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

Optimal. Leaf size=72 \[ \frac{b x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)}-\frac{d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0258087, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {391, 245} \[ \frac{b x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)}-\frac{d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^n)*(c + d*x^n)),x]

[Out]

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

Rule 391

Int[1/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^n),
x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0]

Rule 245

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

Rubi steps

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

Mathematica [A]  time = 0.0340111, size = 64, normalized size = 0.89 \[ \frac{x \left (a d \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )-b c \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )\right )}{a c (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.711, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \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(1/(a+b*x^n)/(c+d*x^n),x)

[Out]

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

[Out]

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

[Out]

integral(1/(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{1}{\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(1/(a+b*x**n)/(c+d*x**n),x)

[Out]

Integral(1/((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{1}{{\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(1/(a+b*x^n)/(c+d*x^n),x, algorithm="giac")

[Out]

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