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

Optimal. Leaf size=181 \[ -\frac{3}{4} b^2 m n^2 \text{PolyLog}\left (4,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} m \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{3}{4} b m n \text{PolyLog}\left (3,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{3}{8} b^3 m n^3 \text{PolyLog}\left (5,-\frac{f x^2}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^2\right )^m\right )}{4 b n}-\frac{m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]

[Out]

((a + b*Log[c*x^n])^4*Log[d*(e + f*x^2)^m])/(4*b*n) - (m*(a + b*Log[c*x^n])^4*Log[1 + (f*x^2)/e])/(4*b*n) - (m
*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*x^2)/e)])/2 + (3*b*m*n*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*x^2)/e)])/4
- (3*b^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[4, -((f*x^2)/e)])/4 + (3*b^3*m*n^3*PolyLog[5, -((f*x^2)/e)])/8

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Rubi [A]  time = 0.214528, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2375, 2337, 2374, 2383, 6589} \[ -\frac{3}{4} b^2 m n^2 \text{PolyLog}\left (4,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} m \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{3}{4} b m n \text{PolyLog}\left (3,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{3}{8} b^3 m n^3 \text{PolyLog}\left (5,-\frac{f x^2}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^2\right )^m\right )}{4 b n}-\frac{m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*Log[c*x^n])^4*Log[d*(e + f*x^2)^m])/(4*b*n) - (m*(a + b*Log[c*x^n])^4*Log[1 + (f*x^2)/e])/(4*b*n) - (m
*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*x^2)/e)])/2 + (3*b*m*n*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*x^2)/e)])/4
- (3*b^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[4, -((f*x^2)/e)])/4 + (3*b^3*m*n^3*PolyLog[5, -((f*x^2)/e)])/8

Rule 2375

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

Rule 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

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 2383

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

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{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x} \, dx &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^2\right )^m\right )}{4 b n}-\frac{(f m) \int \frac{x \left (a+b \log \left (c x^n\right )\right )^4}{e+f x^2} \, dx}{2 b n}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^2\right )^m\right )}{4 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x^2}{e}\right )}{4 b n}+m \int \frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x^2}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^2\right )^m\right )}{4 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x^2}{e}\right )}{4 b n}-\frac{1}{2} m \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x^2}{e}\right )+\frac{1}{2} (3 b m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x^2}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^2\right )^m\right )}{4 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x^2}{e}\right )}{4 b n}-\frac{1}{2} m \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x^2}{e}\right )+\frac{3}{4} b m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-\frac{f x^2}{e}\right )-\frac{1}{2} \left (3 b^2 m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x^2}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^2\right )^m\right )}{4 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x^2}{e}\right )}{4 b n}-\frac{1}{2} m \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x^2}{e}\right )+\frac{3}{4} b m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-\frac{f x^2}{e}\right )-\frac{3}{4} b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_4\left (-\frac{f x^2}{e}\right )+\frac{1}{4} \left (3 b^3 m n^3\right ) \int \frac{\text{Li}_4\left (-\frac{f x^2}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^2\right )^m\right )}{4 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x^2}{e}\right )}{4 b n}-\frac{1}{2} m \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x^2}{e}\right )+\frac{3}{4} b m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-\frac{f x^2}{e}\right )-\frac{3}{4} b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_4\left (-\frac{f x^2}{e}\right )+\frac{3}{8} b^3 m n^3 \text{Li}_5\left (-\frac{f x^2}{e}\right )\\ \end{align*}

Mathematica [C]  time = 0.365217, size = 1348, normalized size = 7.45 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*m*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]]) + (3*a^2*b*m*n*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]])/2 - a*b
^2*m*n^2*Log[x]^3*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (b^3*m*n^3*Log[x]^4*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]])/4 - 3*a
^2*b*m*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 3*a*b^2*m*n*Log[x]^2*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x
)/Sqrt[e]] - b^3*m*n^2*Log[x]^3*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 3*a*b^2*m*Log[x]*Log[c*x^n]^2*Log[
1 - (I*Sqrt[f]*x)/Sqrt[e]] + (3*b^3*m*n*Log[x]^2*Log[c*x^n]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]])/2 - b^3*m*Log[x]
*Log[c*x^n]^3*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - a^3*m*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (3*a^2*b*m*n*Log[
x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]])/2 - a*b^2*m*n^2*Log[x]^3*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (b^3*m*n^3*Log[
x]^4*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]])/4 - 3*a^2*b*m*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 3*a*b^2*
m*n*Log[x]^2*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - b^3*m*n^2*Log[x]^3*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/S
qrt[e]] - 3*a*b^2*m*Log[x]*Log[c*x^n]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (3*b^3*m*n*Log[x]^2*Log[c*x^n]^2*Log[
1 + (I*Sqrt[f]*x)/Sqrt[e]])/2 - b^3*m*Log[x]*Log[c*x^n]^3*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + a^3*Log[x]*Log[d*(e
+ f*x^2)^m] - (3*a^2*b*n*Log[x]^2*Log[d*(e + f*x^2)^m])/2 + a*b^2*n^2*Log[x]^3*Log[d*(e + f*x^2)^m] - (b^3*n^
3*Log[x]^4*Log[d*(e + f*x^2)^m])/4 + 3*a^2*b*Log[x]*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 3*a*b^2*n*Log[x]^2*Log[c
*x^n]*Log[d*(e + f*x^2)^m] + b^3*n^2*Log[x]^3*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 3*a*b^2*Log[x]*Log[c*x^n]^2*Lo
g[d*(e + f*x^2)^m] - (3*b^3*n*Log[x]^2*Log[c*x^n]^2*Log[d*(e + f*x^2)^m])/2 + b^3*Log[x]*Log[c*x^n]^3*Log[d*(e
+ f*x^2)^m] - m*(a + b*Log[c*x^n])^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] - m*(a + b*Log[c*x^n])^3*PolyLog[2,
(I*Sqrt[f]*x)/Sqrt[e]] + 3*a^2*b*m*n*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 6*a*b^2*m*n*Log[c*x^n]*PolyLog[3,
((-I)*Sqrt[f]*x)/Sqrt[e]] + 3*b^3*m*n*Log[c*x^n]^2*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 3*a^2*b*m*n*PolyLog
[3, (I*Sqrt[f]*x)/Sqrt[e]] + 6*a*b^2*m*n*Log[c*x^n]*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] + 3*b^3*m*n*Log[c*x^n]^2
*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] - 6*a*b^2*m*n^2*PolyLog[4, ((-I)*Sqrt[f]*x)/Sqrt[e]] - 6*b^3*m*n^2*Log[c*x^
n]*PolyLog[4, ((-I)*Sqrt[f]*x)/Sqrt[e]] - 6*a*b^2*m*n^2*PolyLog[4, (I*Sqrt[f]*x)/Sqrt[e]] - 6*b^3*m*n^2*Log[c*
x^n]*PolyLog[4, (I*Sqrt[f]*x)/Sqrt[e]] + 6*b^3*m*n^3*PolyLog[5, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 6*b^3*m*n^3*PolyLo
g[5, (I*Sqrt[f]*x)/Sqrt[e]]

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Maple [F]  time = 2.873, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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 )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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