3.3.59 \(\int \frac {\sqrt {d+e x^2} (a+b \log (c x^n))}{x^2} \, dx\) [259]

3.3.59.1 Optimal result

Integrand size = 25, antiderivative size = 345 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {b n \sqrt {d+e x^2}}{x}+\frac {b \sqrt {e} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {1+\frac {e x^2}{d}}}+\frac {b \sqrt {e} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {d} \sqrt {1+\frac {e x^2}{d}}}-\frac {b \sqrt {e} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {d} \sqrt {1+\frac {e x^2}{d}}}-\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {\sqrt {e} \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {1+\frac {e x^2}{d}}}-\frac {b \sqrt {e} n \sqrt {d+e x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {d} \sqrt {1+\frac {e x^2}{d}}} \]

output

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

3.3.59.2 Mathematica [C] (verified)

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

Time = 0.33 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {b n \sqrt {d+e x^2} \left (-\, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};-\frac {e x^2}{d}\right )-\sqrt {1+\frac {e x^2}{d}} \log (x)+\frac {\sqrt {e} x \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)}{\sqrt {d}}\right )}{x \sqrt {1+\frac {e x^2}{d}}}-\frac {\sqrt {d+e x^2} \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{x}+\sqrt {e} \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right ) \]

input

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

output

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

3.3.59.3 Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.71, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2786, 2792, 25, 2010, 2009}

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.

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

\(\Big \downarrow \) 2786

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

\(\Big \downarrow \) 2792

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.3.59.3.1 Defintions of rubi rules used

rule 25

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

rule 2786

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^ 
(q_), x_Symbol] :> Simp[d^IntPart[q]*((d + e*x^2)^FracPart[q]/(1 + (e/d)*x^ 
2)^FracPart[q])   Int[x^m*(1 + (e/d)*x^2)^q*(a + b*Log[c*x^n]), x], x] /; F 
reeQ[{a, b, c, d, e, n}, x] && IntegerQ[m/2] && IntegerQ[q - 1/2] &&  !(LtQ 
[m + 2*q, -2] || GtQ[d, 0])
 

rule 2792

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x] 
}, Simp[(a + b*Log[c*x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, 
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2] 
) || InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x 
] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])
 

3.3.59.4 Maple [F]

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

input

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

output

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

3.3.59.5 Fricas [F]

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

input

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

output

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

3.3.59.6 Sympy [F]

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

input

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

output

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

3.3.59.7 Maxima [F(-2)]

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

input

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

3.3.59.8 Giac [F]

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

input

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

output

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

3.3.59.9 Mupad [F(-1)]

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

input

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

output

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