3.5.25 \(\int \frac {a+b \log (c x^n)}{x (d+e x^r)^2} \, dx\) [425]

Optimal. Leaf size=102 \[ -\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^2 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^2 r}+\frac {b n \log \left (d+e x^r\right )}{d^2 r^2}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^2 r^2} \]

[Out]

-e*x^r*(a+b*ln(c*x^n))/d^2/r/(d+e*x^r)-(a+b*ln(c*x^n))*ln(1+d/e/(x^r))/d^2/r+b*n*ln(d+e*x^r)/d^2/r^2+b*n*polyl
og(2,-d/e/(x^r))/d^2/r^2

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Rubi [A]
time = 0.16, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2391, 2379, 2438, 2373, 266} \begin {gather*} \frac {b n \text {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^2 r^2}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 r}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^2 r \left (d+e x^r\right )}+\frac {b n \log \left (d+e x^r\right )}{d^2 r^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*x^r*(a + b*Log[c*x^n]))/(d^2*r*(d + e*x^r))) - ((a + b*Log[c*x^n])*Log[1 + d/(e*x^r)])/(d^2*r) + (b*n*Log
[d + e*x^r])/(d^2*r^2) + (b*n*PolyLog[2, -(d/(e*x^r))])/(d^2*r^2)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2373

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Dist[b*(n/(d*(m + 1))), Int[(f*x)^
m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[
m, -1]

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2391

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_))/(x_), x_Symbol] :> Dist[1/d,
Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[x^(r - 1)*(d + e*x^r)^q*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0] && ILtQ[q, -1]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^2} \, dx &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx}{d}-\frac {e \int \frac {x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx}{d}\\ &=-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^2 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^2 r}+\frac {(b n) \int \frac {\log \left (1+\frac {d x^{-r}}{e}\right )}{x} \, dx}{d^2 r}+\frac {(b e n) \int \frac {x^{-1+r}}{d+e x^r} \, dx}{d^2 r}\\ &=-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^2 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^2 r}+\frac {b n \log \left (d+e x^r\right )}{d^2 r^2}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^2 r^2}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 132, normalized size = 1.29 \begin {gather*} \frac {\frac {d r \left (a+b \log \left (c x^n\right )\right )}{d+e x^r}+b n \log \left (d-d x^r\right )-a r \log \left (d-d x^r\right )+b r \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+b n \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\text {Li}_2\left (1+\frac {e x^r}{d}\right )\right )}{d^2 r^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((d*r*(a + b*Log[c*x^n]))/(d + e*x^r) + b*n*Log[d - d*x^r] - a*r*Log[d - d*x^r] + b*r*(n*Log[x] - Log[c*x^n])*
Log[d - d*x^r] + b*n*((r^2*Log[x]^2)/2 + (-(r*Log[x]) + Log[-((e*x^r)/d)])*Log[d + e*x^r] + PolyLog[2, 1 + (e*
x^r)/d]))/(d^2*r^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 715, normalized size = 7.01

method	result	size
risch	\(\frac {b \ln \left (d +e \,x^{r}\right ) n \ln \left (x \right )}{r \,d^{2}}-\frac {b n \ln \left (x \right )}{r d \left (d +e \,x^{r}\right )}-\frac {b \ln \left (x^{r}\right ) n \ln \left (x \right )}{r \,d^{2}}-\frac {b n \dilog \left (\frac {d +e \,x^{r}}{d}\right )}{r^{2} d^{2}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {d +e \,x^{r}}{d}\right )}{r \,d^{2}}+\frac {b n \ln \left (x \right )^{2}}{2 d^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 r d \left (d +e \,x^{r}\right )}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (x^{r}\right )}{2 r \,d^{2}}+\frac {b \ln \left (c \right )}{r d \left (d +e \,x^{r}\right )}+\frac {b \ln \left (c \right ) \ln \left (x^{r}\right )}{r \,d^{2}}-\frac {b \ln \left (c \right ) \ln \left (d +e \,x^{r}\right )}{r \,d^{2}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (d +e \,x^{r}\right )}{2 r \,d^{2}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (x^{r}\right )}{2 r \,d^{2}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 r d \left (d +e \,x^{r}\right )}-\frac {b \ln \left (d +e \,x^{r}\right ) \ln \left (x^{n}\right )}{r \,d^{2}}+\frac {b \ln \left (x^{n}\right )}{r d \left (d +e \,x^{r}\right )}+\frac {b \ln \left (x^{r}\right ) \ln \left (x^{n}\right )}{r \,d^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 r d \left (d +e \,x^{r}\right )}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d +e \,x^{r}\right )}{2 r \,d^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d +e \,x^{r}\right )}{2 r \,d^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x^{r}\right )}{2 r \,d^{2}}-\frac {b n e \ln \left (x \right ) x^{r}}{r \,d^{2} \left (d +e \,x^{r}\right )}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x^{r}\right )}{2 r \,d^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 r d \left (d +e \,x^{r}\right )}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (d +e \,x^{r}\right )}{2 r \,d^{2}}+\frac {a \ln \left (x^{r}\right )}{r \,d^{2}}-\frac {a \ln \left (d +e \,x^{r}\right )}{r \,d^{2}}+\frac {a}{r d \left (d +e \,x^{r}\right )}+\frac {b n \ln \left (d +e \,x^{r}\right )}{d^{2} r^{2}}\)	\(715\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x/(d+e*x^r)^2,x,method=_RETURNVERBOSE)

