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

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))

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Rubi [A]  time = 0.0139553, 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, Rules used = {379} \[ \frac{x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a} \]

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))

Rule 379

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*x*Hypergeometric2F1[1/
n, -p, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(c^(p + 1)*(c + d*x^n)^(1/n)), x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && ILtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (c+d x^n\right )^{-1/n}}{a+b x^n} \, dx &=\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 (c+d x^n\right )}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0129253, size = 52, normalized size = 0.98 \[ \frac{x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};\frac{(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[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))

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Maple [F]  time = 0.714, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{x}^{n} \right ) \sqrt [n]{c+d{x}^{n}}}}\, dx \end{align*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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