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

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Rubi [A]  time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [F]  time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b d x^{2 \, n} + a c + {\left (b c + a d\right )} x^{n}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}}\,{d x} \]

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)

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maple [F]  time = 0.91, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{n}+a \right ) \left (d \,x^{n}+c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\left (a+b\,x^n\right )\,\left (c+d\,x^n\right )} \,d x \]

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x^{n}\right ) \left (c + d x^{n}\right )}\, dx \]

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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