Optimal. Leaf size=122 \[ \frac{b x (a d (1-2 n)-b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n (b c-a d)^2}+\frac{d^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c (b c-a d)^2}+\frac{b x}{a n (b c-a d) \left (a+b x^n\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.341763, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{b x (a d (1-2 n)-b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n (b c-a d)^2}+\frac{d^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c (b c-a d)^2}+\frac{b x}{a n (b c-a d) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^n)^2*(c + d*x^n)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 53.7515, size = 100, normalized size = 0.82 \[ \frac{d^{2} x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c \left (a d - b c\right )^{2}} - \frac{b x}{a n \left (a + b x^{n}\right ) \left (a d - b c\right )} + \frac{b x \left (- 2 a d n + a d + b c n - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} n \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*x**n)**2/(c+d*x**n),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.272287, size = 108, normalized size = 0.89 \[ \frac{x \left (\frac{b^2 c-a b d}{a^2 n+a b n x^n}+\frac{b (a d (1-2 n)+b c (n-1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n}+\frac{d^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c}\right )}{(b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^n)^2*(c + d*x^n)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0., size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*x^n)^2/(c+d*x^n),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ d^{2} \int \frac{1}{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{n}}\,{d x} -{\left (a b d{\left (2 \, n - 1\right )} - b^{2} c{\left (n - 1\right )}\right )} \int \frac{1}{a^{2} b^{2} c^{2} n - 2 \, a^{3} b c d n + a^{4} d^{2} n +{\left (a b^{3} c^{2} n - 2 \, a^{2} b^{2} c d n + a^{3} b d^{2} n\right )} x^{n}}\,{d x} + \frac{b x}{a^{2} b c n - a^{3} d n +{\left (a b^{2} c n - a^{2} b d n\right )} x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b^{2} d x^{3 \, n} + a^{2} c +{\left (b^{2} c + 2 \, a b d\right )} x^{2 \, n} +{\left (2 \, a b c + a^{2} d\right )} x^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="giac")
[Out]