\(\int \frac {(a+b \log (c (d+e \sqrt {x})^n))^2}{x} \, dx\) [411]

Optimal result

Integrand size = 24, antiderivative size = 93 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x} \, dx=2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt {x}}{d}\right )+4 b n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )-4 b^2 n^2 \operatorname {PolyLog}\left (3,1+\frac {e \sqrt {x}}{d}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2504, 2443, 2481, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x} \, dx=4 b n \operatorname {PolyLog}\left (2,\frac {\sqrt {x} e}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-4 b^2 n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {x} e}{d}+1\right ) \]

[In]

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

[Out]

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

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 2443

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

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 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*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt {x}}{d}\right )-(4 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt {x}}{d}\right )-(4 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+e \sqrt {x}\right ) \\ & = 2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt {x}}{d}\right )+4 b n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )-\left (4 b^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt {x}\right ) \\ & = 2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt {x}}{d}\right )+4 b n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )-4 b^2 n^2 \text {Li}_3\left (1+\frac {e \sqrt {x}}{d}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(195\) vs. \(2(93)=186\).

Time = 0.10 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.10 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x} \, dx=\left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log (x)+2 b n \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (\left (\log \left (d+e \sqrt {x}\right )-\log \left (1+\frac {e \sqrt {x}}{d}\right )\right ) \log (x)-2 \operatorname {PolyLog}\left (2,-\frac {e \sqrt {x}}{d}\right )\right )+2 b^2 n^2 \left (\log ^2\left (d+e \sqrt {x}\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+2 \log \left (d+e \sqrt {x}\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e \sqrt {x}}{d}\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{2}}{x}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{2}}{x}\, dx \]

[In]

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

[Out]

Integral((a + b*log(c*(d + e*sqrt(x))**n))**2/x, x)

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^2}{x} \,d x \]

[In]

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

[Out]

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