[Out]

1/2*I/r*b*Pi*csgn(I*c*x^n)^3/d^2*ln(d+e*x^r)+b/r/d^2*ln(d+e*x^r)*n*ln(x)-b/r/d/(d+e*x^r)*n*ln(x)-b/r/d^2*ln(x^
r)*n*ln(x)-1/2*I/r*b*Pi*csgn(I*c*x^n)^3/d^2*ln(x^r)+1/2*I/r*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d/(d+e*x^r)+1/2*I
/r*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^2*ln(x^r)-1/2*I/r*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^2*ln(d+e*x^r)+1/2*I/r
*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d/(d+e*x^r)+1/2*I/r*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^2*ln(d+e*x^r)-b
/r*n/d^2*ln(x)*ln((d+e*x^r)/d)+1/2*b*n/d^2*ln(x)^2-1/2*I/r*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d/(d+e*x^r
)-1/2*I/r*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^2*ln(x^r)+1/r*b*ln(c)/d/(d+e*x^r)+1/r*b*ln(c)/d^2*ln(x^r)
-1/r*b*ln(c)/d^2*ln(d+e*x^r)-b/r/d^2*ln(d+e*x^r)*ln(x^n)+b/r/d/(d+e*x^r)*ln(x^n)+b/r/d^2*ln(x^r)*ln(x^n)-b/r^2
*n/d^2*dilog((d+e*x^r)/d)-1/2*I/r*b*Pi*csgn(I*c*x^n)^3/d/(d+e*x^r)-1/2*I/r*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^
2*ln(d+e*x^r)-b/r*n*e/d^2*ln(x)*x^r/(d+e*x^r)+1/2*I/r*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^2*ln(x^r)+a/r/d^2*ln(x^
r)-a/r/d^2*ln(d+e*x^r)+a/r/d/(d+e*x^r)+b*n*ln(d+e*x^r)/d^2/r^2

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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((a+b*log(c*x^n))/x/(d+e*x^r)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (102) = 204\).
time = 0.36, size = 229, normalized size = 2.25 \begin {gather*} \frac {b d n r^{2} \log \left (x\right )^{2} + 2 \, b d r \log \left (c\right ) + 2 \, a d r + {\left (b n r^{2} e \log \left (x\right )^{2} + 2 \, {\left (b r^{2} e \log \left (c\right ) - {\left (b n r - a r^{2}\right )} e\right )} \log \left (x\right )\right )} x^{r} - 2 \, {\left (b n x^{r} e + b d n\right )} {\rm Li}_2\left (-\frac {x^{r} e + d}{d} + 1\right ) - 2 \, {\left (b d r \log \left (c\right ) - b d n + a d r + {\left (b r e \log \left (c\right ) - {\left (b n - a r\right )} e\right )} x^{r}\right )} \log \left (x^{r} e + d\right ) + 2 \, {\left (b d r^{2} \log \left (c\right ) + a d r^{2}\right )} \log \left (x\right ) - 2 \, {\left (b n r x^{r} e \log \left (x\right ) + b d n r \log \left (x\right )\right )} \log \left (\frac {x^{r} e + d}{d}\right )}{2 \, {\left (d^{2} r^{2} x^{r} e + d^{3} r^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*d*n*r^2*log(x)^2 + 2*b*d*r*log(c) + 2*a*d*r + (b*n*r^2*e*log(x)^2 + 2*(b*r^2*e*log(c) - (b*n*r - a*r^2)
*e)*log(x))*x^r - 2*(b*n*x^r*e + b*d*n)*dilog(-(x^r*e + d)/d + 1) - 2*(b*d*r*log(c) - b*d*n + a*d*r + (b*r*e*l
og(c) - (b*n - a*r)*e)*x^r)*log(x^r*e + d) + 2*(b*d*r^2*log(c) + a*d*r^2)*log(x) - 2*(b*n*r*x^r*e*log(x) + b*d
*n*r*log(x))*log((x^r*e + d)/d))/(d^2*r^2*x^r*e + d^3*r^2)

