3.11.29 \(\int \frac {1}{x (a+b x^n) (c+d x^n)} \, dx\) [1029]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 63, 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 = {457, 84} \begin {gather*} -\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} \end {gather*}

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 84

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]

Rule 457

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]]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 58, normalized size = 0.92 \begin {gather*} \frac {b c \log \left (x^n\right )-a d \log \left (x^n\right )-b c \log \left (a+b x^n\right )+a d \log \left (c+d x^n\right )}{a b c^2 n-a^2 c d n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*Log[x^n] - a*d*Log[x^n] - 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 = 0.39, size = 64, normalized size = 1.02

method	result	size
derivativedivides	\(\frac {-\frac {d \ln \left (c +d \,x^{n}\right )}{c \left (a d -b c \right )}+\frac {b \ln \left (a +b \,x^{n}\right )}{a \left (a d -b c \right )}+\frac {\ln \left (x^{n}\right )}{a c}}{n}\)	\(64\)
default	\(\frac {-\frac {d \ln \left (c +d \,x^{n}\right )}{c \left (a d -b c \right )}+\frac {b \ln \left (a +b \,x^{n}\right )}{a \left (a d -b c \right )}+\frac {\ln \left (x^{n}\right )}{a c}}{n}\)	\(64\)
norman	\(\frac {\ln \left (x \right )}{a c}+\frac {b \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{\left (a d -b c \right ) a n}-\frac {d \ln \left (c +d \,{\mathrm e}^{n \ln \left (x \right )}\right )}{c n \left (a d -b c \right )}\)	\(68\)
risch	\(-\frac {\ln \left (x \right ) b}{\left (a d -b c \right ) a}+\frac {\ln \left (x \right ) d}{c \left (a d -b c \right )}+\frac {b \ln \left (x^{n}+\frac {a}{b}\right )}{\left (a d -b c \right ) a n}-\frac {d \ln \left (x^{n}+\frac {c}{d}\right )}{c n \left (a d -b c \right )}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 69, normalized size = 1.10 \begin {gather*} -\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 {gather*}

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 = 1.03, size = 58, normalized size = 0.92 \begin {gather*} -\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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (46) = 92\).
time = 1.38, size = 332, normalized size = 5.27 \begin {gather*} \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 {c}{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}} & \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 {gather*}

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)),
((-1/(c*n*x**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) + c/(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), Eq(b, a*d/c)), ((-1/(a*n*x**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 = 0.00, 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/(a+b*x^n)/(c+d*x^n),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 5.72, size = 162, normalized size = 2.57 \begin {gather*} \frac {b\,\ln \left (-\frac {1}{b\,d\,x}-\frac {2\,a\,c\,n+a\,d\,n\,x^n+b\,c\,n\,x^n}{d\,x\,\left (a^2\,d\,n-a\,b\,c\,n\right )}\right )}{a^2\,d\,n-a\,b\,c\,n}+\frac {d\,\ln \left (-\frac {1}{b\,d\,x}-\frac {2\,a\,c\,n+a\,d\,n\,x^n+b\,c\,n\,x^n}{b\,x\,\left (b\,c^2\,n-a\,c\,d\,n\right )}\right )}{b\,c^2\,n-a\,c\,d\,n}+\frac {\ln \left (x\right )\,\left (n-1\right )}{a\,c\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^n)*(c + d*x^n)),x)

[Out]

(b*log(- 1/(b*d*x) - (2*a*c*n + a*d*n*x^n + b*c*n*x^n)/(d*x*(a^2*d*n - a*b*c*n))))/(a^2*d*n - a*b*c*n) + (d*lo
g(- 1/(b*d*x) - (2*a*c*n + a*d*n*x^n + b*c*n*x^n)/(b*x*(b*c^2*n - a*c*d*n))))/(b*c^2*n - a*c*d*n) + (log(x)*(n
- 1))/(a*c*n)

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