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

Optimal. Leaf size=91 \[ -\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac {d p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{e^2}-\frac {p x}{e} \]

[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

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Rubi [A]  time = 0.11, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {43, 2416, 2389, 2295, 2394, 2393, 2391} \[ -\frac {d p \text {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^2}-\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac {p x}{e} \]

Antiderivative was successfully verified.

[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 43

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 2295

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

Rule 2389

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 2391

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 2393

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 2394

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 2416

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

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Mathematica [A]  time = 0.03, size = 79, normalized size = 0.87 \[ \frac {\log \left (c (a+b x)^p\right ) \left (-b d \log \left (\frac {b (d+e x)}{b d-a e}\right )+a e+b e x\right )-b d p \text {Li}_2\left (\frac {e (a+b x)}{a e-b d}\right )-b e p x}{b e^2} \]

Antiderivative was successfully verified.

[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)

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.32, size = 427, normalized size = 4.69 \[ \frac {i \pi d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{2 e^{2}}-\frac {i \pi d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 e^{2}}-\frac {i \pi d \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 e^{2}}+\frac {i \pi d \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3} \ln \left (e x +d \right )}{2 e^{2}}-\frac {i \pi x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )}{2 e}+\frac {i \pi x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2 e}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2 e}-\frac {i \pi x \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2 e}+\frac {d p \ln \left (\frac {a e -b d +\left (e x +d \right ) b}{a e -b d}\right ) \ln \left (e x +d \right )}{e^{2}}+\frac {a p \ln \left (a e -b d +\left (e x +d \right ) b \right )}{b e}+\frac {d p \dilog \left (\frac {a e -b d +\left (e x +d \right ) b}{a e -b d}\right )}{e^{2}}-\frac {d \ln \relax (c ) \ln \left (e x +d \right )}{e^{2}}-\frac {d \ln \left (\left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{e^{2}}-\frac {p x}{e}+\frac {x \ln \relax (c )}{e}+\frac {x \ln \left (\left (b x +a \right )^{p}\right )}{e}-\frac {d p}{e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{d+e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \log {\left (c \left (a + b x\right )^{p} \right )}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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