Integrand size = 30, antiderivative size = 330 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\frac {8 b^2 \sqrt {f} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {8 b \sqrt {f} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}-\frac {16 b^2 \sqrt {f} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}}-\frac {8 b^2 \sqrt {f} p^2 q^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}} \]
8*b^2*p^2*q^2*arctanh(f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))^2*f^(1/2)/h/ (-e*h+f*g)^(1/2)-8*b*p*q*arctanh(f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))*( a+b*ln(c*(d*(f*x+e)^p)^q))*f^(1/2)/h/(-e*h+f*g)^(1/2)-16*b^2*p^2*q^2*arcta nh(f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))*ln(2/(1-f^(1/2)*(h*x+g)^(1/2)/( -e*h+f*g)^(1/2)))*f^(1/2)/h/(-e*h+f*g)^(1/2)-8*b^2*p^2*q^2*polylog(2,1-2/( 1-f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2)))*f^(1/2)/h/(-e*h+f*g)^(1/2)-2*(a +b*ln(c*(d*(f*x+e)^p)^q))^2/h/(h*x+g)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(785\) vs. \(2(330)=660\).
Time = 2.54 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.38 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=-\frac {2 \left (a^2 \sqrt {f g-e h} \sqrt {f (g+h x)}+4 b^2 f g p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \log (e+f x)+4 b^2 f h p^2 q^2 x \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \log (e+f x)-4 b^2 f g p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \log \left (\frac {h (e+f x)}{-f g+e h}\right )-4 b^2 f h p^2 q^2 x \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \log \left (\frac {h (e+f x)}{-f g+e h}\right )-b^2 \sqrt {f g-e h} p^2 q^2 \sqrt {f (g+h x)} \sqrt {\frac {f (g+h x)}{f g-e h}} \log ^2\left (\frac {h (e+f x)}{-f g+e h}\right )+2 a b \sqrt {f g-e h} \sqrt {f (g+h x)} \log \left (c \left (d (e+f x)^p\right )^q\right )+b^2 \sqrt {f g-e h} \sqrt {f (g+h x)} \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+4 b \sqrt {f} p q \sqrt {g+h x} \sqrt {f (g+h x)} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+4 b^2 \sqrt {f g-e h} p^2 q^2 \sqrt {f (g+h x)} \sqrt {\frac {f (g+h x)}{f g-e h}} \log \left (\frac {h (e+f x)}{-f g+e h}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )-2 b^2 \sqrt {f g-e h} p^2 q^2 \sqrt {f (g+h x)} \sqrt {\frac {f (g+h x)}{f g-e h}} \log ^2\left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )+4 b^2 \sqrt {f g-e h} p^2 q^2 \sqrt {f (g+h x)} \sqrt {\frac {f (g+h x)}{f g-e h}} \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )}{h \sqrt {f g-e h} \sqrt {g+h x} \sqrt {f (g+h x)}} \]
(-2*(a^2*Sqrt[f*g - e*h]*Sqrt[f*(g + h*x)] + 4*b^2*f*g*p^2*q^2*ArcTanh[Sqr t[f*(g + h*x)]/Sqrt[f*g - e*h]]*Log[e + f*x] + 4*b^2*f*h*p^2*q^2*x*ArcTanh [Sqrt[f*(g + h*x)]/Sqrt[f*g - e*h]]*Log[e + f*x] - 4*b^2*f*g*p^2*q^2*ArcTa nh[Sqrt[f*(g + h*x)]/Sqrt[f*g - e*h]]*Log[(h*(e + f*x))/(-(f*g) + e*h)] - 4*b^2*f*h*p^2*q^2*x*ArcTanh[Sqrt[f*(g + h*x)]/Sqrt[f*g - e*h]]*Log[(h*(e + f*x))/(-(f*g) + e*h)] - b^2*Sqrt[f*g - e*h]*p^2*q^2*Sqrt[f*(g + h*x)]*Sqr t[(f*(g + h*x))/(f*g - e*h)]*Log[(h*(e + f*x))/(-(f*g) + e*h)]^2 + 2*a*b*S qrt[f*g - e*h]*Sqrt[f*(g + h*x)]*Log[c*(d*(e + f*x)^p)^q] + b^2*Sqrt[f*g - e*h]*Sqrt[f*(g + h*x)]*Log[c*(d*(e + f*x)^p)^q]^2 + 4*b*Sqrt[f]*p*q*Sqrt[ g + h*x]*Sqrt[f*(g + h*x)]*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h] ]*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q]) + 4*b^2*Sqrt[f*g - e*h]*p^2*q^2*Sqrt[f*(g + h*x)]*Sqrt[(f*(g + h*x))/(f*g - e*h)]*Log[(h*(e + f*x))/(-(f*g) + e*h)]*Log[(1 + Sqrt[(f*(g + h*x))/(f*g - e*h)])/2] - 2*b ^2*Sqrt[f*g - e*h]*p^2*q^2*Sqrt[f*(g + h*x)]*Sqrt[(f*(g + h*x))/(f*g - e*h )]*Log[(1 + Sqrt[(f*(g + h*x))/(f*g - e*h)])/2]^2 + 4*b^2*Sqrt[f*g - e*h]* p^2*q^2*Sqrt[f*(g + h*x)]*Sqrt[(f*(g + h*x))/(f*g - e*h)]*PolyLog[2, 1/2 - Sqrt[(f*(g + h*x))/(f*g - e*h)]/2]))/(h*Sqrt[f*g - e*h]*Sqrt[g + h*x]*Sqr t[f*(g + h*x)])
Time = 2.42 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.21, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2895, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}}dx\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {4 b f p q \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(e+f x) \sqrt {g+h x}}dx}{h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {4 b p q \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)}{h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 2790 |
