3.3.79 \(\int \frac {a+b \log (c x^n)}{x \sqrt {d+e x^2}} \, dx\) [279]

3.3.79.1 Optimal result

Integrand size = 25, antiderivative size = 166 \[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x^2}} \, dx=\frac {b n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2}{2 \sqrt {d}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}-\frac {b n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )}{\sqrt {d}}-\frac {b n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )}{2 \sqrt {d}} \]

output

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

3.3.79.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x^2}} \, dx=\frac {b n \sqrt {1+\frac {d}{e x^2}} \left (-\, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {d}{e x^2}\right )-\frac {\sqrt {e} x \text {arcsinh}\left (\frac {\sqrt {d}}{\sqrt {e} x}\right ) \log (x)}{\sqrt {d}}\right )}{\sqrt {d+e x^2}}-\frac {\log (x) \left (-a-b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{\sqrt {d}}+\frac {\left (-a-b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{\sqrt {d}} \]

input

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

output

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

3.3.79.3 Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.93, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2790, 25, 27, 7282, 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.

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

\(\Big \downarrow \) 2790

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7282

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

input

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

output

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

3.3.79.3.1 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]]
 

rule 7282

Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 
/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( 
lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ 
u] &&  !RationalFunctionQ[u, x]
 

3.3.79.4 Maple [F]

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

input

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

output

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

3.3.79.5 Fricas [F]

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

input

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

output

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

3.3.79.6 Sympy [F]

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

input

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

output

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

3.3.79.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.3.79.8 Giac [F]

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

input

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

output

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

3.3.79.9 Mupad [F(-1)]

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

input

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

output

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