\(\int \frac {\log (d (a+b x+c x^2)^n)}{d+e x} \, dx\) [87]

Optimal result

Integrand size = 23, antiderivative size = 228 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{d+e x} \, dx=-\frac {n \log \left (-\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right ) \log (d+e x)}{e}-\frac {n \log \left (-\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (d \left (a+b x+c x^2\right )^n\right )}{e}-\frac {n \operatorname {PolyLog}\left (2,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{e}-\frac {n \operatorname {PolyLog}\left (2,\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e} \]

[Out]

ln(e*x+d)*ln(d*(c*x^2+b*x+a)^n)/e-n*ln(e*x+d)*ln(-e*(b+2*c*x-(-4*a*c+b^2)^(1/2))/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2
))))/e-n*ln(e*x+d)*ln(-e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))/e-n*polylog(2,2*c*(e*x
+d)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2))))/e-n*polylog(2,2*c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))/e

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2604, 2465, 2441, 2440, 2438} \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{d+e x} \, dx=-\frac {n \operatorname {PolyLog}\left (2,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{e}-\frac {n \operatorname {PolyLog}\left (2,\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e}-\frac {n \log (d+e x) \log \left (-\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )}{e}-\frac {n \log (d+e x) \log \left (-\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )}{e}+\frac {\log (d+e x) \log \left (d \left (a+b x+c x^2\right )^n\right )}{e} \]

[In]

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

[Out]

-((n*Log[-((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e))]*Log[d + e*x])/e) - (n*Log
[-((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))]*Log[d + e*x])/e + (Log[d + e*x]*L
og[d*(a + b*x + c*x^2)^n])/e - (n*PolyLog[2, (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)])/e - (n*Poly
Log[2, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/e

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 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2604

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

Rubi steps \begin{align*} \text {integral}& = \frac {\log (d+e x) \log \left (d \left (a+b x+c x^2\right )^n\right )}{e}-\frac {n \int \frac {(b+2 c x) \log (d+e x)}{a+b x+c x^2} \, dx}{e} \\ & = \frac {\log (d+e x) \log \left (d \left (a+b x+c x^2\right )^n\right )}{e}-\frac {n \int \left (\frac {2 c \log (d+e x)}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {2 c \log (d+e x)}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{e} \\ & = \frac {\log (d+e x) \log \left (d \left (a+b x+c x^2\right )^n\right )}{e}-\frac {(2 c n) \int \frac {\log (d+e x)}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{e}-\frac {(2 c n) \int \frac {\log (d+e x)}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{e} \\ & = -\frac {n \log \left (-\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right ) \log (d+e x)}{e}-\frac {n \log \left (-\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (d \left (a+b x+c x^2\right )^n\right )}{e}+n \int \frac {\log \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{d+e x} \, dx+n \int \frac {\log \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{d+e x} \, dx \\ & = -\frac {n \log \left (-\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right ) \log (d+e x)}{e}-\frac {n \log \left (-\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (d \left (a+b x+c x^2\right )^n\right )}{e}+\frac {n \text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{x} \, dx,x,d+e x\right )}{e}+\frac {n \text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{x} \, dx,x,d+e x\right )}{e} \\ & = -\frac {n \log \left (-\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right ) \log (d+e x)}{e}-\frac {n \log \left (-\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (d \left (a+b x+c x^2\right )^n\right )}{e}-\frac {n \text {Li}_2\left (\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{e}-\frac {n \text {Li}_2\left (\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.88 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{d+e x} \, dx=\frac {\log (d+e x) \left (-n \log \left (\frac {e \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{2 c d-b e+\sqrt {b^2-4 a c} e}\right )-n \log \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}\right )+\log \left (d (a+x (b+c x))^n\right )\right )-n \operatorname {PolyLog}\left (2,\frac {2 c (d+e x)}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )-n \operatorname {PolyLog}\left (2,\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e} \]

[In]

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

[Out]

(Log[d + e*x]*(-(n*Log[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)]) - n*Log[(e*(
b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)] + Log[d*(a + x*(b + c*x))^n]) - n*PolyLo
g[2, (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)] - n*PolyLog[2, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b
^2 - 4*a*c])*e)])/e

Maple [A] (verified)

Time = 1.37 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.32

method	result	size
parts	\(\frac {\ln \left (e x +d \right ) \ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right )}{e}-\frac {n \left (\ln \left (e x +d \right ) \ln \left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}{-b e +2 c d +\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}\right )+\ln \left (e x +d \right ) \ln \left (\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}{b e -2 c d +\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}\right )+\operatorname {dilog}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}{-b e +2 c d +\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}\right )+\operatorname {dilog}\left (\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}{b e -2 c d +\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}\right )\right )}{e}\)	\(300\)
risch	\(\frac {\ln \left (\left (c \,x^{2}+b x +a \right )^{n}\right ) \ln \left (e x +d \right )}{e}-\frac {n \ln \left (e x +d \right ) \ln \left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}{-b e +2 c d +\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}\right )}{e}-\frac {n \ln \left (e x +d \right ) \ln \left (\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}{b e -2 c d +\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}\right )}{e}-\frac {n \operatorname {dilog}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}{-b e +2 c d +\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}\right )}{e}-\frac {n \operatorname {dilog}\left (\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}{b e -2 c d +\sqrt {-4 a c \,e^{2}+e^{2} b^{2}}}\right )}{e}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) {\operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) \operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right ) \operatorname {csgn}\left (i d \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )}^{2} \operatorname {csgn}\left (i d \right )}{2}+\ln \left (d \right )\right ) \ln \left (e x +d \right )}{e}\)	\(458\)

[In]

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

[Out]

ln(e*x+d)*ln(d*(c*x^2+b*x+a)^n)/e-1/e*n*(ln(e*x+d)*ln((-b*e+2*c*d-2*c*(e*x+d)+(-4*a*c*e^2+b^2*e^2)^(1/2))/(-b*
e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))+ln(e*x+d)*ln((b*e-2*c*d+2*c*(e*x+d)+(-4*a*c*e^2+b^2*e^2)^(1/2))/(b*e-2*c*
d+(-4*a*c*e^2+b^2*e^2)^(1/2)))+dilog((-b*e+2*c*d-2*c*(e*x+d)+(-4*a*c*e^2+b^2*e^2)^(1/2))/(-b*e+2*c*d+(-4*a*c*e
^2+b^2*e^2)^(1/2)))+dilog((b*e-2*c*d+2*c*(e*x+d)+(-4*a*c*e^2+b^2*e^2)^(1/2))/(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(
1/2))))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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