3.2.49 \(\int \frac {a+b \log (c x^n)}{x^2 \sqrt {d+e x}} \, dx\) [149]

3.2.49.1 Optimal result

Integrand size = 23, antiderivative size = 226 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \sqrt {d+e x}} \, dx=-\frac {b n \sqrt {d+e x}}{d x}-\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac {e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}+\frac {2 b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}}+\frac {b e n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}} \]

output

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

3.2.49.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.73 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \sqrt {d+e x}} \, dx=-\frac {4 a \sqrt {d} \sqrt {d+e x}+4 b \sqrt {d} n \sqrt {d+e x}+4 b e n x \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+4 b \sqrt {d} \sqrt {d+e x} \log \left (c x^n\right )+2 a e x \log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 b e x \log \left (c x^n\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-b e n x \log ^2\left (\sqrt {d}-\sqrt {d+e x}\right )-2 a e x \log \left (\sqrt {d}+\sqrt {d+e x}\right )-2 b e x \log \left (c x^n\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )+b e n x \log ^2\left (\sqrt {d}+\sqrt {d+e x}\right )+2 b e n x \log \left (\sqrt {d}+\sqrt {d+e x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )-2 b e n x \log \left (\sqrt {d}-\sqrt {d+e x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )-2 b e n x \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+2 b e n x \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{4 d^{3/2} x} \]

input

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

output

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

3.2.49.3 Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {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.

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

\(\Big \downarrow \) 2792

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.2.49.3.1 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 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])
 

3.2.49.4 Maple [F]

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

input

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

output

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

3.2.49.5 Fricas [F]

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

input

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

3.2.49.6 Sympy [F]

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

input

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

output

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

3.2.49.7 Maxima [F]

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

input

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

output

-1/2*a*(2*sqrt(e*x + d)*e/((e*x + d)*d - d^2) + e*log((sqrt(e*x + d) - sqr 
t(d))/(sqrt(e*x + d) + sqrt(d)))/d^(3/2)) + b*integrate((log(c) + log(x^n) 
)/(sqrt(e*x + d)*x^2), x)
 

3.2.49.8 Giac [F]

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

input

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

output

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

3.2.49.9 Mupad [F(-1)]

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

input

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

output

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