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Sympy [A]
time = 216.37, size = 360, normalized size = 3.53 \begin {gather*} - \frac {a e \left (\begin {cases} \frac {x^{r}}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x^{r}} & \text {otherwise} \end {cases}\right )}{d r} - \frac {a e \left (\begin {cases} \frac {x^{r}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{r} \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{2} r} + \frac {a \log {\left (x^{r} \right )}}{d^{2} r} + \frac {b e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r}}{r} & \text {for}\: r \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{d^{2}} & \text {for}\: e = 0 \\- \begin {cases} \frac {\log {\left (x \right )}}{e^{2}} & \text {for}\: d = 0 \wedge r = 0 \\- \frac {x^{- r}}{e^{2} r} & \text {for}\: d = 0 \\\frac {\log {\left (x \right )}}{d e + e^{2}} & \text {for}\: r = 0 \\\frac {\log {\left (x \right )}}{d e} - \frac {\log {\left (\frac {d}{e} + x^{r} \right )}}{d e r} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases}\right )}{d r} - \frac {b e \left (\begin {cases} \frac {x^{r}}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x^{r}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d r} + \frac {b e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r}}{r} & \text {for}\: r \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{r} e^{i \pi }}{d}\right )}{r} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{r} e^{i \pi }}{d}\right )}{r} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{r} e^{i \pi }}{d}\right )}{r} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{r} e^{i \pi }}{d}\right )}{r} & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{d^{2} r} - \frac {b e \left (\begin {cases} \frac {x^{r}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{r} \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{2} r} + \frac {b n \left (\begin {cases} 0 & \text {for}\: r = 0 \\- \frac {\log {\left (x^{r} \right )}^{2}}{2 r} & \text {otherwise} \end {cases}\right )}{d^{2} r} + \frac {b \log {\left (x^{r} \right )} \log {\left (c x^{n} \right )}}{d^{2} r} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x/(d+e*x**r)**2,x)

[Out]

-a*e*Piecewise((x**r/d**2, Eq(e, 0)), (-1/(d*e + e**2*x**r), True))/(d*r) - a*e*Piecewise((x**r/d, Eq(e, 0)),
(log(d + e*x**r)/e, True))/(d**2*r) + a*log(x**r)/(d**2*r) + b*e*n*Piecewise((Piecewise((x**r/r, Ne(r, 0)), (l
og(x), True))/d**2, Eq(e, 0)), (-Piecewise((log(x)/e**2, Eq(d, 0) & Eq(r, 0)), (-1/(e**2*r*x**r), Eq(d, 0)), (
log(x)/(d*e + e**2), Eq(r, 0)), (log(x)/(d*e) - log(d/e + x**r)/(d*e*r), True)), True))/(d*r) - b*e*Piecewise(
(x**r/d**2, Eq(e, 0)), (-1/(d*e + e**2*x**r), True))*log(c*x**n)/(d*r) + b*e*n*Piecewise((Piecewise((x**r/r, N
e(r, 0)), (log(x), True))/d, Eq(e, 0)), (Piecewise((-polylog(2, e*x**r*exp_polar(I*pi)/d)/r, (Abs(x) < 1) & (1
/Abs(x) < 1)), (log(d)*log(x) - polylog(2, e*x**r*exp_polar(I*pi)/d)/r, Abs(x) < 1), (-log(d)*log(1/x) - polyl
og(2, e*x**r*exp_polar(I*pi)/d)/r, 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(d) + meijerg(((
1, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x**r*exp_polar(I*pi)/d)/r, True))/e, True))/(d**2*r) - b*e*
Piecewise((x**r/d, Eq(e, 0)), (log(d + e*x**r)/e, True))*log(c*x**n)/(d**2*r) + b*n*Piecewise((0, Eq(r, 0)), (
-log(x**r)**2/(2*r), True))/(d**2*r) + b*log(x**r)*log(c*x**n)/(d**2*r)

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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((a+b*log(c*x^n))/x/(d+e*x^r)^2,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((x^r*e + d)^2*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(x*(d + e*x^r)^2),x)

[Out]

int((a + b*log(c*x^n))/(x*(d + e*x^r)^2), x)

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