\(\int \frac {x (a+b \log (c x^n))}{d+e x} \, dx\) [33]

Optimal result

Integrand size = 19, antiderivative size = 69 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\frac {a x}{e}-\frac {b n x}{e}+\frac {b x \log \left (c x^n\right )}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {45, 2393, 2332, 2354, 2438} \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {a x}{e}+\frac {b x \log \left (c x^n\right )}{e}-\frac {b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2}-\frac {b n x}{e} \]

[In]

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

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2354

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

Rule 2393

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\frac {a e x-b e n x-a d \log \left (1+\frac {e x}{d}\right )+b \log \left (c x^n\right ) \left (e x-d \log \left (1+\frac {e x}{d}\right )\right )-b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.39 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.72

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) x}{e}-\frac {b \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{2}}-\frac {b n x}{e}-\frac {b n d}{e^{2}}+\frac {b n d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{2}}+\frac {b n d \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{e}-\frac {d \ln \left (e x +d \right )}{e^{2}}\right )\)	\(188\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{e x + d} \,d x } \]

[In]

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

[Out]

integral((b*x*log(c*x^n) + a*x)/(e*x + d), x)

Sympy [A] (verification not implemented)

Time = 8.06 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.36 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=- \frac {a d \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e} + \frac {a x}{e} + \frac {b d n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{e} - \frac {b d \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e} - \frac {b n x}{e} + \frac {b x \log {\left (c x^{n} \right )}}{e} \]

[In]

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

[Out]

-a*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e + a*x/e + b*d*n*Piecewise((x/d, Eq(e, 0)), (Piecewis
e((-polylog(2, e*x*exp_polar(I*pi)/d), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*log(x) - polylog(2, e*x*exp_pol
ar(I*pi)/d), Abs(x) < 1), (-log(d)*log(1/x) - polylog(2, e*x*exp_polar(I*pi)/d), 1/Abs(x) < 1), (-meijerg(((),
(1, 1)), ((0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x*exp_polar(I
*pi)/d), True))/e, True))/e - b*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))*log(c*x**n)/e - b*n*x/e +
b*x*log(c*x**n)/e

Maxima [F]

\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{e x + d} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{e x + d} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,x} \,d x \]

[In]

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

[Out]

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