3.1.85 \(\int \frac {\log (e (\frac {a+b x}{c+d x})^n)}{f+g x+h x^2} \, dx\) [85]

Optimal. Leaf size=401 \[ -\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {2 \left (d^2 f-c d g+c^2 h\right ) (a+b x)}{\left (2 b d f-b c g-a d g+2 a c h-(b c-a d) \sqrt {g^2-4 f h}\right ) (c+d x)}\right )}{\sqrt {g^2-4 f h}}+\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {2 \left (d^2 f-c d g+c^2 h\right ) (a+b x)}{\left (2 b d f-b c g-a d g+2 a c h+(b c-a d) \sqrt {g^2-4 f h}\right ) (c+d x)}\right )}{\sqrt {g^2-4 f h}}-\frac {n \text {Li}_2\left (\frac {2 \left (d^2 f-c d g+c^2 h\right ) (a+b x)}{\left (2 b d f-b c g-a d g+2 a c h-(b c-a d) \sqrt {g^2-4 f h}\right ) (c+d x)}\right )}{\sqrt {g^2-4 f h}}+\frac {n \text {Li}_2\left (\frac {2 \left (d^2 f-c d g+c^2 h\right ) (a+b x)}{\left (2 b d f-b c g-a d g+2 a c h+(b c-a d) \sqrt {g^2-4 f h}\right ) (c+d x)}\right )}{\sqrt {g^2-4 f h}} \]

[Out]

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

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Rubi [A]
time = 0.53, antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2576, 2404, 2354, 2438} \begin {gather*} -\frac {n \text {PolyLog}\left (2,\frac {2 (a+b x) \left (c^2 h-c d g+d^2 f\right )}{(c+d x) \left (-\sqrt {g^2-4 f h} (b c-a d)+2 a c h-a d g-b c g+2 b d f\right )}\right )}{\sqrt {g^2-4 f h}}+\frac {n \text {PolyLog}\left (2,\frac {2 (a+b x) \left (c^2 h-c d g+d^2 f\right )}{(c+d x) \left (\sqrt {g^2-4 f h} (b c-a d)+2 a c h-a d g-b c g+2 b d f\right )}\right )}{\sqrt {g^2-4 f h}}-\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {2 (a+b x) \left (c^2 h-c d g+d^2 f\right )}{(c+d x) \left (-\sqrt {g^2-4 f h} (b c-a d)+2 a c h-a d g-b c g+2 b d f\right )}\right )}{\sqrt {g^2-4 f h}}+\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {2 (a+b x) \left (c^2 h-c d g+d^2 f\right )}{(c+d x) \left (\sqrt {g^2-4 f h} (b c-a d)+2 a c h-a d g-b c g+2 b d f\right )}\right )}{\sqrt {g^2-4 f h}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Log[e*((a + b*x)/(c + d*x))^n]*Log[1 - (2*(d^2*f - c*d*g + c^2*h)*(a + b*x))/((2*b*d*f - b*c*g - a*d*g + 2*
a*c*h - (b*c - a*d)*Sqrt[g^2 - 4*f*h])*(c + d*x))])/Sqrt[g^2 - 4*f*h]) + (Log[e*((a + b*x)/(c + d*x))^n]*Log[1
- (2*(d^2*f - c*d*g + c^2*h)*(a + b*x))/((2*b*d*f - b*c*g - a*d*g + 2*a*c*h + (b*c - a*d)*Sqrt[g^2 - 4*f*h])*
(c + d*x))])/Sqrt[g^2 - 4*f*h] - (n*PolyLog[2, (2*(d^2*f - c*d*g + c^2*h)*(a + b*x))/((2*b*d*f - b*c*g - a*d*g
+ 2*a*c*h - (b*c - a*d)*Sqrt[g^2 - 4*f*h])*(c + d*x))])/Sqrt[g^2 - 4*f*h] + (n*PolyLog[2, (2*(d^2*f - c*d*g +
c^2*h)*(a + b*x))/((2*b*d*f - b*c*g - a*d*g + 2*a*c*h + (b*c - a*d)*Sqrt[g^2 - 4*f*h])*(c + d*x))])/Sqrt[g^2
- 4*f*h]

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 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

