3.326 \(\int \frac{(c+d x^n)^{1-\frac{1}{n}}}{(a+b x^n)^2} \, dx\)

Optimal. Leaf size=54 \[ \frac{c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0136726, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {379} \[ \frac{c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(a^2*(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-\frac{1}{n}}}{\left (a+b x^n\right )^2} \, dx &=\frac{c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0132257, size = 53, normalized size = 0.98 \[ \frac{c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};\frac{(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.7, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{2}} \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)^2,x)

[Out]

int((c+d*x^n)^(1-1/n)/(a+b*x^n)^2,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}}{{\left (b x^{n} + a\right )}^{2}}\,{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)^2,x, algorithm="maxima")

[Out]

integrate((d*x^n + c)^(-1/n + 1)/(b*x^n + a)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d x^{n} + c\right )}^{\frac{n - 1}{n}}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, 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)^2,x, algorithm="fricas")

[Out]

integral((d*x^n + c)^((n - 1)/n)/(b^2*x^(2*n) + 2*a*b*x^n + a^2), 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)**2,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}}{{\left (b x^{n} + a\right )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

integrate((d*x^n + c)^(-1/n + 1)/(b*x^n + a)^2, x)