Integrand size = 30, antiderivative size = 537 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=-\frac {16 b^2 f^2 p^2 q^2}{15 h (f g-e h)^2 \sqrt {g+h x}}+\frac {64 b^2 f^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{15 h (f g-e h)^{5/2}}+\frac {8 b^2 f^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 h (f g-e h)^{5/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {8 b f^{5/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}-\frac {16 b^2 f^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 h (f g-e h)^{5/2}}-\frac {8 b^2 f^{5/2} p^2 q^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 h (f g-e h)^{5/2}} \]
64/15*b^2*f^(5/2)*p^2*q^2*arctanh(f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))/ h/(-e*h+f*g)^(5/2)+8/5*b^2*f^(5/2)*p^2*q^2*arctanh(f^(1/2)*(h*x+g)^(1/2)/( -e*h+f*g)^(1/2))^2/h/(-e*h+f*g)^(5/2)+8/15*b*f*p*q*(a+b*ln(c*(d*(f*x+e)^p) ^q))/h/(-e*h+f*g)/(h*x+g)^(3/2)-8/5*b*f^(5/2)*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))/h/(-e*h+f*g)^(5/2)-2/5 *(a+b*ln(c*(d*(f*x+e)^p)^q))^2/h/(h*x+g)^(5/2)-16/5*b^2*f^(5/2)*p^2*q^2*ar ctanh(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)))/h/(-e*h+f*g)^(5/2)-8/5*b^2*f^(5/2)*p^2*q^2*polylog(2, 1-2/(1-f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2)))/h/(-e*h+f*g)^(5/2)-16/15*b ^2*f^2*p^2*q^2/h/(-e*h+f*g)^2/(h*x+g)^(1/2)+8/5*b*f^2*p*q*(a+b*ln(c*(d*(f* x+e)^p)^q))/h/(-e*h+f*g)^2/(h*x+g)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(1349\) vs. \(2(537)=1074\).
Time = 9.70 (sec) , antiderivative size = 1349, normalized size of antiderivative = 2.51 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx =\text {Too large to display} \]
(4*a*b*f^(5/2)*p*q*((-6*ArcTanh[(Sqrt[f]*Sqrt[(f*g - e*h + h*(e + f*x))/f] )/Sqrt[f*g - e*h]])/(f*g - e*h)^(5/2) + (Sqrt[f]*Sqrt[(f*g - e*h + h*(e + f*x))/f]*(2*(f*g - e*h)*(f*g + f*h*x) + 6*(f*g + f*h*x)^2 - 3*(f*g - e*h)^ 2*Log[e + f*x]))/((f*g - e*h)^2*(f*g + f*h*x)^3)))/(15*h) + (4*b^2*f^(5/2) *p*q^2*((-6*ArcTanh[(Sqrt[f]*Sqrt[(f*g - e*h + h*(e + f*x))/f])/Sqrt[f*g - e*h]])/(f*g - e*h)^(5/2) + (Sqrt[f]*Sqrt[(f*g - e*h + h*(e + f*x))/f]*(2* (f*g - e*h)*(f*g + f*h*x) + 6*(f*g + f*h*x)^2 - 3*(f*g - e*h)^2*Log[e + f* x]))/((f*g - e*h)^2*(f*g + f*h*x)^3))*(-(p*Log[e + f*x]) + Log[d*(e + f*x) ^p]))/(15*h) + (4*b^2*f^(5/2)*p*q*((-6*ArcTanh[(Sqrt[f]*Sqrt[(f*g - e*h + h*(e + f*x))/f])/Sqrt[f*g - e*h]])/(f*g - e*h)^(5/2) + (Sqrt[f]*Sqrt[(f*g - e*h + h*(e + f*x))/f]*(2*(f*g - e*h)*(f*g + f*h*x) + 6*(f*g + f*h*x)^2 - 3*(f*g - e*h)^2*Log[e + f*x]))/((f*g - e*h)^2*(f*g + f*h*x)^3))*(-(q*(-(p *Log[e + f*x]) + Log[d*(e + f*x)^p])) - Log[d*(e + f*x)^p]*(q - (q*(-(p*Lo g[e + f*x]) + Log[d*(e + f*x)^p]))/Log[d*(e + f*x)^p]) + Log[c*E^(q*(-(p*L og[e + f*x]) + Log[d*(e + f*x)^p]))*(d*(e + f*x)^p)^(q - (q*(-(p*Log[e + f *x]) + Log[d*(e + f*x)^p]))/Log[d*(e + f*x)^p])]))/(15*h) - (2*(a + b*q*(- (p*Log[e + f*x]) + Log[d*(e + f*x)^p]) + b*(-(q*(-(p*Log[e + f*x]) + Log[d *(e + f*x)^p])) - Log[d*(e + f*x)^p]*(q - (q*(-(p*Log[e + f*x]) + Log[d*(e + f*x)^p]))/Log[d*(e + f*x)^p]) + Log[c*E^(q*(-(p*Log[e + f*x]) + Log[d*( e + f*x)^p]))*(d*(e + f*x)^p)^(q - (q*(-(p*Log[e + f*x]) + Log[d*(e + f...
