3.2.48 \(\int \frac {a+b \log (c x^n)}{x \sqrt {d+e x}} \, dx\) [148]

3.2.48.1 Optimal result

Integrand size = 23, antiderivative size = 152 \[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx=\frac {2 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}-\frac {4 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}}-\frac {2 b n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}} \]

output

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

3.2.48.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.64 \[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx=\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-2 \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )-b n \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+b n \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right )}{2 \sqrt {d}} \]

input

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

output

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

3.2.48.3 Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2790, 27, 7267, 25, 6546, 27, 6470, 27, 2849, 2752}

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.

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

\(\Big \downarrow \) 2790

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7267

\(\displaystyle \frac {4 b n \int \frac {\sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e x}d\sqrt {d+e x}}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 b n \int -\frac {\sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e x}d\sqrt {d+e x}}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\)

\(\Big \downarrow \) 6546

\(\displaystyle \frac {4 b n \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\frac {\int \frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}-\sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 b n \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\int \frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}-\sqrt {d+e x}}d\sqrt {d+e x}\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {4 b n \left (\frac {\int -\frac {d \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{e x}d\sqrt {d+e x}}{\sqrt {d}}+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 b n \left (\sqrt {d} \int -\frac {\log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{e x}d\sqrt {d+e x}+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\)

\(\Big \downarrow \) 2849

\(\displaystyle \frac {4 b n \left (-\sqrt {d} \int \frac {\log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}}d\frac {1}{\sqrt {d}-\sqrt {d+e x}}+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\)

\(\Big \downarrow \) 2752

\(\displaystyle \frac {4 b n \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\)

input

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

output

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

3.2.48.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2790

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.)) 
/(x_), x_Symbol] :> With[{u = IntHide[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*L 
og[c*x^n]), x] - Simp[b*n   Int[1/x   u, x], x]] /; FreeQ[{a, b, c, d, e, n 
, r}, x] && IntegerQ[q - 1/2]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 
2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 
, 0]
 

rule 6546

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

3.2.48.4 Maple [F]

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

input

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

output

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

3.2.48.5 Fricas [F]

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

input

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

output

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

3.2.48.6 Sympy [F]

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

input

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

output

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

3.2.48.7 Maxima [F]

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

input

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

output

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

3.2.48.8 Giac [F]

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

input

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

output

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

3.2.48.9 Mupad [F(-1)]

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

input

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

output

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