\(\int \frac {(a+b \log (c (d+e x)^n))^2}{(f+g x)^{3/2}} \, dx\) [148]

Optimal result

Integrand size = 26, antiderivative size = 312 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{3/2}} \, dx=\frac {8 b^2 \sqrt {e} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{g \sqrt {e f-d g}}-\frac {8 b \sqrt {e} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {e f-d g}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g \sqrt {f+g x}}-\frac {16 b^2 \sqrt {e} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{g \sqrt {e f-d g}}-\frac {8 b^2 \sqrt {e} n^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{g \sqrt {e f-d g}} \] Output:

8*b^2*e^(1/2)*n^2*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))^2/g/(-d* 
g+e*f)^(1/2)-8*b*e^(1/2)*n*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2)) 
*(a+b*ln(c*(e*x+d)^n))/g/(-d*g+e*f)^(1/2)-2*(a+b*ln(c*(e*x+d)^n))^2/g/(g*x 
+f)^(1/2)-16*b^2*e^(1/2)*n^2*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2 
))*ln(2/(1-e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2)))/g/(-d*g+e*f)^(1/2)-8*b 
^2*e^(1/2)*n^2*polylog(2,1-2/(1-e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2)))/g 
/(-d*g+e*f)^(1/2)
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{3/2}} \, dx=\frac {2 \left (-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {f+g x}}+\frac {b \sqrt {e} n \left (2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )-2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )-b n \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+b n \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )\right )}{\sqrt {e f-d g}}\right )}{g} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.40 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.25, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2845, 2858, 2790, 27, 7267, 2092, 6546, 6470, 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.

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

\(\Big \downarrow \) 2845

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 2790

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7267

\(\displaystyle \frac {4 b n \left (\frac {4 b e^{3/2} n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{d g-e \left (\frac {d g}{e}-\frac {g (d+e x)}{e}\right )}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g \sqrt {f+g x}}\)

\(\Big \downarrow \) 2092

\(\displaystyle \frac {4 b n \left (\frac {4 b e^{3/2} n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{-e f+d g+e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g \sqrt {f+g x}}\)

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

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

Defintions of rubi rules used

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 2092

Int[(Px_)*(u_)^(p_.)*(z_)^(q_.), x_Symbol] :> Int[Px*ExpandToSum[z, x]^q*Ex 
pandToSum[u, x]^p, x] /; FreeQ[{p, q}, x] && BinomialQ[z, x] && BinomialQ[u 
, x] &&  !(BinomialMatchQ[z, x] && BinomialMatchQ[u, 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 2845

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. 
)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ 
n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1)))   Int[(f + g*x)^(q + 1) 
*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, 
d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In 
tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
 

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 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

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]]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((a+b*log(c*(e*x+d)^n))^2/(g*x+f)^(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*(d*g-e*f)>0)', see `assume?` f 
or more de
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{3/2}} \, dx=\frac {8 \sqrt {e}\, \sqrt {g x +f}\, \sqrt {d g -e f}\, \mathit {atan} \left (\frac {\sqrt {g x +f}\, e}{\sqrt {e}\, \sqrt {d g -e f}}\right ) a b d g n +16 \sqrt {e}\, \sqrt {g x +f}\, \sqrt {d g -e f}\, \mathit {atan} \left (\frac {\sqrt {g x +f}\, e}{\sqrt {e}\, \sqrt {d g -e f}}\right ) b^{2} e f \,n^{2}+4 \sqrt {g x +f}\, \left (\int \frac {\sqrt {g x +f}\, \mathrm {log}\left (\left (e x +d \right )^{n} c \right ) x}{e \,g^{2} x^{3}+d \,g^{2} x^{2}+2 e f g \,x^{2}+2 d f g x +e \,f^{2} x +d \,f^{2}}d x \right ) b^{2} d^{2} e \,g^{3} n -8 \sqrt {g x +f}\, \left (\int \frac {\sqrt {g x +f}\, \mathrm {log}\left (\left (e x +d \right )^{n} c \right ) x}{e \,g^{2} x^{3}+d \,g^{2} x^{2}+2 e f g \,x^{2}+2 d f g x +e \,f^{2} x +d \,f^{2}}d x \right ) b^{2} d \,e^{2} f \,g^{2} n +4 \sqrt {g x +f}\, \left (\int \frac {\sqrt {g x +f}\, \mathrm {log}\left (\left (e x +d \right )^{n} c \right ) x}{e \,g^{2} x^{3}+d \,g^{2} x^{2}+2 e f g \,x^{2}+2 d f g x +e \,f^{2} x +d \,f^{2}}d x \right ) b^{2} e^{3} f^{2} g n -2 \mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} b^{2} d^{2} g^{2}+2 \mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} b^{2} d e f g -4 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) a b \,d^{2} g^{2}+4 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) a b d e f g -8 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b^{2} d e f g n +8 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b^{2} e^{2} f^{2} n -2 a^{2} d^{2} g^{2}+2 a^{2} d e f g}{\sqrt {g x +f}\, d \,g^{2} \left (d g -e f \right )} \] Input:

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

Output:

(2*(4*sqrt(e)*sqrt(f + g*x)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e 
)*sqrt(d*g - e*f)))*a*b*d*g*n + 8*sqrt(e)*sqrt(f + g*x)*sqrt(d*g - e*f)*at 
an((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*b**2*e*f*n**2 + 2*sqrt(f + 
 g*x)*int((sqrt(f + g*x)*log((d + e*x)**n*c)*x)/(d*f**2 + 2*d*f*g*x + d*g* 
*2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*b**2*d**2*e*g**3*n - 4 
*sqrt(f + g*x)*int((sqrt(f + g*x)*log((d + e*x)**n*c)*x)/(d*f**2 + 2*d*f*g 
*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*b**2*d*e**2*f 
*g**2*n + 2*sqrt(f + g*x)*int((sqrt(f + g*x)*log((d + e*x)**n*c)*x)/(d*f** 
2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*b* 
*2*e**3*f**2*g*n - log((d + e*x)**n*c)**2*b**2*d**2*g**2 + log((d + e*x)** 
n*c)**2*b**2*d*e*f*g - 2*log((d + e*x)**n*c)*a*b*d**2*g**2 + 2*log((d + e* 
x)**n*c)*a*b*d*e*f*g - 4*log((d + e*x)**n*c)*b**2*d*e*f*g*n + 4*log((d + e 
*x)**n*c)*b**2*e**2*f**2*n - a**2*d**2*g**2 + a**2*d*e*f*g))/(sqrt(f + g*x 
)*d*g**2*(d*g - e*f))