Optimal. Leaf size=127 \[ \frac{b x \left (c+d x^n\right )^{-\frac{1-n}{n}}}{a n (b c-a d) \left (a+b x^n\right )}-\frac{x \left (c+d x^n\right )^{-1/n} (a d n+b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2 n (b c-a d)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.151078, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{b x \left (c+d x^n\right )^{-\frac{1-n}{n}}}{a n (b c-a d) \left (a+b x^n\right )}-\frac{x \left (c+d x^n\right )^{-1/n} (a d n+b (c-c n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2 n (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^n)^2*(c + d*x^n)^n^(-1)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 19.2463, size = 95, normalized size = 0.75 \[ - \frac{b x \left (c + d x^{n}\right )^{\frac{n - 1}{n}}}{a n \left (a + b x^{n}\right ) \left (a d - b c\right )} + \frac{x \left (c + d x^{n}\right )^{- \frac{1}{n}} \left (a d n - b c n + b c\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, 1 \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{x^{n} \left (- a d + b c\right )}{a \left (c + d x^{n}\right )}} \right )}}{a^{2} n \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*x**n)**2/((c+d*x**n)**(1/n)),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 62.1905, size = 1070, normalized size = 8.43 \[ \frac{c^2 (2 n+1) (3 n+1) x \left (b x^n+a\right ) \left (d x^n+c\right )^{-1/n} \left (\frac{d x^n}{c}+1\right ) \Gamma \left (2+\frac{1}{n}\right ) \Gamma \left (3+\frac{1}{n}\right ) \left (\frac{2 (b c-a d) n \left (d x^n+c\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n}{\left (b x^n+a\right ) \Gamma \left (3+\frac{1}{n}\right )}+\frac{c \left (d n x^n+c+c n\right ) \, _2F_1\left (1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )}{\Gamma \left (2+\frac{1}{n}\right )}\right )}{-c d (1-n) (2 n+1) (3 n+1) \left (b x^n+a\right )^2 \left (2 (b c-a d) n \left (d x^n+c\right ) \Gamma \left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n+c \left (b x^n+a\right ) \left (d n x^n+c+c n\right ) \Gamma \left (3+\frac{1}{n}\right ) \, _2F_1\left (1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )\right ) x^n-2 b c n (2 n+1) (3 n+1) \left (b x^n+a\right ) \left (d x^n+c\right ) \left (2 (b c-a d) n \left (d x^n+c\right ) \Gamma \left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n+c \left (b x^n+a\right ) \left (d n x^n+c+c n\right ) \Gamma \left (3+\frac{1}{n}\right ) \, _2F_1\left (1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )\right ) x^n+n^2 \left (d x^n+c\right ) \left (2 c d (b c-a d) (2 n+1) (3 n+1) \left (b x^n+a\right )^2 \Gamma \left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n-2 b c (b c-a d) (2 n+1) (3 n+1) \left (b x^n+a\right ) \left (d x^n+c\right ) \Gamma \left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n+12 a (b c-a d)^2 n (2 n+1) \left (d x^n+c\right ) \Gamma \left (2+\frac{1}{n}\right ) \, _2F_1\left (3,4;4+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n+c^2 d (2 n+1) (3 n+1) \left (b x^n+a\right )^3 \Gamma \left (3+\frac{1}{n}\right ) \, _2F_1\left (1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )+2 c (b c-a d) (2 n+1) (3 n+1) \left (b x^n+a\right )^2 \left (d x^n+c\right ) \Gamma \left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )+2 a c (b c-a d) (3 n+1) \left (b x^n+a\right ) \left (d n x^n+c+c n\right ) \Gamma \left (3+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )\right ) x^n+c (2 n+1) (3 n+1) \left (b x^n+a\right )^2 \left (d x^n+c\right ) \left (2 (b c-a d) n \left (d x^n+c\right ) \Gamma \left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n+c \left (b x^n+a\right ) \left (d n x^n+c+c n\right ) \Gamma \left (3+\frac{1}{n}\right ) \, _2F_1\left (1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((a + b*x^n)^2*(c + d*x^n)^n^(-1)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.116, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{2}\sqrt [n]{c+d{x}^{n}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*x^n)^2/((c+d*x^n)^(1/n)),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{n} + c\right )}^{-\frac{1}{n}}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^2*(d*x^n + c)^(1/n)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}{\left (d x^{n} + c\right )}^{\left (\frac{1}{n}\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^2*(d*x^n + c)^(1/n)),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)**(1/n)),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )}^{\left (\frac{1}{n}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^2*(d*x^n + c)^(1/n)),x, algorithm="giac")
[Out]