3.2.30 \(\int \frac {\log (d (e+f \sqrt {x})) (a+b \log (c x^n))^3}{x} \, dx\) [130]

Optimal. Leaf size=178 \[ \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )-48 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )+96 b^3 n^3 \text {Li}_5\left (-\frac {f \sqrt {x}}{e}\right ) \]

[Out]

1/4*(a+b*ln(c*x^n))^4*ln(d*(e+f*x^(1/2)))/b/n-1/4*(a+b*ln(c*x^n))^4*ln(1+f*x^(1/2)/e)/b/n-2*(a+b*ln(c*x^n))^3*
polylog(2,-f*x^(1/2)/e)+12*b*n*(a+b*ln(c*x^n))^2*polylog(3,-f*x^(1/2)/e)-48*b^2*n^2*(a+b*ln(c*x^n))*polylog(4,
-f*x^(1/2)/e)+96*b^3*n^3*polylog(5,-f*x^(1/2)/e)

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Rubi [A]
time = 0.16, antiderivative size = 178, normalized size of antiderivative = 1.00, 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 = {2422, 2375, 2421, 2430, 6724} \begin {gather*} -48 b^2 n^2 \text {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-2 \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+12 b n \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+96 b^3 n^3 \text {PolyLog}\left (5,-\frac {f \sqrt {x}}{e}\right )+\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^4)/(4*b*n) - (Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^4)/(4*b*n)
- 2*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*Sqrt[x])/e)] + 12*b*n*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*Sqrt[x])/e
)] - 48*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[4, -((f*Sqrt[x])/e)] + 96*b^3*n^3*PolyLog[5, -((f*Sqrt[x])/e)]

Rule 2375

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2422

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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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 6724

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 (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {f \int \frac {\left (a+b \log \left (c x^n\right )\right )^4}{\left (e+f \sqrt {x}\right ) \sqrt {x}} \, dx}{8 b n}\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}+\int \frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+(6 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )-\left (24 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )-48 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )+\left (48 b^3 n^3\right ) \int \frac {\text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )-48 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )+96 b^3 n^3 \text {Li}_5\left (-\frac {f \sqrt {x}}{e}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(403\) vs. \(2(178)=356\).
time = 0.24, size = 403, normalized size = 2.26 \begin {gather*} \frac {1}{8} \left (-2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log (x) \left (b^3 n^3 \log ^3(x)-4 b^2 n^2 \log ^2(x) \left (a+b \log \left (c x^n\right )\right )+6 b n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-4 \left (a+b \log \left (c x^n\right )\right )^3\right )-8 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3 \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)+2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )\right )-12 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+4 \log (x) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )-8 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )\right )-8 b^2 n^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^3(x)+6 \log ^2(x) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )-24 \log (x) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )+48 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )\right )-2 b^3 n^3 \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^4(x)+8 \log ^3(x) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )-48 \log ^2(x) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )+192 \log (x) \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )-384 \text {Li}_5\left (-\frac {f \sqrt {x}}{e}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Log[d*(e + f*Sqrt[x])]*Log[x]*(b^3*n^3*Log[x]^3 - 4*b^2*n^2*Log[x]^2*(a + b*Log[c*x^n]) + 6*b*n*Log[x]*(a
+ b*Log[c*x^n])^2 - 4*(a + b*Log[c*x^n])^3) - 8*(a - b*n*Log[x] + b*Log[c*x^n])^3*(Log[1 + (f*Sqrt[x])/e]*Log[
x] + 2*PolyLog[2, -((f*Sqrt[x])/e)]) - 12*b*n*(a - b*n*Log[x] + b*Log[c*x^n])^2*(Log[1 + (f*Sqrt[x])/e]*Log[x]
^2 + 4*Log[x]*PolyLog[2, -((f*Sqrt[x])/e)] - 8*PolyLog[3, -((f*Sqrt[x])/e)]) - 8*b^2*n^2*(a - b*n*Log[x] + b*L
og[c*x^n])*(Log[1 + (f*Sqrt[x])/e]*Log[x]^3 + 6*Log[x]^2*PolyLog[2, -((f*Sqrt[x])/e)] - 24*Log[x]*PolyLog[3, -
((f*Sqrt[x])/e)] + 48*PolyLog[4, -((f*Sqrt[x])/e)]) - 2*b^3*n^3*(Log[1 + (f*Sqrt[x])/e]*Log[x]^4 + 8*Log[x]^3*
PolyLog[2, -((f*Sqrt[x])/e)] - 48*Log[x]^2*PolyLog[3, -((f*Sqrt[x])/e)] + 192*Log[x]*PolyLog[4, -((f*Sqrt[x])/
e)] - 384*PolyLog[5, -((f*Sqrt[x])/e)]))/8

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (e +f \sqrt {x}\right )\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(e+f*x^(1/2)))/x,x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(e+f*x^(1/2)))/x,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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