∫ (a+b log(c(d+ex)n))2 _______________________________

3.3.27 \(\int \frac {(a+b \log (c (d+e x)^n))^2}{(f+g x) (h+i x)} \, dx\) [227]

Optimal. Leaf size=264 \[ \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g h-f i}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (h+i x)}{e h-d i}\right )}{g h-f i}+\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}+\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i} \]

[Out]

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

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

polylog(2,-i*(e*x+d)/(-d*i+e*h))/(-f*i+g*h)-2*b^2*n^2*polylog(3,-g*(e*x+d)/(-d*g+e*f))/(-f*i+g*h)+2*b^2*n^2*po

lylog(3,-i*(e*x+d)/(-d*i+e*h))/(-f*i+g*h)

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Rubi [A]
time = 0.26, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2465, 2443, 2481, 2421, 6724} \begin {gather*} \frac {2 b n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}-\frac {2 b n \text {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}-\frac {2 b^2 n^2 \text {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}+\frac {2 b^2 n^2 \text {PolyLog}\left (3,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}+\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g h-f i}-\frac {\log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g h-f i} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

- (2*b^2*n^2*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i) + (2*b^2*n^2*PolyLog[3, -((i*(d + e*x))/(e*

h - d*i))])/(g*h - f*i)

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp

[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L

og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2443

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((

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

*g)]*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*

f - d*g, 0] && IGtQ[p, 1]

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 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*

(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo

g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},

x] && EqQ[e*k - d*l, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a

+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(h+227 x) (f+g x)} \, dx &=\int \left (\frac {227 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(227 f-g h) (h+227 x)}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(227 f-g h) (f+g x)}\right ) \, dx\\ &=\frac {227 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{h+227 x} \, dx}{227 f-g h}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx}{227 f-g h}\\ &=\frac {\log \left (-\frac {e (h+227 x)}{227 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{227 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{227 f-g h}-\frac {(2 b e n) \int \frac {\log \left (\frac {e (h+227 x)}{-227 d+e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{227 f-g h}+\frac {(2 b e n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{227 f-g h}\\ &=\frac {\log \left (-\frac {e (h+227 x)}{227 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{227 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{227 f-g h}-\frac {(2 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {-227 d+e h}{e}+\frac {227 x}{e}\right )}{-227 d+e h}\right )}{x} \, dx,x,d+e x\right )}{227 f-g h}+\frac {(2 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{227 f-g h}\\ &=\frac {\log \left (-\frac {e (h+227 x)}{227 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{227 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{227 f-g h}-\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{227 f-g h}+\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {227 (d+e x)}{227 d-e h}\right )}{227 f-g h}+\frac {\left (2 b^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{227 f-g h}-\frac {\left (2 b^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {227 x}{-227 d+e h}\right )}{x} \, dx,x,d+e x\right )}{227 f-g h}\\ &=\frac {\log \left (-\frac {e (h+227 x)}{227 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{227 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{227 f-g h}-\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{227 f-g h}+\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {227 (d+e x)}{227 d-e h}\right )}{227 f-g h}+\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{227 f-g h}-\frac {2 b^2 n^2 \text {Li}_3\left (\frac {227 (d+e x)}{227 d-e h}\right )}{227 f-g h}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 353, normalized size = 1.34 \begin {gather*} \frac {\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)-\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (h+i x)+2 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log (d+e x) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right )-\log \left (\frac {e (h+i x)}{e h-d i}\right )\right )+\text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )-\text {Li}_2\left (\frac {i (d+e x)}{-e h+d i}\right )\right )+b^2 n^2 \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )-\log ^2(d+e x) \log \left (\frac {e (h+i x)}{e h-d i}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )-2 \log (d+e x) \text {Li}_2\left (\frac {i (d+e x)}{-e h+d i}\right )-2 \text {Li}_3\left (\frac {g (d+e x)}{-e f+d g}\right )+2 \text {Li}_3\left (\frac {i (d+e x)}{-e h+d i}\right )\right )}{g h-f i} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g*x] - (a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^

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

*g)] - Log[(e*(h + i*x))/(e*h - d*i)]) + PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - PolyLog[2, (i*(d + e*x))/(

-(e*h) + d*i)]) + b^2*n^2*(Log[d + e*x]^2*Log[(e*(f + g*x))/(e*f - d*g)] - Log[d + e*x]^2*Log[(e*(h + i*x))/(e

*h - d*i)] + 2*Log[d + e*x]*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*Log[d + e*x]*PolyLog[2, (i*(d + e*x))

