3.11.27 \(\int \frac {x}{(a+b x^n) (c+d x^n)} \, dx\) [1027]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 371} \begin {gather*} \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)} \end {gather*}

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 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 {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*}

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Mathematica [A]
time = 0.07, size = 78, normalized size = 0.88 \begin {gather*} \frac {b c x^2 \, _2F_1\left (1,\frac {2}{n};\frac {2+n}{n};-\frac {b x^n}{a}\right )-a d x^2 \, _2F_1\left (1,\frac {2}{n};\frac {2+n}{n};-\frac {d x^n}{c}\right )}{2 a b c^2-2 a^2 c d} \end {gather*}

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 = 0.12, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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 = 0.00, 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(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 = 0.00, 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(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\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(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 = 0.00, 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(x/(a+b*x^n)/(c+d*x^n),x, algorithm="giac")

[Out]

integrate(x/((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 {x}{\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(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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