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

Optimal. Leaf size=147 \[ -\frac{1}{2} m \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} b m n \text{PolyLog}\left (3,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b^2 m n^2 \text{PolyLog}\left (4,-\frac{f x^2}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 b n}-\frac{m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]

[Out]

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

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Rubi [A]  time = 0.175081, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ -\frac{1}{2} m \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} b m n \text{PolyLog}\left (3,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b^2 m n^2 \text{PolyLog}\left (4,-\frac{f x^2}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 b n}-\frac{m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.233063, size = 736, normalized size = 5.01 \[ -m \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2-m \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 a b m n \text{PolyLog}\left (3,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 a b m n \text{PolyLog}\left (3,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 b^2 m n \log \left (c x^n\right ) \text{PolyLog}\left (3,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 b^2 m n \log \left (c x^n\right ) \text{PolyLog}\left (3,\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 b^2 m n^2 \text{PolyLog}\left (4,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 b^2 m n^2 \text{PolyLog}\left (4,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+a^2 \log (x) \log \left (d \left (e+f x^2\right )^m\right )-a^2 m \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-a^2 m \log (x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 a b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 a b m \log (x) \log \left (c x^n\right ) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 a b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )-a b n \log ^2(x) \log \left (d \left (e+f x^2\right )^m\right )+a b m n \log ^2(x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+a b m n \log ^2(x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )-b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )-b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )+\frac{1}{3} b^2 n^2 \log ^3(x) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} b^2 m n^2 \log ^3(x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-\frac{1}{3} b^2 m n^2 \log ^3(x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.58, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\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))^2*ln(d*(f*x^2+e)^m)/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^2*n^2*log(x)^3 + 3*b^2*log(x)*log(x^n)^2 - 3*(b^2*n*log(c) + a*b*n)*log(x)^2 - 3*(b^2*n*log(x)^2 - 2*(b
^2*log(c) + a*b)*log(x))*log(x^n) + 3*(b^2*log(c)^2 + 2*a*b*log(c) + a^2)*log(x))*log((f*x^2 + e)^m) - integra
te(1/3*(2*b^2*f*m*n^2*x^2*log(x)^3 - 3*b^2*e*log(c)^2*log(d) - 6*a*b*e*log(c)*log(d) - 6*(b^2*f*m*n*log(c) + a
*b*f*m*n)*x^2*log(x)^2 - 3*a^2*e*log(d) + 6*(b^2*f*m*log(c)^2 + 2*a*b*f*m*log(c) + a^2*f*m)*x^2*log(x) - 3*(b^
2*f*log(c)^2*log(d) + 2*a*b*f*log(c)*log(d) + a^2*f*log(d))*x^2 + 3*(2*b^2*f*m*x^2*log(x) - b^2*f*x^2*log(d) -
b^2*e*log(d))*log(x^n)^2 - 6*(b^2*f*m*n*x^2*log(x)^2 + b^2*e*log(c)*log(d) + a*b*e*log(d) - 2*(b^2*f*m*log(c)
+ a*b*f*m)*x^2*log(x) + (b^2*f*log(c)*log(d) + a*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^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\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))^2*log(d*(f*x^2+e)^m)/x,x, algorithm="fricas")

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)*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))**2*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 )}^{2} \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))^2*log(d*(f*x^2+e)^m)/x,x, algorithm="giac")

[Out]

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