3.109 \(\int \frac{x (a+b \log (c x^n))^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=112 \[ -\frac{b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d e^2}-\frac{b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )+b n\right )}{d e^2}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.356785, antiderivative size = 176, normalized size of antiderivative = 1.57, number of steps used = 13, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {2353, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2318} \[ -\frac{b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d e^2}-\frac{b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d e^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}-\frac{b^2 n^2 \log (d+e x)}{d e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

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 2318

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

Rubi steps

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

Mathematica [A]  time = 0.219712, size = 155, normalized size = 1.38 \[ \frac{-\frac{2 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-\frac{2 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac{2 b^2 n^2 (\log (x)-\log (d+e x))}{d}}{2 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.275, size = 1199, normalized size = 10.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*ln(c*x^n))^2/(e*x+d)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n))^2/(e*x+d)^3,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n))^2/(e*x+d)^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*ln(c*x**n))**2/(e*x+d)**3,x)

[Out]

Integral(x*(a + b*log(c*x**n))**2/(d + e*x)**3, 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} x}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n))^2/(e*x+d)^3,x, algorithm="giac")

[Out]

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