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

Optimal. Leaf size=95 \[ \frac{b x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a (b c-a d)}-\frac{d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0313336, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {382, 379} \[ \frac{b x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a (b c-a d)}-\frac{d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -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-\frac{1}{n}}}{a+b x^n} \, dx &=-\frac{d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)}+\frac{b \int \frac{\left (c+d x^n\right )^{-1/n}}{a+b x^n} \, dx}{b c-a d}\\ &=-\frac{d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)}+\frac{b 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 (b c-a d)}\\ \end{align*}

Mathematica [C]  time = 6.44092, size = 153, normalized size = 1.61 \[ \frac{x \left (c+d x^n\right )^{-\frac{n+1}{n}} \left (\frac{b n x^{2 n} (a d-b c) \, _2F_1\left (2,2+\frac{1}{n};3+\frac{1}{n};\frac{(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{a^2 (2 n+1) \left (c+d x^n\right )}+\frac{b x^n \Phi \left (\frac{(a d-b c) x^n}{a \left (d x^n+c\right )},1,1+\frac{1}{n}\right )}{a}+\frac{a \left (c+d x^n\right )}{c \left (a+b x^n\right )}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*((a*(c + d*x^n))/(c*(a + b*x^n)) + (b*x^n*HurwitzLerchPhi[((-(b*c) + a*d)*x^n)/(a*(c + d*x^n)), 1, 1 + n^(-
1)])/a + (b*(-(b*c) + a*d)*n*x^(2*n)*Hypergeometric2F1[2, 2 + n^(-1), 3 + n^(-1), ((-(b*c) + a*d)*x^n)/(a*(c +
d*x^n))])/(a^2*(1 + 2*n)*(c + d*x^n))))/(a*(c + d*x^n)^((1 + n)/n))

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Maple [F]  time = 0.696, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b{x}^{n}} \left ( c+d{x}^{n} \right ) ^{-1-{n}^{-1}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x^n + c)^(-1/n - 1)/(b*x^n + a), 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 )}^{\frac{n + 1}{n}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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