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

Optimal. Leaf size=53 \[ \frac{x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0478586, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.25931, size = 39, normalized size = 0.74 \[ \frac{x \left (c + d x^{n}\right )^{- \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, 1 \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{x^{n} \left (- a d + b c\right )}{a \left (c + d x^{n}\right )}} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.300655, size = 79, normalized size = 1.49 \[ \frac{x \left (c+d x^n\right )^{-1/n} \left (\frac{a \left (c+d x^n\right )}{c \left (a+b x^n\right )}\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n},\frac{1}{n};1+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.12, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) \sqrt [n]{c+d{x}^{n}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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