3.869 \(\int \frac{1}{x^3 \left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx\)

Optimal. Leaf size=145 \[ \frac{b (2 a d (n+1)-b c (n+2)) \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^2 n x^2 (b c-a d)^2}-\frac{d^2 \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 c x^2 (b c-a d)^2}+\frac{b}{a n x^2 (b c-a d) \left (a+b x^n\right )} \]

[Out]

b/(a*(b*c - a*d)*n*x^2*(a + b*x^n)) + (b*(2*a*d*(1 + n) - b*c*(2 + n))*Hypergeom
etric2F1[1, -2/n, -((2 - n)/n), -((b*x^n)/a)])/(2*a^2*(b*c - a*d)^2*n*x^2) - (d^
2*Hypergeometric2F1[1, -2/n, -((2 - n)/n), -((d*x^n)/c)])/(2*c*(b*c - a*d)^2*x^2
)

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Rubi [A]  time = 0.589429, antiderivative size = 145, 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 \[ \frac{b (2 a d (n+1)-b c (n+2)) \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^2 n x^2 (b c-a d)^2}-\frac{d^2 \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 c x^2 (b c-a d)^2}+\frac{b}{a n x^2 (b c-a d) \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

b/(a*(b*c - a*d)*n*x^2*(a + b*x^n)) + (b*(2*a*d*(1 + n) - b*c*(2 + n))*Hypergeom
etric2F1[1, -2/n, -((2 - n)/n), -((b*x^n)/a)])/(2*a^2*(b*c - a*d)^2*n*x^2) - (d^
2*Hypergeometric2F1[1, -2/n, -((2 - n)/n), -((d*x^n)/c)])/(2*c*(b*c - a*d)^2*x^2
)

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Rubi in Sympy [A]  time = 156.965, size = 214, normalized size = 1.48 \[ - \frac{b}{a n x^{2} \left (a + b x^{n}\right ) \left (a d - b c\right )} + \frac{b d^{2} x^{n - 2} \left (n + 2\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n - 2}{n} \\ 2 - \frac{2}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{a c n \left (- n + 2\right ) \left (a d - b c\right )^{2}} - \frac{d \left (a d n - b c \left (n + 2\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 a c n x^{2} \left (a d - b c\right )^{2}} - \frac{b^{2} d x^{n - 2} \left (n + 2\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n - 2}{n} \\ 2 - \frac{2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} n \left (- n + 2\right ) \left (a d - b c\right )^{2}} + \frac{b \left (a d n - b c \left (n + 2\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{2} n x^{2} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a+b*x**n)**2/(c+d*x**n),x)

[Out]

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

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Mathematica [A]  time = 0.275758, size = 136, normalized size = 0.94 \[ \frac{b c \left (a+b x^n\right ) (2 a d (n+1)-b c (n+2)) \, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )-a \left (a d^2 n \left (a+b x^n\right ) \, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{d x^n}{c}\right )+2 b c (a d-b c)\right )}{2 a^2 c n x^2 (b c-a d)^2 \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

(b*c*(2*a*d*(1 + n) - b*c*(2 + n))*(a + b*x^n)*Hypergeometric2F1[1, -2/n, (-2 +
n)/n, -((b*x^n)/a)] - a*(2*b*c*(-(b*c) + a*d) + a*d^2*n*(a + b*x^n)*Hypergeometr
ic2F1[1, -2/n, (-2 + n)/n, -((d*x^n)/c)]))/(2*a^2*c*(b*c - a*d)^2*n*x^2*(a + b*x
^n))

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Maple [F]  time = 0.165, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^2*(d*x^n + c)*x^3),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^2*(d*x^n + c)*x^3),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^2*(d*x^n + c)*x^3),x, algorithm="giac")

[Out]

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