3.2.55 \(\int \frac {a+b \log (c x^n)}{x (d+e x)^{3/2}} \, dx\) [155]

3.2.55.1 Optimal result

Integrand size = 23, antiderivative size = 201 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx=\frac {4 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac {4 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}}-\frac {2 b n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}} \]

output

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

3.2.55.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.47 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx=\frac {8 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+\frac {4 \sqrt {d} \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x}}+2 \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-2 \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )-b n \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+b n \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right )}{2 d^{3/2}} \]

input

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

output

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

3.2.55.3 Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2789, 2756, 73, 221, 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.

\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx\)

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 2790

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

input

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

output

-((e*((-4*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(Sqrt[d]*e) - (2*(a + b*Log[ 
c*x^n]))/(e*Sqrt[d + e*x])))/d) + ((-2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + 
 b*Log[c*x^n]))/Sqrt[d] + (4*b*n*(ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2/2 - Arc 
Tanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])] - P 
olyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])]/2))/Sqrt[d])/d
 

3.2.55.3.1 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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

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 2756

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

rule 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

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]]
 

3.2.55.4 Maple [F]

\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x \left (e x +d \right )^{\frac {3}{2}}}d x\]

input

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

output

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

3.2.55.5 Fricas [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x} \,d x } \]

input

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

output

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

3.2.55.6 Sympy [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{x \left (d + e x\right )^{\frac {3}{2}}}\, dx \]

input

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

output

Integral((a + b*log(c*x**n))/(x*(d + e*x)**(3/2)), x)
 

3.2.55.7 Maxima [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x} \,d x } \]

input

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

output

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

3.2.55.8 Giac [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x} \,d x } \]

input

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

output

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

3.2.55.9 Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x\right )}^{3/2}} \,d x \]

input

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

output

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