/(-(e*h) + d*i)] - 2*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)] + 2*PolyLog[3, (i*(d + e*x))/(-(e*h) + d*i)]))/(

g*h - f*i)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.61, size = 4712, normalized size = 17.85


method	result	size

risch	\(\text {Expression too large to display}\)	\(4712\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*n/(f*i-g*h)*ln(i*x+h)*ln(((i*x+h)*e+d*i-e*h)/(d*i-e*h))*b^2*ln(c)+2*n/(f*i-g*h)*ln(g*x+f)*ln(((g*x+f)*e+d*g

-e*f)/(d*g-e*f))*b^2*ln(c)-I/(f*i-g*h)*ln(i*x+h)*ln((e*x+d)^n)*b^2*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*

x+d)^n)-I*n/(f*i-g*h)*ln(i*x+h)*ln(((i*x+h)*e+d*i-e*h)/(d*i-e*h))*b^2*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-I*n/(

f*i-g*h)*ln(i*x+h)*ln(((i*x+h)*e+d*i-e*h)/(d*i-e*h))*b^2*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*n/(f*i-g

*h)*dilog(((g*x+f)*e+d*g-e*f)/(d*g-e*f))*b^2*Pi*csgn(I*c*(e*x+d)^n)^3-I/(f*i-g*h)*ln(i*x+h)*ln(c)*Pi*b^2*csgn(

I*c*(e*x+d)^n)^3+I*n/(f*i-g*h)*dilog(((i*x+h)*e+d*i-e*h)/(d*i-e*h))*b^2*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*

c*(e*x+d)^n)+I/(f*i-g*h)*ln(i*x+h)*ln(c)*Pi*b^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+I/(f*i-g*h)*ln(i*x+h)*

Pi*a*b*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I/(f*i-g*h)*ln(i*x+h)*ln((e*x+d)^n)*b^2*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n

)^2+I/(f*i-g*h)*ln(i*x+h)*ln((e*x+d)^n)*b^2*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+I*n/(f*i-g*h)*ln(i*x+h)

*ln(((i*x+h)*e+d*i-e*h)/(d*i-e*h))*b^2*Pi*csgn(I*c*(e*x+d)^n)^3+I*n/(f*i-g*h)*dilog(((g*x+f)*e+d*g-e*f)/(d*g-e

*f))*b^2*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-I*n/(f*i-g*h)*dilog(((g*x+f)*e+d*g-e*f)/(d*g-e*f))*b^2*Pi*csgn(I*c

)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-I/(f*i-g*h)*ln(i*x+h)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*

c*(e*x+d)^n)-I/(f*i-g*h)*ln(i*x+h)*Pi*a*b*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-a^2/(f*i-g*h)*ln(g*x

+f)+a^2/(f*i-g*h)*ln(i*x+h)+1/4/(f*i-g*h)*ln(g*x+f)*Pi^2*b^2*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^

n)^2+1/4/(f*i-g*h)*ln(g*x+f)*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4+I*n/(f*i-g*h)*dilog(((g*x+f)*e+d*g-e*f

)/(d*g-e*f))*b^2*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I/(f*i-g*h)*ln(g*x+f)*ln(c)*Pi*b^2*csgn(I*c)*csgn(

I*c*(e*x+d)^n)^2-I/(f*i-g*h)*ln(g*x+f)*ln(c)*Pi*b^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I/(f*i-g*h)*ln(g*x

+f)*Pi*a*b*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-I/(f*i-g*h)*ln(i*x+h)*ln((e*x+d)^n)*b^2*Pi*csgn(I*c*(e*x+d)^n)^3+I/

(f*i-g*h)*ln(i*x+h)*Pi*a*b*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+I/(f*i-g*h)*ln(i*x+h)*ln(c)*Pi*b^2*csgn(I*c

)*csgn(I*c*(e*x+d)^n)^2-I/(f*i-g*h)*ln(i*x+h)*Pi*a*b*csgn(I*c*(e*x+d)^n)^3+1/(f*i-g*h)*ln(g*x+f)*Pi^2*b^2*csgn

(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^4+I*n/(f*i-g*h)*ln(g*x+f)*ln(((g*x+f)*e+d*g-e*f)/(d*g-e*f))*b^2*Pi

*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*n/(f*i-g*h)*ln(g*x+f)*ln(((g*x+f)*e+d*g-e*f)/(d*g-e*f))*b^2*Pi*csgn(I*(e*x+

d)^n)*csgn(I*c*(e*x+d)^n)^2+I*ln((e*x+d)^n)/(f*i-g*h)*ln(g*x+f)*b^2*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e

*x+d)^n)+I/(f*i-g*h)*ln(g*x+f)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/4/(f*i-g*h)*ln(g

