3.5.32 \(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^2}{x} \, dx\) [432]

3.5.32.1 Optimal result

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

output

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

3.5.32.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(386\) vs. \(2(93)=186\).

Time = 0.23 (sec) , antiderivative size = 386, normalized size of antiderivative = 4.15 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x} \, dx=\left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \log (x)+2 b n \left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (\left (\log \left (d+\frac {e}{\sqrt {x}}\right )-\log \left (1+\frac {e}{d \sqrt {x}}\right )\right ) \log (x)+2 \operatorname {PolyLog}\left (2,-\frac {e}{d \sqrt {x}}\right )\right )+\frac {1}{12} b^2 n^2 \left (24 \log ^2\left (\frac {e}{d}+\sqrt {x}\right ) \log \left (-\frac {d \sqrt {x}}{e}\right )+12 \log ^2\left (d+\frac {e}{\sqrt {x}}\right ) \log (x)-12 \log ^2\left (\frac {e}{d}+\sqrt {x}\right ) \log (x)-24 \log \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (1+\frac {d \sqrt {x}}{e}\right ) \log (x)+24 \log \left (\frac {e}{d}+\sqrt {x}\right ) \log \left (1+\frac {d \sqrt {x}}{e}\right ) \log (x)+6 \log \left (d+\frac {e}{\sqrt {x}}\right ) \log ^2(x)-6 \log \left (1+\frac {d \sqrt {x}}{e}\right ) \log ^2(x)+\log ^3(x)+48 \log \left (\frac {e}{d}+\sqrt {x}\right ) \operatorname {PolyLog}\left (2,1+\frac {d \sqrt {x}}{e}\right )-48 \left (\log \left (d+\frac {e}{\sqrt {x}}\right )-\log \left (\frac {e}{d}+\sqrt {x}\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d \sqrt {x}}{e}\right )-48 \operatorname {PolyLog}\left (3,1+\frac {d \sqrt {x}}{e}\right )-48 \operatorname {PolyLog}\left (3,-\frac {d \sqrt {x}}{e}\right )\right ) \]

input

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

output

(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/Sqr 
t[x]] - Log[1 + e/(d*Sqrt[x])])*Log[x] + 2*PolyLog[2, -(e/(d*Sqrt[x]))]) + 
 (b^2*n^2*(24*Log[e/d + Sqrt[x]]^2*Log[-((d*Sqrt[x])/e)] + 12*Log[d + e/Sq 
rt[x]]^2*Log[x] - 12*Log[e/d + Sqrt[x]]^2*Log[x] - 24*Log[d + e/Sqrt[x]]*L 
og[1 + (d*Sqrt[x])/e]*Log[x] + 24*Log[e/d + Sqrt[x]]*Log[1 + (d*Sqrt[x])/e 
]*Log[x] + 6*Log[d + e/Sqrt[x]]*Log[x]^2 - 6*Log[1 + (d*Sqrt[x])/e]*Log[x] 
^2 + Log[x]^3 + 48*Log[e/d + Sqrt[x]]*PolyLog[2, 1 + (d*Sqrt[x])/e] - 48*( 
Log[d + e/Sqrt[x]] - Log[e/d + Sqrt[x]])*PolyLog[2, -((d*Sqrt[x])/e)] - 48 
*PolyLog[3, 1 + (d*Sqrt[x])/e] - 48*PolyLog[3, -((d*Sqrt[x])/e)]))/12
 

3.5.32.3 Rubi [A] (warning: unable to verify)

Time = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2904, 2843, 2881, 2821, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2904

\(\displaystyle -2 \int \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 2843

\(\displaystyle -2 \left (\log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-2 b e n \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )}{d+\frac {e}{\sqrt {x}}}d\frac {1}{\sqrt {x}}\right )\)

\(\Big \downarrow \) 2881

\(\displaystyle -2 \left (\log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-2 b n \int \sqrt {x} \log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )d\left (d+\frac {e}{\sqrt {x}}\right )\right )\)

\(\Big \downarrow \) 2821

\(\displaystyle -2 \left (\log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-2 b n \left (b n \int \sqrt {x} \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt {x}}}{d}\right )d\left (d+\frac {e}{\sqrt {x}}\right )-\operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt {x}}}{d}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle -2 \left (\log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-2 b n \left (b n \operatorname {PolyLog}\left (3,\frac {d+\frac {e}{\sqrt {x}}}{d}\right )-\operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt {x}}}{d}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )\right )\right )\)

input

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

output

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

3.5.32.3.1 Defintions of rubi rules used

rule 2821

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] + Simp[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 2843

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] - Simp[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 2881

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym 
bol] :> Simp[1/e   Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[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 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

3.5.32.4 Maple [F]

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

input

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

output

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

3.5.32.5 Fricas [F]

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

input

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

output

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

3.5.32.6 Sympy [F]

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

input

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

output

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

3.5.32.7 Maxima [F]

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

input

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

output

b^2*log((d*sqrt(x) + e)^n)^2*log(x) - integrate(-((b^2*d*x + b^2*e*sqrt(x) 
)*log(x^(1/2*n))^2 + (b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x - (b^2*d* 
n*x*log(x) - 2*(b^2*d*log(c) + a*b*d)*x + 2*(b^2*d*x + b^2*e*sqrt(x))*log( 
x^(1/2*n)) - 2*(b^2*e*log(c) + a*b*e)*sqrt(x))*log((d*sqrt(x) + e)^n) - 2* 
((b^2*d*log(c) + a*b*d)*x + (b^2*e*log(c) + a*b*e)*sqrt(x))*log(x^(1/2*n)) 
 + (b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*sqrt(x))/(d*x^2 + e*x^(3/2)), 
 x)
 

3.5.32.8 Giac [F]

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

input

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

output

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

3.5.32.9 Mupad [F(-1)]

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

input

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

output

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