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

Optimal. Leaf size=143 \[ \frac{b x^2 (2 a d (1-n)-b c (2-n)) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^2 n (b c-a d)^2}+\frac{d^2 x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )}{2 c (b c-a d)^2}+\frac{b x^2}{a n (b c-a d) \left (a+b x^n\right )} \]

[Out]

(b*x^2)/(a*(b*c - a*d)*n*(a + b*x^n)) + (b*(2*a*d*(1 - n) - b*c*(2 - n))*x^2*Hyp
ergeometric2F1[1, 2/n, (2 + n)/n, -((b*x^n)/a)])/(2*a^2*(b*c - a*d)^2*n) + (d^2*
x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, -((d*x^n)/c)])/(2*c*(b*c - a*d)^2)

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Rubi [A]  time = 0.494396, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{b x^2 (2 a d (1-n)-b c (2-n)) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^2 n (b c-a d)^2}+\frac{d^2 x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )}{2 c (b c-a d)^2}+\frac{b x^2}{a n (b c-a d) \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

(b*x^2)/(a*(b*c - a*d)*n*(a + b*x^n)) + (b*(2*a*d*(1 - n) - b*c*(2 - n))*x^2*Hyp
ergeometric2F1[1, 2/n, (2 + n)/n, -((b*x^n)/a)])/(2*a^2*(b*c - a*d)^2*n) + (d^2*
x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, -((d*x^n)/c)])/(2*c*(b*c - a*d)^2)

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Rubi in Sympy [A]  time = 145.688, size = 218, normalized size = 1.52 \[ - \frac{b x^{2}}{a n \left (a + b x^{n}\right ) \left (a d - b c\right )} + \frac{b d^{2} x^{n + 2} \left (- n + 2\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + 2}{n} \\ 2 + \frac{2}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{a c n \left (n + 2\right ) \left (a d - b c\right )^{2}} + \frac{d x^{2} \left (a d n - b c n + 2 b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 a c n \left (a d - b c\right )^{2}} - \frac{b^{2} d x^{n + 2} \left (- n + 2\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + 2}{n} \\ 2 + \frac{2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} n \left (n + 2\right ) \left (a d - b c\right )^{2}} - \frac{b x^{2} \left (a d n - b c n + 2 b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{2} n \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.273947, size = 134, normalized size = 0.94 \[ \frac{x^2 \left (a \left (a d^2 n \left (a+b x^n\right ) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )+2 b c (b c-a d)\right )+b c \left (a+b x^n\right ) (b c (n-2)-2 a d (n-1)) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )\right )}{2 a^2 c n (b c-a d)^2 \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(b*c*(b*c*(-2 + n) - 2*a*d*(-1 + 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)*Hypergeome
tric2F1[1, 2/n, (2 + n)/n, -((d*x^n)/c)])))/(2*a^2*c*(b*c - a*d)^2*n*(a + b*x^n)
)

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Maple [F]  time = 0.163, size = 0, normalized size = 0. \[ \int{\frac{x}{ \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^n)^2/(c+d*x^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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