3.61 \(\int \frac{\log (d (\frac{1}{d}+f \sqrt{x})) (a+b \log (c x^n))^3}{x} \, dx\)

Optimal. Leaf size=101 \[ -48 b^2 n^2 \text{PolyLog}\left (4,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+12 b n \text{PolyLog}\left (3,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+96 b^3 n^3 \text{PolyLog}\left (5,-d f \sqrt{x}\right ) \]

[Out]

-2*(a + b*Log[c*x^n])^3*PolyLog[2, -(d*f*Sqrt[x])] + 12*b*n*(a + b*Log[c*x^n])^2*PolyLog[3, -(d*f*Sqrt[x])] -
48*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[4, -(d*f*Sqrt[x])] + 96*b^3*n^3*PolyLog[5, -(d*f*Sqrt[x])]

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Rubi [A]  time = 0.0992244, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2374, 2383, 6589} \[ -48 b^2 n^2 \text{PolyLog}\left (4,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+12 b n \text{PolyLog}\left (3,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+96 b^3 n^3 \text{PolyLog}\left (5,-d f \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/x,x]

[Out]

-2*(a + b*Log[c*x^n])^3*PolyLog[2, -(d*f*Sqrt[x])] + 12*b*n*(a + b*Log[c*x^n])^2*PolyLog[3, -(d*f*Sqrt[x])] -
48*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[4, -(d*f*Sqrt[x])] + 96*b^3*n^3*PolyLog[5, -(d*f*Sqrt[x])]

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{\log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx &=-2 \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-d f \sqrt{x}\right )+(6 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-2 \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-d f \sqrt{x}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-d f \sqrt{x}\right )-\left (24 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-2 \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-d f \sqrt{x}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-d f \sqrt{x}\right )-48 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_4\left (-d f \sqrt{x}\right )+\left (48 b^3 n^3\right ) \int \frac{\text{Li}_4\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-2 \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-d f \sqrt{x}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-d f \sqrt{x}\right )-48 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_4\left (-d f \sqrt{x}\right )+96 b^3 n^3 \text{Li}_5\left (-d f \sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.205817, size = 98, normalized size = 0.97 \[ 12 b n \left (\text{PolyLog}\left (3,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+4 b n \left (2 b n \text{PolyLog}\left (5,-d f \sqrt{x}\right )-\text{PolyLog}\left (4,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )\right )\right )-2 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3 \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/x,x]

[Out]

-2*(a + b*Log[c*x^n])^3*PolyLog[2, -(d*f*Sqrt[x])] + 12*b*n*((a + b*Log[c*x^n])^2*PolyLog[3, -(d*f*Sqrt[x])] +
4*b*n*(-((a + b*Log[c*x^n])*PolyLog[4, -(d*f*Sqrt[x])]) + 2*b*n*PolyLog[5, -(d*f*Sqrt[x])]))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}{x}\ln \left ( d \left ({d}^{-1}+f\sqrt{x} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^(1/2)))/x,x)

[Out]

int((a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^(1/2)))/x,x)

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Maxima [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 \sqrt{x} + \frac{1}{d}\right )} 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*(1/d+f*x^(1/2)))/x,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)^3*log((f*sqrt(x) + 1/d)*d)/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 (d f \sqrt{x} + 1\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*(1/d+f*x^(1/2)))/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(d*f*sqrt(x) + 1)/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*(1/d+f*x**(1/2)))/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 \sqrt{x} + \frac{1}{d}\right )} 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*(1/d+f*x^(1/2)))/x,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^3*log((f*sqrt(x) + 1/d)*d)/x, x)