\(\int \frac {a+b \log (c x^n)}{x (d+e x^2)^{3/2}} \, dx\) [290]

Optimal result

Integrand size = 25, antiderivative size = 209 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^{3/2}} \, dx=\frac {b n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {b n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2}{2 d^{3/2}}+\left (\frac {1}{d \sqrt {d+e x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )}{d^{3/2}}-\frac {b n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )}{2 d^{3/2}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.51 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.15 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^{3/2}} \, dx=\frac {-b d^{3/2} n \sqrt {1+\frac {d}{e x^2}} \, _3F_2\left (\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};-\frac {d}{e x^2}\right )+9 e x^2 \left (-b \sqrt {e} n \sqrt {1+\frac {d}{e x^2}} x \text {arcsinh}\left (\frac {\sqrt {d}}{\sqrt {e} x}\right ) \log (x)-b n \sqrt {d+e x^2} \log ^2(x)+\sqrt {d+e x^2} \log (x) \left (a+b \log \left (c x^n\right )+b n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )\right )+\left (a+b \log \left (c x^n\right )\right ) \left (\sqrt {d}-\sqrt {d+e x^2} \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )\right )\right )}{9 d^{3/2} e x^2 \sqrt {d+e x^2}} \] Input:

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

Output:

(-(b*d^(3/2)*n*Sqrt[1 + d/(e*x^2)]*HypergeometricPFQ[{3/2, 3/2, 3/2}, {5/2 
, 5/2}, -(d/(e*x^2))]) + 9*e*x^2*(-(b*Sqrt[e]*n*Sqrt[1 + d/(e*x^2)]*x*ArcS 
inh[Sqrt[d]/(Sqrt[e]*x)]*Log[x]) - b*n*Sqrt[d + e*x^2]*Log[x]^2 + Sqrt[d + 
 e*x^2]*Log[x]*(a + b*Log[c*x^n] + b*n*Log[d + Sqrt[d]*Sqrt[d + e*x^2]]) + 
 (a + b*Log[c*x^n])*(Sqrt[d] - Sqrt[d + e*x^2]*Log[d + Sqrt[d]*Sqrt[d + e* 
x^2]])))/(9*d^(3/2)*e*x^2*Sqrt[d + e*x^2])
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2790, 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.

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

\(\Big \downarrow \) 2790

\(\displaystyle \left (\frac {1}{d \sqrt {d+e x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \left (\frac {1}{d x \sqrt {e x^2+d}}-\frac {\text {arctanh}\left (\frac {\sqrt {e x^2+d}}{\sqrt {d}}\right )}{d^{3/2} x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \left (\frac {1}{d \sqrt {d+e x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \left (-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2}{2 d^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )}{d^{3/2}}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {e x^2+d}}\right )}{2 d^{3/2}}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((sqrt(e*x^2 + d)*b*log(c*x^n) + sqrt(e*x^2 + d)*a)/(e^2*x^5 + 2*d 
*e*x^3 + d^2*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d + e*x**2)*a*d + sqrt(d)*log((sqrt(d + e*x**2) - sqrt(d) + sqrt(e)* 
x)/sqrt(d))*a*d + sqrt(d)*log((sqrt(d + e*x**2) - sqrt(d) + sqrt(e)*x)/sqr 
t(d))*a*e*x**2 - sqrt(d)*log((sqrt(d + e*x**2) + sqrt(d) + sqrt(e)*x)/sqrt 
(d))*a*d - sqrt(d)*log((sqrt(d + e*x**2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*a 
*e*x**2 + int(log(x**n*c)/(sqrt(d + e*x**2)*d*x + sqrt(d + e*x**2)*e*x**3) 
,x)*b*d**3 + int(log(x**n*c)/(sqrt(d + e*x**2)*d*x + sqrt(d + e*x**2)*e*x* 
*3),x)*b*d**2*e*x**2)/(d**2*(d + e*x**2))