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

Optimal. Leaf size=281 \[ \frac{6 b d^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{2 b^2 d^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^4}-\frac{6 b^2 d^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e^4}+\frac{2 b d^2 n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac{3 d^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac{2 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac{b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac{4 a b d n x}{e^3}+\frac{4 b^2 d n x \log \left (c x^n\right )}{e^3}-\frac{4 b^2 d n^2 x}{e^3}+\frac{b^2 n^2 x^2}{4 e^2} \]

[Out]

(4*a*b*d*n*x)/e^3 - (4*b^2*d*n^2*x)/e^3 + (b^2*n^2*x^2)/(4*e^2) + (4*b^2*d*n*x*Log[c*x^n])/e^3 - (b*n*x^2*(a +
b*Log[c*x^n]))/(2*e^2) - (2*d*x*(a + b*Log[c*x^n])^2)/e^3 + (x^2*(a + b*Log[c*x^n])^2)/(2*e^2) - (d^2*x*(a +
b*Log[c*x^n])^2)/(e^3*(d + e*x)) + (2*b*d^2*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^4 + (3*d^2*(a + b*Log[c*x
^n])^2*Log[1 + (e*x)/d])/e^4 + (2*b^2*d^2*n^2*PolyLog[2, -((e*x)/d)])/e^4 + (6*b*d^2*n*(a + b*Log[c*x^n])*Poly
Log[2, -((e*x)/d)])/e^4 - (6*b^2*d^2*n^2*PolyLog[3, -((e*x)/d)])/e^4

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Rubi [A]  time = 0.309721, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {2353, 2296, 2295, 2305, 2304, 2318, 2317, 2391, 2374, 6589} \[ \frac{6 b d^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{2 b^2 d^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^4}-\frac{6 b^2 d^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e^4}+\frac{2 b d^2 n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac{3 d^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac{2 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac{b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac{4 a b d n x}{e^3}+\frac{4 b^2 d n x \log \left (c x^n\right )}{e^3}-\frac{4 b^2 d n^2 x}{e^3}+\frac{b^2 n^2 x^2}{4 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*b*d*n*x)/e^3 - (4*b^2*d*n^2*x)/e^3 + (b^2*n^2*x^2)/(4*e^2) + (4*b^2*d*n*x*Log[c*x^n])/e^3 - (b*n*x^2*(a +
b*Log[c*x^n]))/(2*e^2) - (2*d*x*(a + b*Log[c*x^n])^2)/e^3 + (x^2*(a + b*Log[c*x^n])^2)/(2*e^2) - (d^2*x*(a +
b*Log[c*x^n])^2)/(e^3*(d + e*x)) + (2*b*d^2*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^4 + (3*d^2*(a + b*Log[c*x
^n])^2*Log[1 + (e*x)/d])/e^4 + (2*b^2*d^2*n^2*PolyLog[2, -((e*x)/d)])/e^4 + (6*b*d^2*n*(a + b*Log[c*x^n])*Poly
Log[2, -((e*x)/d)])/e^4 - (6*b^2*d^2*n^2*PolyLog[3, -((e*x)/d)])/e^4

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 2296

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2305

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

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]

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 6589

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{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx &=\int \left (-\frac{2 d \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^2}+\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac{(2 d) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^3}+\frac{\left (3 d^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^3}-\frac{d^3 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e^3}+\frac{\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^2}\\ &=-\frac{2 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac{d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^4}-\frac{\left (6 b d^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^4}+\frac{(4 b d n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^3}+\frac{\left (2 b d^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^3}-\frac{(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}\\ &=\frac{4 a b d n x}{e^3}+\frac{b^2 n^2 x^2}{4 e^2}-\frac{b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{2 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac{d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac{2 b d^2 n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^4}+\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^4}+\frac{6 b d^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^4}+\frac{\left (4 b^2 d n\right ) \int \log \left (c x^n\right ) \, dx}{e^3}-\frac{\left (2 b^2 d^2 n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^4}-\frac{\left (6 b^2 d^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{e^4}\\ &=\frac{4 a b d n x}{e^3}-\frac{4 b^2 d n^2 x}{e^3}+\frac{b^2 n^2 x^2}{4 e^2}+\frac{4 b^2 d n x \log \left (c x^n\right )}{e^3}-\frac{b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{2 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac{d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac{2 b d^2 n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^4}+\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^4}+\frac{2 b^2 d^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{e^4}+\frac{6 b d^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^4}-\frac{6 b^2 d^2 n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{e^4}\\ \end{align*}

Mathematica [A]  time = 0.20345, size = 240, normalized size = 0.85 \[ \frac{4 d^2 \left (2 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )-\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (\frac{e x}{d}+1\right )\right )\right )+24 b d^2 n \left (\text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{PolyLog}\left (3,-\frac{e x}{d}\right )\right )+\frac{4 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+12 d^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-8 d e x \left (a+b \log \left (c x^n\right )\right )^2+16 b d e n x \left (a+b \log \left (c x^n\right )-b n\right )+2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+b e^2 n x^2 \left (b n-2 \left (a+b \log \left (c x^n\right )\right )\right )}{4 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*d*e*x*(a + b*Log[c*x^n])^2 + 2*e^2*x^2*(a + b*Log[c*x^n])^2 + (4*d^3*(a + b*Log[c*x^n])^2)/(d + e*x) + 16*
b*d*e*n*x*(a - b*n + b*Log[c*x^n]) + b*e^2*n*x^2*(b*n - 2*(a + b*Log[c*x^n])) + 12*d^2*(a + b*Log[c*x^n])^2*Lo
g[1 + (e*x)/d] + 4*d^2*(-((a + b*Log[c*x^n])*(a + b*Log[c*x^n] - 2*b*n*Log[1 + (e*x)/d])) + 2*b^2*n^2*PolyLog[
2, -((e*x)/d)]) + 24*b*d^2*n*((a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] - b*n*PolyLog[3, -((e*x)/d)]))/(4*e^4)

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Maple [F]  time = 0.688, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{ \left ( ex+d \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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