∫ (a+b log(cxn))2 ______________________

3.5.30 \(\int \frac {(a+b \log (c x^n))^2}{x (d+e x^r)} \, dx\) [430]

Optimal. Leaf size=94 \[ -\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {d x^{-r}}{e}\right )}{d r}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d r^2}+\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {d x^{-r}}{e}\right )}{d r^3} \]

[Out]

-(a+b*ln(c*x^n))^2*ln(1+d/e/(x^r))/d/r+2*b*n*(a+b*ln(c*x^n))*polylog(2,-d/e/(x^r))/d/r^2+2*b^2*n^2*polylog(3,-

d/e/(x^r))/d/r^3

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Rubi [A]
time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2379, 2421, 6724} \begin {gather*} \frac {2 b n \text {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{d r^2}+\frac {2 b^2 n^2 \text {PolyLog}\left (3,-\frac {d x^{-r}}{e}\right )}{d r^3}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d r} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(((a + b*Log[c*x^n])^2*Log[1 + d/(e*x^r)])/(d*r)) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x^r))])/(d*r^

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

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +

d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^

(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

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 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 x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx &=-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {d x^{-r}}{e}\right )}{d r}+\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{x} \, dx}{d r}\\ &=-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {d x^{-r}}{e}\right )}{d r}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d r^2}-\frac {\left (2 b^2 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{x} \, dx}{d r^2}\\ &=-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {d x^{-r}}{e}\right )}{d r}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d r^2}+\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {d x^{-r}}{e}\right )}{d r^3}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(94)=188\).
time = 0.18, size = 270, normalized size = 2.87 \begin {gather*} -\frac {a^2 r^2 \log \left (d-d x^r\right )-2 a b r^2 \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+b^2 r^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2 \log \left (d-d x^r\right )-2 a b n r \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\text {Li}_2\left (1+\frac {e x^r}{d}\right )\right )+2 b^2 n r \left (n \log (x)-\log \left (c x^n\right )\right ) \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\text {Li}_2\left (1+\frac {e x^r}{d}\right )\right )+b^2 n^2 \left (r^2 \log ^2(x) \log \left (1+\frac {d x^{-r}}{e}\right )-2 r \log (x) \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )-2 \text {Li}_3\left (-\frac {d x^{-r}}{e}\right )\right )}{d r^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

n])^2*Log[d - d*x^r] - 2*a*b*n*r*((r^2*Log[x]^2)/2 + (-(r*Log[x]) + Log[-((e*x^r)/d)])*Log[d + e*x^r] + PolyLo

g[2, 1 + (e*x^r)/d]) + 2*b^2*n*r*(n*Log[x] - Log[c*x^n])*((r^2*Log[x]^2)/2 + (-(r*Log[x]) + Log[-((e*x^r)/d)])

*Log[d + e*x^r] + PolyLog[2, 1 + (e*x^r)/d]) + b^2*n^2*(r^2*Log[x]^2*Log[1 + d/(e*x^r)] - 2*r*Log[x]*PolyLog[2

, -(d/(e*x^r))] - 2*PolyLog[3, -(d/(e*x^r))]))/(d*r^3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.18, size = 3012, normalized size = 32.04


method	result	size

risch	\(\text {Expression too large to display}\)	\(3012\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^2/r/d*ln(d+e*x^r)*ln(x)^2*n^2+1/2*I*n/d*ln(x)^2*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I/r/d*ln(x^r)*Pi*a*b*csg

n(I*c*x^n)^3-I/r/d*ln(x^r)*ln(c)*Pi*b^2*csgn(I*c*x^n)^3+1/2*I*n/d*ln(x)^2*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2

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

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

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

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

^n)^3+2*b/r/d*ln(x^r)*ln(x^n)*a-2*b/r^2*n/d*polylog(2,-e*x^r/d)*a+1/r/d*ln(d+e*x^r)*Pi^2*b^2*csgn(I*c)*csgn(I*

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

I*c*x^n)^5+I/r/d*ln(x^r)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*c*x^n)^2+I/r*n/d*ln(x)*ln(1+e*x^r/d)*b^2*Pi*csgn(I*c*x^

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

b^2/r/d*ln(x^r)*ln(x^n)^2+2*b^2/r^3*n^2/d*polylog(3,-e*x^r/d)+I/r/d*ln(x^r)*ln(x^n)*b^2*Pi*csgn(I*x^n)*csgn(I*

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

x^n)*csgn(I*c*x^n)^2-2*b/r/d*ln(d+e*x^r)*ln(x^n)*a+I/r^2*n/d*polylog(2,-e*x^r/d)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*

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

(I*x^n)*csgn(I*c*x^n)^2+I/r/d*ln(d+e*x^r)*Pi*a*b*csgn(I*c*x^n)^3+I/r*n/d*ln(x)*ln(1+e*x^r/d)*b^2*Pi*csgn(I*c)*

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

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

x)*b^2*ln(c)+I/r^2*n/d*polylog(2,-e*x^r/d)*b^2*Pi*csgn(I*c*x^n)^3+I/r/d*ln(d+e*x^r)*ln(x^n)*b^2*Pi*csgn(I*c*x^

n)^3-2*b^2/r/d*ln(x^r)*ln(x)*ln(x^n)*n-2*b^2/r*n/d*ln(x)*ln(1+e*x^r/d)*ln(x^n)+a^2/r/d*ln(x^r)-a^2/r/d*ln(d+e*

