\(\int \frac {x^2 (a+b \log (c x^n))}{(d+e x)^2} \, dx\) [40]

Optimal result

Integrand size = 21, antiderivative size = 98 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=-\frac {b n x}{e^2}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}-\frac {d \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}-\frac {2 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3} \]

[Out]

-b*n*x/e^2+2*x*(a+b*ln(c*x^n))/e^2-x^2*(a+b*ln(c*x^n))/e/(e*x+d)-d*(2*a+b*n+2*b*ln(c*x^n))*ln(1+e*x/d)/e^3-2*b
*d*n*polylog(2,-e*x/d)/e^3

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2384, 45, 2393, 2332, 2354, 2438} \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {x (2 a+b n)}{e^2}+\frac {2 b x \log \left (c x^n\right )}{e^2}-\frac {2 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}-\frac {2 b n x}{e^2} \]

[In]

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

[Out]

(-2*b*n*x)/e^2 + ((2*a + b*n)*x)/e^2 + (2*b*x*Log[c*x^n])/e^2 - (x^2*(a + b*Log[c*x^n]))/(e*(d + e*x)) - (d*(2
*a + b*n + 2*b*Log[c*x^n])*Log[1 + (e*x)/d])/e^3 - (2*b*d*n*PolyLog[2, -((e*x)/d)])/e^3

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 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 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}& = -\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {\int \frac {x \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{d+e x} \, dx}{e} \\ & = -\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {\int \left (\frac {2 a+b n+2 b \log \left (c x^n\right )}{e}-\frac {d \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{e (d+e x)}\right ) \, dx}{e} \\ & = -\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {\int \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \, dx}{e^2}-\frac {d \int \frac {2 a+b n+2 b \log \left (c x^n\right )}{d+e x} \, dx}{e^2} \\ & = \frac {(2 a+b n) x}{e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}-\frac {d \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {(2 b) \int \log \left (c x^n\right ) \, dx}{e^2}+\frac {(2 b d n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^3} \\ & = -\frac {2 b n x}{e^2}+\frac {(2 a+b n) x}{e^2}+\frac {2 b x \log \left (c x^n\right )}{e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}-\frac {d \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}-\frac {2 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\frac {a e x-b e n x+b e x \log \left (c x^n\right )-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x}+b d n (\log (x)-\log (d+e x))-2 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-2 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3} \]

[In]

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

[Out]

(a*e*x - b*e*n*x + b*e*x*Log[c*x^n] - (d^2*(a + b*Log[c*x^n]))/(d + e*x) + b*d*n*(Log[x] - Log[d + e*x]) - 2*d
*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] - 2*b*d*n*PolyLog[2, -((e*x)/d)])/e^3

Maple [C] (warning: unable to verify)

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

Time = 0.41 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.55

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) x}{e^{2}}-\frac {2 b \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{3}}-\frac {b \ln \left (x^{n}\right ) d^{2}}{e^{3} \left (e x +d \right )}-\frac {b n d \ln \left (e x +d \right )}{e^{3}}+\frac {b n d \ln \left (e x \right )}{e^{3}}-\frac {b n x}{e^{2}}-\frac {b n d}{e^{3}}+\frac {2 b n d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}+\frac {2 b n d \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{3}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{e^{2}}-\frac {2 d \ln \left (e x +d \right )}{e^{3}}-\frac {d^{2}}{e^{3} \left (e x +d \right )}\right )\)	\(250\)

[In]

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

[Out]

b*ln(x^n)/e^2*x-2*b*ln(x^n)/e^3*d*ln(e*x+d)-b*ln(x^n)/e^3*d^2/(e*x+d)-b*n/e^3*d*ln(e*x+d)+b*n/e^3*d*ln(e*x)-b*
n*x/e^2-b*n/e^3*d+2*b*n/e^3*d*ln(e*x+d)*ln(-e*x/d)+2*b*n/e^3*d*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^2-2/e^3*d*ln(e*x+d)-1/e^3*d^2/(e*x+d))

Fricas [F]

\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 12.08 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.74 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\frac {a d^{2} \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {2 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^{2}} + \frac {a x}{e^{2}} - \frac {b d^{2} n \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {\log {\left (x \right )}}{d e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{d e} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {b d^{2} \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{2}} + \frac {2 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^{2}} - \frac {2 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^{2}} - \frac {b n x}{e^{2}} + \frac {b x \log {\left (c x^{n} \right )}}{e^{2}} \]

[In]

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

[Out]

a*d**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/e**2 - 2*a*d*Piecewise((x/d, Eq(e, 0)), (log(d
+ e*x)/e, True))/e**2 + a*x/e**2 - b*d**2*n*Piecewise((x/d**2, Eq(e, 0)), (-log(x)/(d*e) + log(d/e + x)/(d*e)
, True))/e**2 + b*d**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))*log(c*x**n)/e**2 + 2*b*d*n*Pie
cewise((x/d, Eq(e, 0)), (Piecewise((-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_polar(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)*lo
g(d) - polylog(2, e*x*exp_polar(I*pi)/d), True))/e, True))/e**2 - 2*b*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*
x)/e, True))*log(c*x**n)/e**2 - b*n*x/e**2 + b*x*log(c*x**n)/e**2

Maxima [F]

\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

-a*(d^2/(e^4*x + d*e^3) - x/e^2 + 2*d*log(e*x + d)/e^3) + b*integrate((x^2*log(c) + x^2*log(x^n))/(e^2*x^2 + 2
*d*e*x + d^2), x)

Giac [F]

\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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