3.2.42 \(\int \frac {(d+e x)^{3/2} (a+b \log (c x^n))}{x^2} \, dx\) [142]

3.2.42.1 Optimal result

Integrand size = 23, antiderivative size = 259 \[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-4 b e n \sqrt {d+e x}-\frac {b d n \sqrt {d+e x}}{x}+3 b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+3 b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-6 b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-3 b \sqrt {d} e n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \]

output

-(e*x+d)^(3/2)*(a+b*ln(c*x^n))/x+3*b*e*n*arctanh((e*x+d)^(1/2)/d^(1/2))*d^ 
(1/2)+3*b*e*n*arctanh((e*x+d)^(1/2)/d^(1/2))^2*d^(1/2)-3*e*arctanh((e*x+d) 
^(1/2)/d^(1/2))*(a+b*ln(c*x^n))*d^(1/2)-6*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^(1/2)-3*b*e*n*polylog(2,1-2*d 
^(1/2)/(d^(1/2)-(e*x+d)^(1/2)))*d^(1/2)-4*b*e*n*(e*x+d)^(1/2)-b*d*n*(e*x+d 
)^(1/2)/x+3*e*(a+b*ln(c*x^n))*(e*x+d)^(1/2)
 

3.2.42.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {-4 a d \sqrt {d+e x}-4 b d n \sqrt {d+e x}+8 a e x \sqrt {d+e x}-16 b e n x \sqrt {d+e x}+12 b \sqrt {d} e n x \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-4 b d \sqrt {d+e x} \log \left (c x^n\right )+8 b e x \sqrt {d+e x} \log \left (c x^n\right )+6 a \sqrt {d} e x \log \left (\sqrt {d}-\sqrt {d+e x}\right )+6 b \sqrt {d} e x \log \left (c x^n\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-3 b \sqrt {d} e n x \log ^2\left (\sqrt {d}-\sqrt {d+e x}\right )-6 a \sqrt {d} e x \log \left (\sqrt {d}+\sqrt {d+e x}\right )-6 b \sqrt {d} e x \log \left (c x^n\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )+3 b \sqrt {d} e n x \log ^2\left (\sqrt {d}+\sqrt {d+e x}\right )+6 b \sqrt {d} e n x \log \left (\sqrt {d}+\sqrt {d+e x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )-6 b \sqrt {d} e n x \log \left (\sqrt {d}-\sqrt {d+e x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )-6 b \sqrt {d} e n x \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+6 b \sqrt {d} e n x \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{4 x} \]

input

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

output

(-4*a*d*Sqrt[d + e*x] - 4*b*d*n*Sqrt[d + e*x] + 8*a*e*x*Sqrt[d + e*x] - 16 
*b*e*n*x*Sqrt[d + e*x] + 12*b*Sqrt[d]*e*n*x*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] 
 - 4*b*d*Sqrt[d + e*x]*Log[c*x^n] + 8*b*e*x*Sqrt[d + e*x]*Log[c*x^n] + 6*a 
*Sqrt[d]*e*x*Log[Sqrt[d] - Sqrt[d + e*x]] + 6*b*Sqrt[d]*e*x*Log[c*x^n]*Log 
[Sqrt[d] - Sqrt[d + e*x]] - 3*b*Sqrt[d]*e*n*x*Log[Sqrt[d] - Sqrt[d + e*x]] 
^2 - 6*a*Sqrt[d]*e*x*Log[Sqrt[d] + Sqrt[d + e*x]] - 6*b*Sqrt[d]*e*x*Log[c* 
x^n]*Log[Sqrt[d] + Sqrt[d + e*x]] + 3*b*Sqrt[d]*e*n*x*Log[Sqrt[d] + Sqrt[d 
 + e*x]]^2 + 6*b*Sqrt[d]*e*n*x*Log[Sqrt[d] + Sqrt[d + e*x]]*Log[1/2 - Sqrt 
[d + e*x]/(2*Sqrt[d])] - 6*b*Sqrt[d]*e*n*x*Log[Sqrt[d] - Sqrt[d + e*x]]*Lo 
g[(1 + Sqrt[d + e*x]/Sqrt[d])/2] - 6*b*Sqrt[d]*e*n*x*PolyLog[2, 1/2 - Sqrt 
[d + e*x]/(2*Sqrt[d])] + 6*b*Sqrt[d]*e*n*x*PolyLog[2, (1 + Sqrt[d + e*x]/S 
qrt[d])/2])/(4*x)
 

3.2.42.3 Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 251, 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.

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

\(\Big \downarrow \) 2792

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

\(\displaystyle -3 \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+b n \left (3 \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+3 \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-6 \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-3 \sqrt {d} e \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-4 e \sqrt {d+e x}-\frac {d \sqrt {d+e x}}{x}\right )\)

input

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

output

3*e*Sqrt[d + e*x]*(a + b*Log[c*x^n]) - ((d + e*x)^(3/2)*(a + b*Log[c*x^n]) 
)/x - 3*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]) + b*n* 
(-4*e*Sqrt[d + e*x] - (d*Sqrt[d + e*x])/x + 3*Sqrt[d]*e*ArcTanh[Sqrt[d + e 
*x]/Sqrt[d]] + 3*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2 - 6*Sqrt[d]*e* 
ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])] 
- 3*Sqrt[d]*e*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])
 

3.2.42.3.1 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])
 

3.2.42.4 Maple [F]

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

input

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

output

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

3.2.42.5 Fricas [F]

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

input

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

output

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

3.2.42.6 Sympy [F]

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

input

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

output

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

3.2.42.7 Maxima [F]

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

input

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

output

1/2*(3*sqrt(d)*e*log((sqrt(e*x + d) - sqrt(d))/(sqrt(e*x + d) + sqrt(d))) 
+ 4*sqrt(e*x + d)*e - 2*sqrt(e*x + d)*d/x)*a + b*integrate((e*x*log(c) + d 
*log(c) + (e*x + d)*log(x^n))*sqrt(e*x + d)/x^2, x)
 

3.2.42.8 Giac [F]

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

input

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

output

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

3.2.42.9 Mupad [F(-1)]

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

input

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

output

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