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

x^r)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3+1/2/r/d*ln(x^r)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5+I/r

/d*ln(d+e*x^r)*ln(x^n)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I/r/d*ln(d+e*x^r)*ln(c)*Pi*b^2*csgn(I*c)*csg

n(I*x^n)*csgn(I*c*x^n)+I/r/d*ln(d+e*x^r)*Pi*a*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-I/r*n/d*ln(x)*ln(1+e*x^r/d

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

*b^2*csgn(I*c)*csgn(I*c*x^n)^5+b^2/r/d*ln(x^r)*ln(x)^2*n^2-2/r/d*ln(d+e*x^r)*ln(c)*a*b+2/r/d*ln(x^r)*ln(c)*a*b

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

2*b^2-2/3*b^2/d*ln(x)^3*n^2+1/4/r/d*ln(d+e*x^r)*Pi^2*b^2*csgn(I*c*x^n)^6+I/r/d*ln(x^r)*n*ln(x)*b^2*Pi*csgn(I*c

)*csgn(I*x^n)*csgn(I*c*x^n)+2/r/d*ln(x^r)*ln(x^n)*b^2*ln(c)-2/r^2*n/d*polylog(2,-e*x^r/d)*b^2*ln(c)-2/r/d*ln(d

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

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

(I*x^n)*csgn(I*c*x^n)^5-2*b^2/r^2*n/d*polylog(2,-e*x^r/d)*ln(x^n)-b^2/r/d*ln(d+e*x^r)*ln(x^n)^2+b^2*n/d*ln(x)^

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

(I*c*x^n)^2-I/r/d*ln(x^r)*ln(x^n)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-I/r/d*ln(x^r)*Pi*a*b*csgn(I*c)*cs

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

b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-I/r/d*ln(d+e*x^r)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*c*x^n)^2-I/r^2*n/d*polylog(2,-

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

a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+I/r/d*ln(x^r)*n*ln(x)*b^2*Pi*csgn(I*c*x^n)^3+I/r/d*ln(x^r)*ln(c)*Pi*b^2*csgn(I

*x^n)*csgn(I*c*x^n)^2+I/r/d*ln(x^r)*Pi*a*b*csgn(I*c)*csgn(I*c*x^n)^2-I/r/d*ln(x^r)*n*ln(x)*b^2*Pi*csgn(I*c)*cs

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

*x^n)*csgn(I*c*x^n)-I/r*n/d*ln(x)*ln(1+e*x^r/d)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-I/r^2*n/d*polylog(2,-e*x^r/d)

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

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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*x^n))^2/x/(d+e*x^r),x, algorithm="maxima")

[Out]

a^2*(log(x)/d - log((d + e^(r*log(x) + 1))*e^(-1))/(d*r)) + integrate((b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*l

og(c) + 2*(b^2*log(c) + a*b)*log(x^n))/(d*x + x*e^(r*log(x) + 1)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (90) = 180\).
time = 0.39, size = 232, normalized size = 2.47 \begin {gather*} \frac {b^{2} n^{2} r^{3} \log \left (x\right )^{3} + 6 \, b^{2} n^{2} {\rm polylog}\left (3, -\frac {x^{r} e}{d}\right ) + 3 \, {\left (b^{2} n r^{3} \log \left (c\right ) + a b n r^{3}\right )} \log \left (x\right )^{2} - 6 \, {\left (b^{2} n^{2} r \log \left (x\right ) + b^{2} n r \log \left (c\right ) + a b n r\right )} {\rm Li}_2\left (-\frac {x^{r} e + d}{d} + 1\right ) - 3 \, {\left (b^{2} r^{2} \log \left (c\right )^{2} + 2 \, a b r^{2} \log \left (c\right ) + a^{2} r^{2}\right )} \log \left (x^{r} e + d\right ) + 3 \, {\left (b^{2} r^{3} \log \left (c\right )^{2} + 2 \, a b r^{3} \log \left (c\right ) + a^{2} r^{3}\right )} \log \left (x\right ) - 3 \, {\left (b^{2} n^{2} r^{2} \log \left (x\right )^{2} + 2 \, {\left (b^{2} n r^{2} \log \left (c\right ) + a b n r^{2}\right )} \log \left (x\right )\right )} \log \left (\frac {x^{r} e + d}{d}\right )}{3 \, d r^{3}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^2*n^2*r^3*log(x)^3 + 6*b^2*n^2*polylog(3, -x^r*e/d) + 3*(b^2*n*r^3*log(c) + a*b*n*r^3)*log(x)^2 - 6*(b^

2*n^2*r*log(x) + b^2*n*r*log(c) + a*b*n*r)*dilog(-(x^r*e + d)/d + 1) - 3*(b^2*r^2*log(c)^2 + 2*a*b*r^2*log(c)

+ a^2*r^2)*log(x^r*e + d) + 3*(b^2*r^3*log(c)^2 + 2*a*b*r^3*log(c) + a^2*r^3)*log(x) - 3*(b^2*n^2*r^2*log(x)^2

+ 2*(b^2*n*r^2*log(c) + a*b*n*r^2)*log(x))*log((x^r*e + d)/d))/(d*r^3)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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