3.117 \(\int \frac{(a+b \log (c x^n))^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=203 \[ \frac{2 b^2 n^2 \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{3 d^3 e}-\frac{2 b n \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e}-\frac{2 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)}+\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}-\frac{b^2 n^2}{3 d^2 e (d+e x)}-\frac{b^2 n^2 \log (x)}{3 d^3 e}+\frac{b^2 n^2 \log (d+e x)}{d^3 e} \]

[Out]

-(b^2*n^2)/(3*d^2*e*(d + e*x)) - (b^2*n^2*Log[x])/(3*d^3*e) + (b*n*(a + b*Log[c*x^n]))/(3*d*e*(d + e*x)^2) - (
2*b*n*x*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x)) - (2*b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/(3*d^3*e) - (a + b
*Log[c*x^n])^2/(3*e*(d + e*x)^3) + (b^2*n^2*Log[d + e*x])/(d^3*e) + (2*b^2*n^2*PolyLog[2, -(d/(e*x))])/(3*d^3*
e)

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Rubi [A]  time = 0.312296, antiderivative size = 221, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 44} \[ -\frac{2 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{3 d^3 e}-\frac{2 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)}-\frac{2 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 d^3 e}+\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}-\frac{b^2 n^2}{3 d^2 e (d+e x)}-\frac{b^2 n^2 \log (x)}{3 d^3 e}+\frac{b^2 n^2 \log (d+e x)}{d^3 e} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^2/(d + e*x)^4,x]

[Out]

-(b^2*n^2)/(3*d^2*e*(d + e*x)) - (b^2*n^2*Log[x])/(3*d^3*e) + (b*n*(a + b*Log[c*x^n]))/(3*d*e*(d + e*x)^2) - (
2*b*n*x*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x)) + (a + b*Log[c*x^n])^2/(3*d^3*e) - (a + b*Log[c*x^n])^2/(3*e*(d
+ e*x)^3) + (b^2*n^2*Log[d + e*x])/(d^3*e) - (2*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*d^3*e) - (2*b^2*n^
2*PolyLog[2, -((e*x)/d)])/(3*d^3*e)

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2317

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 2391

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 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}+\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{3 e}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}-\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 d}+\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 d e}\\ &=\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}-\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 d^2}+\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 d^2 e}-\frac{\left (b^2 n^2\right ) \int \frac{1}{x (d+e x)^2} \, dx}{3 d e}\\ &=\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac{2 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}-\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{3 d^3}+\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{3 d^3 e}+\frac{\left (2 b^2 n^2\right ) \int \frac{1}{d+e x} \, dx}{3 d^3}-\frac{\left (b^2 n^2\right ) \int \left (\frac{1}{d^2 x}-\frac{e}{d (d+e x)^2}-\frac{e}{d^2 (d+e x)}\right ) \, dx}{3 d e}\\ &=-\frac{b^2 n^2}{3 d^2 e (d+e x)}-\frac{b^2 n^2 \log (x)}{3 d^3 e}+\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac{2 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 d^3 e}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}+\frac{b^2 n^2 \log (d+e x)}{d^3 e}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{3 d^3 e}+\frac{\left (2 b^2 n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{3 d^3 e}\\ &=-\frac{b^2 n^2}{3 d^2 e (d+e x)}-\frac{b^2 n^2 \log (x)}{3 d^3 e}+\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac{2 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 d^3 e}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}+\frac{b^2 n^2 \log (d+e x)}{d^3 e}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{3 d^3 e}-\frac{2 b^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{3 d^3 e}\\ \end{align*}

Mathematica [A]  time = 0.155564, size = 211, normalized size = 1.04 \[ \frac{2 b n \left (-\frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^3}-\frac{\log \left (\frac{d+e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac{a+b \log \left (c x^n\right )}{d^2 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}+\frac{a+b \log \left (c x^n\right )}{2 d (d+e x)^2}-\frac{b n \left (-\frac{\log (d+e x)}{d^2}+\frac{\log (x)}{d^2}+\frac{1}{d (d+e x)}\right )}{2 d}-\frac{b n \left (\frac{\log (x)}{d}-\frac{\log (d+e x)}{d}\right )}{d^2}\right )}{3 e}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^2/(d + e*x)^4,x]

[Out]

-(a + b*Log[c*x^n])^2/(3*e*(d + e*x)^3) + (2*b*n*((a + b*Log[c*x^n])/(2*d*(d + e*x)^2) + (a + b*Log[c*x^n])/(d
^2*(d + e*x)) + (a + b*Log[c*x^n])^2/(2*b*d^3*n) - (b*n*(1/(d*(d + e*x)) + Log[x]/d^2 - Log[d + e*x]/d^2))/(2*
d) - (b*n*(Log[x]/d - Log[d + e*x]/d))/d^2 - ((a + b*Log[c*x^n])*Log[(d + e*x)/d])/d^3 - (b*n*PolyLog[2, -((e*
x)/d)])/d^3))/(3*e)