| \(\displaystyle \frac {4 b p q \left (-b p q \int -\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h} (e+f x)}d(e+f x)-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 b p q \left (\frac {2 b \sqrt {f} p q \int \frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{e+f x}d(e+f x)}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {4 b p q \left (\frac {4 b f^{3/2} p q \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{e h-f \left (\frac {e h}{f}-\frac {h (e+f x)}{f}\right )}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 2092 |
| \(\displaystyle \frac {4 b p q \left (\frac {4 b f^{3/2} p q \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{-f g+e h+f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {4 b p q \left (\frac {4 b f^{3/2} p q \left (\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )^2}{2 f}-\frac {\int \frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f} \sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {4 b p q \left (\frac {4 b f^{3/2} p q \left (\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )^2}{2 f}-\frac {\frac {\sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}}\right )}{\sqrt {f}}-\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}\right )}{1-\frac {f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )}{f g-e h}}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f} \sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {4 b p q \left (\frac {4 b f^{3/2} p q \left (\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )^2}{2 f}-\frac {\frac {\sqrt {f g-e h} \int \frac {\log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}\right )}{1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}}d\frac {1}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}}{\sqrt {f}}+\frac {\sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}}\right )}{\sqrt {f}}}{\sqrt {f} \sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {4 b p q \left (\frac {4 b f^{3/2} p q \left (\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )^2}{2 f}-\frac {\frac {\sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}}\right )}{\sqrt {f}}+\frac {\sqrt {f g-e h} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}\right )}{2 \sqrt {f}}}{\sqrt {f} \sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}\) |
(-2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(h*Sqrt[g + h*x]) + (4*b*p*q*((-2* Sqrt[f]*ArcTanh[(Sqrt[f]*Sqrt[g - (e*h)/f + (h*(e + f*x))/f])/Sqrt[f*g - e *h]]*(a + b*Log[c*d^q*(e + f*x)^(p*q)]))/Sqrt[f*g - e*h] + (4*b*f^(3/2)*p* q*(ArcTanh[(Sqrt[f]*Sqrt[g - (e*h)/f + (h*(e + f*x))/f])/Sqrt[f*g - e*h]]^ 2/(2*f) - ((Sqrt[f*g - e*h]*ArcTanh[(Sqrt[f]*Sqrt[g - (e*h)/f + (h*(e + f* x))/f])/Sqrt[f*g - e*h]]*Log[2/(1 - (Sqrt[f]*Sqrt[g - (e*h)/f + (h*(e + f* x))/f])/Sqrt[f*g - e*h])])/Sqrt[f] + (Sqrt[f*g - e*h]*PolyLog[2, 1 - 2/(1 - (Sqrt[f]*Sqrt[g - (e*h)/f + (h*(e + f*x))/f])/Sqrt[f*g - e*h])])/(2*Sqrt [f]))/(Sqrt[f]*Sqrt[f*g - e*h])))/Sqrt[f*g - e*h]))/h
3.5.92.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]))
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]
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]
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
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]
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]
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]]
\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}}{\left (h x +g \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{\frac {3}{2}}} \,d x } \]
integral((sqrt(h*x + g)*b^2*log(((f*x + e)^p*d)^q*c)^2 + 2*sqrt(h*x + g)*a *b*log(((f*x + e)^p*d)^q*c) + sqrt(h*x + g)*a^2)/(h^2*x^2 + 2*g*h*x + g^2) , x)
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\left (g + h x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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*h-f*g>0)', see `assume?` for m ore detail
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{{\left (g+h\,x\right )}^{3/2}} \,d x \]