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

3.1.48 \(\int \frac {(a+b \log (c (d+e x)^n))^2}{f+g x} \, dx\) [48]

Optimal. Leaf size=111 \[ \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g} \]

[Out]

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

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

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {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}-\frac {2 b^2 n^2 \text {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g}+\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} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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 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}{f+g x} \, dx &=\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}-\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}{g}\\ &=\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}-\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 )}{g}\\ &=\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}+\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}-\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 )}{g}\\ &=\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}+\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}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.07, size = 194, normalized size = 1.75 \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)+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) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )\right )+b^2 n^2 \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )-2 \text {Li}_3\left (\frac {g (d+e x)}{-e f+d g}\right )\right )}{g} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

g[3, (g*(d + e*x))/(-(e*f) + d*g)]))/g

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.75, size = 2018, normalized size = 18.18


method	result	size

risch	\(\text {Expression too large to display}\)	\(2018\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

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

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

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

d)^n)*csgn(I*c*(e*x+d)^n)-I*ln(g*x+f)/g*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

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

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

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

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

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

*c*(e*x+d)^n)^4+b^2*ln(g*(e*x+d)-d*g+e*f)/g*ln(e*x+d)^2*n^2-2*b^2*ln(g*(e*x+d)-d*g+e*f)/g*ln(e*x+d)*ln((e*x+d)

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

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

og(((g*x+f)*e+d*g-e*f)/(d*g-e*f))*a+2*ln(g*x+f)/g*ln((e*x+d)^n)*b^2*ln(c)+2*b*ln(g*x+f)/g*ln((e*x+d)^n)*a+1/2*

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

olylog(3,g/(d*g-e*f)*(e*x+d))+1/2*ln(g*x+f)/g*Pi^2*b^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^5

________________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

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

g((x*e + d)^n))/(g*x + f), 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),x, algorithm="fricas")

[Out]

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

[Out]

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

________________________________________________________________________________________

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),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{f+g\,x} \,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),x)

[Out]

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

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]