{2}} + \frac{b \left (a d n - b c \left (n + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} n x \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.271876, size = 133, normalized size = 0.94 \[ \frac{b c \left (a+b x^n\right ) (a d (2 n+1)-b c (n+1)) \, _2F_1\left (1,-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )-a \left (a d^2 n \left (a+b x^n\right ) \, _2F_1\left (1,-\frac{1}{n};\frac{n-1}{n};-\frac{d x^n}{c}\right )+b c (a d-b c)\right )}{a^2 c n x (b c-a d)^2 \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

(b*c*(-(b*c*(1 + n)) + a*d*(1 + 2*n))*(a + b*x^n)*Hypergeometric2F1[1, -n^(-1),
(-1 + n)/n, -((b*x^n)/a)] - a*(b*c*(-(b*c) + a*d) + a*d^2*n*(a + b*x^n)*Hypergeo
metric2F1[1, -n^(-1), (-1 + n)/n, -((d*x^n)/c)]))/(a^2*c*(b*c - a*d)^2*n*x*(a +
b*x^n))

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Maple [F]  time = 0.192, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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