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 2576

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*(P2x_)^(m_.), x_Symbol]
:> With[{f = Coeff[P2x, x, 0], g = Coeff[P2x, x, 1], h = Coeff[P2x, x, 2]}, Dist[b*c - a*d, Subst[Int[(b^2*f
- a*b*g + a^2*h - (2*b*d*f - b*c*g - a*d*g + 2*a*c*h)*x + (d^2*f - c*d*g + c^2*h)*x^2)^m*((A + B*Log[e*x^n])^p
/(b - d*x)^(2*(m + 1))), x], x, (a + b*x)/(c + d*x)], x]] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && PolyQ[P2x,
x, 2] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]
time = 0.29, size = 604, normalized size = 1.51 \begin {gather*} \frac {-2 \sqrt {g^2-4 f h} n \tan ^{-1}\left (\frac {g+2 h x}{\sqrt {-g^2+4 f h}}\right ) \log \left (\frac {a}{b}+x\right )+2 \sqrt {g^2-4 f h} n \tan ^{-1}\left (\frac {g+2 h x}{\sqrt {-g^2+4 f h}}\right ) \log \left (\frac {c}{d}+x\right )+2 \sqrt {g^2-4 f h} \tan ^{-1}\left (\frac {g+2 h x}{\sqrt {-g^2+4 f h}}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\sqrt {-g^2+4 f h} n \log \left (\frac {a}{b}+x\right ) \log \left (1-\frac {2 h (a+b x)}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\sqrt {-g^2+4 f h} n \log \left (\frac {a}{b}+x\right ) \log \left (1+\frac {2 h (a+b x)}{-2 a h+b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\sqrt {-g^2+4 f h} n \log \left (\frac {c}{d}+x\right ) \log \left (1-\frac {2 h (c+d x)}{-d g+2 c h+d \sqrt {g^2-4 f h}}\right )+\sqrt {-g^2+4 f h} n \log \left (\frac {c}{d}+x\right ) \log \left (1+\frac {2 h (c+d x)}{-2 c h+d \left (g+\sqrt {g^2-4 f h}\right )}\right )+\sqrt {-g^2+4 f h} n \text {Li}_2\left (\frac {2 h (a+b x)}{2 a h+b \left (-g+\sqrt {g^2-4 f h}\right )}\right )-\sqrt {-g^2+4 f h} n \text {Li}_2\left (\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\sqrt {-g^2+4 f h} n \text {Li}_2\left (\frac {2 h (c+d x)}{-d g+2 c h+d \sqrt {g^2-4 f h}}\right )+\sqrt {-g^2+4 f h} n \text {Li}_2\left (\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{\sqrt {-\left (g^2-4 f h\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[g^2 - 4*f*h]*n*ArcTan[(g + 2*h*x)/Sqrt[-g^2 + 4*f*h]]*Log[a/b + x] + 2*Sqrt[g^2 - 4*f*h]*n*ArcTan[(g
+ 2*h*x)/Sqrt[-g^2 + 4*f*h]]*Log[c/d + x] + 2*Sqrt[g^2 - 4*f*h]*ArcTan[(g + 2*h*x)/Sqrt[-g^2 + 4*f*h]]*Log[e*(
(a + b*x)/(c + d*x))^n] + Sqrt[-g^2 + 4*f*h]*n*Log[a/b + x]*Log[1 - (2*h*(a + b*x))/(-(b*g) + 2*a*h + b*Sqrt[g
^2 - 4*f*h])] - Sqrt[-g^2 + 4*f*h]*n*Log[a/b + x]*Log[1 + (2*h*(a + b*x))/(-2*a*h + b*(g + Sqrt[g^2 - 4*f*h]))
] - Sqrt[-g^2 + 4*f*h]*n*Log[c/d + x]*Log[1 - (2*h*(c + d*x))/(-(d*g) + 2*c*h + d*Sqrt[g^2 - 4*f*h])] + Sqrt[-
g^2 + 4*f*h]*n*Log[c/d + x]*Log[1 + (2*h*(c + d*x))/(-2*c*h + d*(g + Sqrt[g^2 - 4*f*h]))] + Sqrt[-g^2 + 4*f*h]
*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h + b*(-g + Sqrt[g^2 - 4*f*h]))] - Sqrt[-g^2 + 4*f*h]*n*PolyLog[2, (2*h*(a
+ b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))] - Sqrt[-g^2 + 4*f*h]*n*PolyLog[2, (2*h*(c + d*x))/(-(d*g) + 2*c*h
+ d*Sqrt[g^2 - 4*f*h])] + Sqrt[-g^2 + 4*f*h]*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h]))
])/Sqrt[-(g^2 - 4*f*h)^2]

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Maple [F]
time = 0.53, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{h \,x^{2}+g x +f}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x)

[Out]

int(ln(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*f*h-g^2>0)', see `assume?` f
or more deta

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{h\,x^2+g\,x+f} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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