3.11.30 \(\int \frac {1}{x^2 (a+b x^n) (c+d x^n)} \, dx\) [1030]

Optimal. Leaf size=90 \[ -\frac {b \, _2F_1\left (1,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a (b c-a d) x}+\frac {d \, _2F_1\left (1,-\frac {1}{n};-\frac {1-n}{n};-\frac {d x^n}{c}\right )}{c (b c-a d) x} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {522, 371} \begin {gather*} \frac {d \, _2F_1\left (1,-\frac {1}{n};-\frac {1-n}{n};-\frac {d x^n}{c}\right )}{c x (b c-a d)}-\frac {b \, _2F_1\left (1,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a x (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 522

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]

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 74, normalized size = 0.82 \begin {gather*} \frac {b c \, _2F_1\left (1,-\frac {1}{n};\frac {-1+n}{n};-\frac {b x^n}{a}\right )-a d \, _2F_1\left (1,-\frac {1}{n};\frac {-1+n}{n};-\frac {d x^n}{c}\right )}{a c (-b c+a d) x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x^2/(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/x^2/(a+b*x^n)/(c+d*x^n),x, algorithm="maxima")

[Out]

integrate(1/((b*x^n + a)*(d*x^n + c)*x^2), 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/x^2/(a+b*x^n)/(c+d*x^n),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \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/x**2/(a+b*x**n)/(c+d*x**n),x)

[Out]

Integral(1/(x**2*(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/x^2/(a+b*x^n)/(c+d*x^n),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^2\,\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/(x^2*(a + b*x^n)*(c + d*x^n)),x)

[Out]

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

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