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Maple [C]  time = 0.277, size = 1227, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b^2*n^2/d^2/e/(e*x+d)+1/3*I/e*n/d^3*ln(x)*b^2*Pi*csgn(I*c*x^n)^2*csgn(I*c)-1/6*I/e*n/d/(e*x+d)^2*b^2*Pi*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/3*I/e*n/d^3*ln(x)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/3*I/e*n/d^
3*ln(e*x+d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/6*I/e*n/d/(e*x+d)^2*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^
2+1/3*I/(e*x+d)^3/e*ln(x^n)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2/3*b^2/e*n^2/d^3*ln(e*x+d)*ln(-e*x/d)+
2/3*b/e*n/d^2/(e*x+d)*a+1/3*b/e*n/d/(e*x+d)^2*a-2/3*b/e*n/d^3*ln(e*x+d)*a+2/3*b/e*n/d^3*ln(x)*a+2/3/e*n/d^2/(e
*x+d)*b^2*ln(c)+1/3/e*n/d/(e*x+d)^2*b^2*ln(c)-2/3/e*n/d^3*ln(e*x+d)*b^2*ln(c)+2/3/e*n/d^3*ln(x)*b^2*ln(c)-2/3*
b^2/e*n*ln(x^n)/d^3*ln(e*x+d)+2/3*b^2/e*n*ln(x^n)/d^2/(e*x+d)+1/3*b^2/e*n*ln(x^n)/d/(e*x+d)^2-1/3*I/e*n/d^2/(e
*x+d)*b^2*Pi*csgn(I*c*x^n)^3-1/3*I/(e*x+d)^3/e*ln(x^n)*b^2*Pi*csgn(I*c*x^n)^2*csgn(I*c)-1/3*I/e*n/d^3*ln(x)*b^
2*Pi*csgn(I*c*x^n)^3+1/3*I/e*n/d^3*ln(e*x+d)*b^2*Pi*csgn(I*c*x^n)^3-1/6*I/e*n/d/(e*x+d)^2*b^2*Pi*csgn(I*c*x^n)
^3-1/3*I/(e*x+d)^3/e*ln(x^n)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/3*b^2/(e*x+d)^3/e*ln(x^n)^2-1/3*I/e*n/d^2/(e
*x+d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2/3*b^2/e*n*ln(x^n)/d^3*ln(x)+1/6*I/e*n/d/(e*x+d)^2*b^2*Pi*cs
gn(I*c*x^n)^2*csgn(I*c)-2/3/(e*x+d)^3/e*ln(x^n)*b^2*ln(c)+1/3*I/(e*x+d)^3/e*ln(x^n)*b^2*Pi*csgn(I*c*x^n)^3-1/1
2*(I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*b*Pi*csgn(I*c*x^n)^3+I*b*Pi
*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a)^2/(e*x+d)^3/e-1/3*I/e*n/d^3*ln(e*x+d)*b^2*Pi*csgn(I*c*x^n)^2*csgn(I*
c)-1/3*I/e*n/d^3*ln(e*x+d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/3*I/e*n/d^2/(e*x+d)*b^2*Pi*csgn(I*c*x^n)^2*csg
n(I*c)+1/3*I/e*n/d^2/(e*x+d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/3*I/e*n/d^3*ln(x)*b^2*Pi*csgn(I*x^n)*csgn(I*
c*x^n)^2-b^2*n^2*ln(x)/d^3/e+b^2*n^2*ln(e*x+d)/d^3/e-1/3*b^2/e*n^2/d^3*ln(x)^2+2/3*b^2/e*n^2/d^3*dilog(-e*x/d)
-2/3*b/(e*x+d)^3/e*ln(x^n)*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a b n{\left (\frac{2 \, e x + 3 \, d}{d^{2} e^{3} x^{2} + 2 \, d^{3} e^{2} x + d^{4} e} - \frac{2 \, \log \left (e x + d\right )}{d^{3} e} + \frac{2 \, \log \left (x\right )}{d^{3} e}\right )} - \frac{1}{3} \, b^{2}{\left (\frac{\log \left (x^{n}\right )^{2}}{e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e} - 3 \, \int \frac{3 \, e x \log \left (c\right )^{2} + 2 \,{\left (d n +{\left (e n + 3 \, e \log \left (c\right )\right )} x\right )} \log \left (x^{n}\right )}{3 \,{\left (e^{5} x^{5} + 4 \, d e^{4} x^{4} + 6 \, d^{2} e^{3} x^{3} + 4 \, d^{3} e^{2} x^{2} + d^{4} e x\right )}}\,{d x}\right )} - \frac{2 \, a b \log \left (c x^{n}\right )}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac{a^{2}}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/3*a*b*n*((2*e*x + 3*d)/(d^2*e^3*x^2 + 2*d^3*e^2*x + d^4*e) - 2*log(e*x + d)/(d^3*e) + 2*log(x)/(d^3*e)) - 1/
3*b^2*(log(x^n)^2/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 3*integrate(1/3*(3*e*x*log(c)^2 + 2*(d*n + (
e*n + 3*e*log(c))*x)*log(x^n))/(e^5*x^5 + 4*d*e^4*x^4 + 6*d^2*e^3*x^3 + 4*d^3*e^2*x^2 + d^4*e*x), x)) - 2/3*a*
b*log(c*x^n)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 1/3*a^2/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^
3*e)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4)
, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**2/(e*x+d)**4,x)

[Out]

Integral((a + b*log(c*x**n))**2/(d + e*x)**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2/(e*x+d)^4,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^2/(e*x + d)^4, x)