*x+f)*Pi^2*b^2*csgn(I*c*(e*x+d)^n)^6-1/4/(f*i-g*h)*ln(i*x+h)*Pi^2*b^2*csgn(I*c*(e*x+d)^n)^6-b^2/(f*i-g*h)*ln(g

*(e*x+d)-d*g+e*f)*ln((e*x+d)^n)^2+b^2/(f*i-g*h)*ln(i*(e*x+d)-d*i+e*h)*ln((e*x+d)^n)^2+1/2/(f*i-g*h)*ln(i*x+h)*

Pi^2*b^2*csgn(I*c)^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^3+1/2/(f*i-g*h)*ln(i*x+h)*Pi^2*b^2*csgn(I*c)*csgn(I

*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3-1/(f*i-g*h)*ln(i*x+h)*Pi^2*b^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d

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

x+d)/(-d*g+e*f))+b^2*n^2/(f*i-g*h)*ln(e*x+d)^2*ln(1+i*(e*x+d)/(-d*i+e*h))+2*b^2*n^2/(f*i-g*h)*ln(e*x+d)*polylo

g(2,-i*(e*x+d)/(-d*i+e*h))+2*b^2*n^2/(f*i-g*h)*dilog((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))*ln(e*x+d)+2*b^2*n^2/(f*i-

g*h)*ln(e*x+d)^2*ln((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))-2*b^2*n^2/(f*i-g*h)*dilog((i*(e*x+d)-d*i+e*h)/(-d*i+e*h))*

ln(e*x+d)-2*b^2*n^2/(f*i-g*h)*ln(e*x+d)^2*ln((i*(e*x+d)-d*i+e*h)/(-d*i+e*h))+1/2/(f*i-g*h)*ln(i*x+h)*Pi^2*b^2*

csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^5-I/(f*i-g*h)*ln(g*x+f)*Pi*a*b*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I

*ln((e*x+d)^n)/(f*i-g*h)*ln(g*x+f)*b^2*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+2*b^2*n^2/(f*i-g*h)*polylog(

3,-g*(e*x+d)/(-d*g+e*f))-2*b^2*n^2/(f*i-g*h)*polylog(3,-i*(e*x+d)/(-d*i+e*h))+I/(f*i-g*h)*ln(g*x+f)*Pi*a*b*csg

n(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*ln((e*x+d)^n)/(f*i-g*h)*ln(g*x+f)*b^2*Pi*csgn(I*c*(e*x+d)^n)^3-

1/2/(f*i-g*h)*ln(g*x+f)*Pi^2*b^2*csgn(I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3-b^2/(f*i-g*h)*ln(g*(e*x+d

)-d*g+e*f)*ln(e*x+d)^2*n^2-2*b^2*n/(f*i-g*h)*dilog((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))*ln((e*x+d)^n)+2*b^2*n/(f*i-

g*h)*dilog((i*(e*x+d)-d*i+e*h)/(-d*i+e*h))*ln((e*x+d)^n)-I*n/(f*i-g*h)*dilog(((i*x+h)*e+d*i-e*h)/(d*i-e*h))*b^

2*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-I*n/(f*i-g*h)*dilog(((i*x+h)*e+d*i-e*h)/(d*i-e*h))*b^2*Pi*csgn(I*(e*x+d)^

n)*csgn(I*c*(e*x+d)^n)^2-I*n/(f*i-g*h)*ln(g*x+f)*ln(((g*x+f)*e+d*g-e*f)/(d*g-e*f))*b^2*Pi*csgn(I*c*(e*x+d)^n)^

3-I*ln((e*x+d)^n)/(f*i-g*h)*ln(g*x+f)*b^2*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/4/(f*i-g*h)*ln(i*x+h)*Pi^2*b^2*

csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2-1/2/(f*i-g*h)*ln(g*x+f)*Pi^2*b^2*csgn(I*c)^2*csgn(I*(e*x

+d)^n)*csgn(I*c*(e*x+d)^n)^3+I*n/(f*i-g*h)*dilog(((i*x+h)*e+d*i-e*h)/(d*i-e*h))*b^2*Pi*csgn(I*c*(e*x+d)^n)^3+I

/(f*i-g*h)*ln(g*x+f)*ln(c)*Pi*b^2*csgn(I*c*(e*x...

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

________________________________________________________________________________________

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

[Out]

integral((-I*b^2*n^2*log(x*e + d)^2 - I*b^2*log(c)^2 - 2*I*a*b*log(c) - I*a^2 - 2*(I*b^2*n*log(c) + I*a*b*n)*l

og(x*e + d))/(g*x^2 - I*f*h + (-I*g*h + f)*x), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}{\left (f + g x\right ) \left (h + i x\right )}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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