\(\int \frac {x \log (c (a+b x)^p)}{d+e x} \, dx\) [221]

Optimal result

Integrand size = 19, antiderivative size = 91 \[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=-\frac {p x}{e}+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}-\frac {d p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {45, 2463, 2436, 2332, 2441, 2440, 2438} \[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=-\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac {d p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^2}-\frac {p x}{e} \]

[In]

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

[Out]

-((p*x)/e) + ((a + b*x)*Log[c*(a + b*x)^p])/(b*e) - (d*Log[c*(a + b*x)^p]*Log[(b*(d + e*x))/(b*d - a*e)])/e^2
- (d*p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/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 2436

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

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]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

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

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.87 \[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\frac {-b e p x+\log \left (c (a+b x)^p\right ) \left (a e+b e x-b d \log \left (\frac {b (d+e x)}{b d-a e}\right )\right )-b d p \operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )}{b e^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.71 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.71

method	result	size
parts	\(\frac {x \ln \left (c \left (b x +a \right )^{p}\right )}{e}-\frac {\ln \left (c \left (b x +a \right )^{p}\right ) d \ln \left (e x +d \right )}{e^{2}}-\frac {p b \left (\frac {e x +d}{e b}-\frac {a \ln \left (\left (e x +d \right ) b +a e -b d \right )}{b^{2}}-\frac {d \left (\frac {\operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{b}+\frac {\ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{b}\right )}{e}\right )}{e}\)	\(156\)
risch	\(\frac {\ln \left (\left (b x +a \right )^{p}\right ) x}{e}-\frac {\ln \left (\left (b x +a \right )^{p}\right ) d \ln \left (e x +d \right )}{e^{2}}-\frac {p x}{e}-\frac {p d}{e^{2}}+\frac {p a \ln \left (\left (e x +d \right ) b +a e -b d \right )}{b e}+\frac {p d \operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e^{2}}+\frac {p d \ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e^{2}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {x}{e}-\frac {d \ln \left (e x +d \right )}{e^{2}}\right )\)	\(270\)

[In]

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

[Out]

x*ln(c*(b*x+a)^p)/e-ln(c*(b*x+a)^p)*d/e^2*ln(e*x+d)-p*b/e*(1/e*(e*x+d)/b-a/b^2*ln((e*x+d)*b+a*e-b*d)-1/e*d*(di
log(((e*x+d)*b+a*e-b*d)/(a*e-b*d))/b+ln(e*x+d)*ln(((e*x+d)*b+a*e-b*d)/(a*e-b*d))/b))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x*log(c*(a + b*x)**p)/(d + e*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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