Time = 4.76 (sec) , antiderivative size = 732, normalized size of antiderivative = 1.36, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2895, 2845, 2858, 2789, 2756, 61, 73, 221, 2789, 2756, 73, 221, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}}dx\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {4 b f p q \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(e+f x) (g+h x)^{5/2}}dx}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {4 b p q \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{5/2}}d(e+f x)}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2}}d(e+f x)}{f g-e h}-\frac {h \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{5/2}}d(e+f x)}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2}}d(e+f x)}{f g-e h}-\frac {h \left (\frac {2 b f p q \int \frac {1}{(e+f x) \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2}}d(e+f x)}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2}}d(e+f x)}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {f \int \frac {1}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)}{f g-e h}+\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2}}d(e+f x)}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f^2 \int \frac {1}{e+\frac {f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )}{h}-\frac {f g}{h}}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{h (f g-e h)}+\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2}}d(e+f x)}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)}{f g-e h}-\frac {h \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2}}d(e+f x)}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)}{f g-e h}-\frac {h \left (\frac {2 b f p q \int \frac {1}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)}{h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}\right )}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)}{f g-e h}-\frac {h \left (\frac {4 b f^2 p q \int \frac {1}{e+\frac {f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )}{h}-\frac {f g}{h}}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{h^2}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}\right )}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)}{f g-e h}-\frac {h \left (-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {4 b f^{3/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}\right )}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 2790 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \left (-b p q \int -\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h} (e+f x)}d(e+f x)-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{f g-e h}-\frac {h \left (-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {4 b f^{3/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}\right )}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \left (\frac {2 b \sqrt {f} p q \int \frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{e+f x}d(e+f x)}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{f g-e h}-\frac {h \left (-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {4 b f^{3/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}\right )}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \left (\frac {4 b f^{3/2} p q \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{e h-f \left (\frac {e h}{f}-\frac {h (e+f x)}{f}\right )}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{f g-e h}-\frac {h \left (-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {4 b f^{3/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}\right )}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 2092 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \left (\frac {4 b f^{3/2} p q \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{-f g+e h+f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{f g-e h}-\frac {h \left (-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {4 b f^{3/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}\right )}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \left (\frac {4 b f^{3/2} p q \left (\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )^2}{2 f}-\frac {\int \frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f} \sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{f g-e h}-\frac {h \left (-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {4 b f^{3/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}\right )}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \left (\frac {4 b f^{3/2} p q \left (\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )^2}{2 f}-\frac {\frac {\sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}}\right )}{\sqrt {f}}-\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}\right )}{1-\frac {f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )}{f g-e h}}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f} \sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{f g-e h}-\frac {h \left (-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {4 b f^{3/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}\right )}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \left (\frac {4 b f^{3/2} p q \left (\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )^2}{2 f}-\frac {\frac {\sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}\right )}{\sqrt {f}}+\frac {\sqrt {f g-e h} \int \frac {\log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}\right )}{1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}}d\frac {1}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}}{\sqrt {f}}}{\sqrt {f} \sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{f g-e h}-\frac {h \left (-\frac {4 b p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right ) f^{3/2}}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) f}{h \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}\right )}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {4 b p q \left (\frac {f \left (\frac {f \left (\frac {4 b f^{3/2} p q \left (\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )^2}{2 f}-\frac {\frac {\sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}}\right )}{\sqrt {f}}+\frac {\sqrt {f g-e h} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}\right )}{2 \sqrt {f}}}{\sqrt {f} \sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )}{f g-e h}-\frac {h \left (-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {4 b f^{3/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}\right )}{f g-e h}\right )}{f g-e h}-\frac {h \left (\frac {2 b f p q \left (\frac {2 f}{(f g-e h) \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}-\frac {2 f^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{(f g-e h)^{3/2}}\right )}{3 h}-\frac {2 f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}}\right )}{f g-e h}\right )}{5 h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}\) |
(-2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(5*h*(g + h*x)^(5/2)) + (4*b*p*q*( -((h*((2*b*f*p*q*((2*f)/((f*g - e*h)*Sqrt[g - (e*h)/f + (h*(e + f*x))/f]) - (2*f^(3/2)*ArcTanh[(Sqrt[f]*Sqrt[g - (e*h)/f + (h*(e + f*x))/f])/Sqrt[f* g - e*h]])/(f*g - e*h)^(3/2)))/(3*h) - (2*f*(a + b*Log[c*d^q*(e + f*x)^(p* q)]))/(3*h*(g - (e*h)/f + (h*(e + f*x))/f)^(3/2))))/(f*g - e*h)) + (f*(-(( 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]])/(h*Sqrt[f*g - e*h]) - (2*f*(a + b*Log[c*d^q*(e + f*x)^( p*q)]))/(h*Sqrt[g - (e*h)/f + (h*(e + f*x))/f])))/(f*g - e*h)) + (f*((-2*S qrt[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]))/(f*g - e*h)))/(f*g - e* h)))/(5*h)
3.5.94.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
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]]
\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}}{\left (h x +g \right )^{\frac {7}{2}}}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{\frac {7}{2}}} \,d x } \]
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^4*x^4 + 4*g*h^3*x^3 + 6*g^2*h^2*x^2 + 4*g^3*h*x + g^4), x)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{{\left (g+h\,x\right )}^{7/2}} \,d x \]