3.1029 \(\int \frac{1}{x (a+b x^n) (c+d x^n)} \, dx\)

Optimal. Leaf size=63 \[ -\frac{b \log \left (a+b x^n\right )}{a n (b c-a d)}+\frac{d \log \left (c+d x^n\right )}{c n (b c-a d)}+\frac{\log (x)}{a c} \]

[Out]

Log[x]/(a*c) - (b*Log[a + b*x^n])/(a*(b*c - a*d)*n) + (d*Log[c + d*x^n])/(c*(b*c - a*d)*n)

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Rubi [A]  time = 0.0678896, antiderivative size = 63, normalized size of antiderivative = 1., 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 = {446, 72} \[ -\frac{b \log \left (a+b x^n\right )}{a n (b c-a d)}+\frac{d \log \left (c+d x^n\right )}{c n (b c-a d)}+\frac{\log (x)}{a c} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^n)*(c + d*x^n)),x]

[Out]

Log[x]/(a*c) - (b*Log[a + b*x^n])/(a*(b*c - a*d)*n) + (d*Log[c + d*x^n])/(c*(b*c - a*d)*n)

Rule 446

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0476577, size = 56, normalized size = 0.89 \[ \frac{-b c \log \left (a+b x^n\right )+a d \log \left (c+d x^n\right )-a d n \log (x)+b c n \log (x)}{a b c^2 n-a^2 c d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 69, normalized size = 1.1 \begin{align*}{\frac{b\ln \left ( a+b{x}^{n} \right ) }{n \left ( ad-bc \right ) a}}-{\frac{d\ln \left ( c+d{x}^{n} \right ) }{nc \left ( ad-bc \right ) }}+{\frac{\ln \left ({x}^{n} \right ) }{nca}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/n*b/(a*d-b*c)/a*ln(a+b*x^n)-1/n*d/c/(a*d-b*c)*ln(c+d*x^n)+1/n/c/a*ln(x^n)

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Maxima [A]  time = 0.947872, size = 93, normalized size = 1.48 \begin{align*} -\frac{b \log \left (\frac{b x^{n} + a}{b}\right )}{a b c n - a^{2} d n} + \frac{d \log \left (\frac{d x^{n} + c}{d}\right )}{b c^{2} n - a c d n} + \frac{\log \left (x\right )}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*log((b*x^n + a)/b)/(a*b*c*n - a^2*d*n) + d*log((d*x^n + c)/d)/(b*c^2*n - a*c*d*n) + log(x)/(a*c)

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Fricas [A]  time = 1.08512, size = 123, normalized size = 1.95 \begin{align*} -\frac{b c \log \left (b x^{n} + a\right ) - a d \log \left (d x^{n} + c\right ) -{\left (b c - a d\right )} n \log \left (x\right )}{{\left (a b c^{2} - a^{2} c d\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*c*log(b*x^n + a) - a*d*log(d*x^n + c) - (b*c - a*d)*n*log(x))/((a*b*c^2 - a^2*c*d)*n)

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Sympy [A]  time = 86.2728, size = 335, normalized size = 5.32 \begin{align*} \begin{cases} \frac{\frac{\log{\left (x \right )}}{c} - \frac{\log{\left (\frac{c}{d} + x^{n} \right )}}{c n}}{a} & \text{for}\: b = 0 \\\frac{\frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{n} \right )}}{a n}}{c} & \text{for}\: d = 0 \\\frac{- \frac{x^{- n}}{c n} + \frac{d \log{\left (x^{- n} + \frac{d}{c} \right )}}{c^{2} n}}{b} & \text{for}\: a = 0 \\\frac{c n \log{\left (x \right )}}{a c^{2} n + a c d n x^{n}} - \frac{c \log{\left (\frac{c}{d} + x^{n} \right )}}{a c^{2} n + a c d n x^{n}} + \frac{d n x^{n} \log{\left (x \right )}}{a c^{2} n + a c d n x^{n}} - \frac{d x^{n} \log{\left (\frac{c}{d} + x^{n} \right )}}{a c^{2} n + a c d n x^{n}} - \frac{d x^{n}}{a c^{2} n + a c d n x^{n}} & \text{for}\: b = \frac{a d}{c} \\\frac{- \frac{x^{- n}}{a n} + \frac{b \log{\left (x^{- n} + \frac{b}{a} \right )}}{a^{2} n}}{d} & \text{for}\: c = 0 \\\frac{\log{\left (x \right )}}{\left (a + b\right ) \left (c + d\right )} & \text{for}\: n = 0 \\\frac{a d n \log{\left (x \right )}}{a^{2} c d n - a b c^{2} n} - \frac{a d \log{\left (\frac{c}{d} + x^{n} \right )}}{a^{2} c d n - a b c^{2} n} - \frac{b c n \log{\left (x \right )}}{a^{2} c d n - a b c^{2} n} + \frac{b c \log{\left (\frac{a}{b} + x^{n} \right )}}{a^{2} c d n - a b c^{2} n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((log(x)/c - log(c/d + x**n)/(c*n))/a, Eq(b, 0)), ((log(x)/a - log(a/b + x**n)/(a*n))/c, Eq(d, 0)),
((-x**(-n)/(c*n) + d*log(x**(-n) + d/c)/(c**2*n))/b, Eq(a, 0)), (c*n*log(x)/(a*c**2*n + a*c*d*n*x**n) - c*log(
c/d + x**n)/(a*c**2*n + a*c*d*n*x**n) + d*n*x**n*log(x)/(a*c**2*n + a*c*d*n*x**n) - d*x**n*log(c/d + x**n)/(a*
c**2*n + a*c*d*n*x**n) - d*x**n/(a*c**2*n + a*c*d*n*x**n), Eq(b, a*d/c)), ((-x**(-n)/(a*n) + b*log(x**(-n) + b
/a)/(a**2*n))/d, Eq(c, 0)), (log(x)/((a + b)*(c + d)), Eq(n, 0)), (a*d*n*log(x)/(a**2*c*d*n - a*b*c**2*n) - a*
d*log(c/d + x**n)/(a**2*c*d*n - a*b*c**2*n) - b*c*n*log(x)/(a**2*c*d*n - a*b*c**2*n) + b*c*log(a/b + x**n)/(a*
*2*c*d*n - a*b*c**2*n), True))

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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 )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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