3.1027 \(\int \frac{x}{(a+b x^n) (c+d x^n)} \, dx\)

Optimal. Leaf size=89 \[ \frac{b x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a (b c-a d)}-\frac{d x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )}{2 c (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.033347, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {508, 364} \[ \frac{b x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a (b c-a d)}-\frac{d x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )}{2 c (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[x/((a + b*x^n)*(c + d*x^n)),x]

[Out]

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

Rule 508

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), I
nt[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
n, m}, x] && NeQ[b*c - a*d, 0]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.051509, size = 78, normalized size = 0.88 \[ \frac{b c x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )-a d x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )}{2 a b c^2-2 a^2 c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/(b*d*x^(2*n) + a*c + (b*c + a*d)*x^n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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