3.4.2 \(\int \frac {1}{(a+b x^n) (c+d x^n)} \, dx\) [302]

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 = 0.02, 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 = {400, 251} \begin {gather*} \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 {gather*}

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 251

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 400

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

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 = 0.11, 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]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x^{n}\right ) \left (c + d x^{n}\right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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