Integrand size = 30, antiderivative size = 547 \[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {64 b^2 (f g-e h) p^2 q^2 \sqrt {g+h x}}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h}-\frac {64 b^2 (f g-e h)^{3/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{9 f^{3/2} h}-\frac {8 b^2 (f g-e h)^{3/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 f^{3/2} h}-\frac {8 b (f g-e h) p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac {8 b (f g-e h)^{3/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\frac {16 b^2 (f g-e h)^{3/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 f^{3/2} h}+\frac {8 b^2 (f g-e h)^{3/2} p^2 q^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 f^{3/2} h} \]
16/27*b^2*p^2*q^2*(h*x+g)^(3/2)/h-64/9*b^2*(-e*h+f*g)^(3/2)*p^2*q^2*arctan h(f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))/f^(3/2)/h-8/3*b^2*(-e*h+f*g)^(3/ 2)*p^2*q^2*arctanh(f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))^2/f^(3/2)/h-8/9 *b*p*q*(h*x+g)^(3/2)*(a+b*ln(c*(d*(f*x+e)^p)^q))/h+8/3*b*(-e*h+f*g)^(3/2)* 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))/f^(3/2)/h+2/3*(h*x+g)^(3/2)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/h+16/3*b^2 *(-e*h+f*g)^(3/2)*p^2*q^2*arctanh(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^(3/2)/h+8/3*b^2*(-e*h+f *g)^(3/2)*p^2*q^2*polylog(2,1-2/(1-f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2)) )/f^(3/2)/h+64/9*b^2*(-e*h+f*g)*p^2*q^2*(h*x+g)^(1/2)/f/h-8/3*b*(-e*h+f*g) *p*q*(a+b*ln(c*(d*(f*x+e)^p)^q))*(h*x+g)^(1/2)/f/h
Leaf count is larger than twice the leaf count of optimal. \(1323\) vs. \(2(547)=1094\).
Time = 11.77 (sec) , antiderivative size = 1323, normalized size of antiderivative = 2.42 \[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {2 \left (-\frac {6 b p q \left (6 (f g-e h)^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )+\sqrt {f} \sqrt {g+h x} (6 e h-2 f (4 g+h x)+3 f (g+h x) \log (e+f x))\right ) \left (-a+b p q \log (e+f x)-b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^{3/2}}+9 (g+h x)^{3/2} \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-\frac {b^2 p^2 q^2 \left (96 f^2 g^2 h^{3/2} \sqrt {f g-e h} (e+f x)^{3/2} \left (\frac {f (g+h x)}{h (e+f x)}\right )^{3/2} \arcsin \left (\frac {\sqrt {-f g+e h}}{\sqrt {h} \sqrt {e+f x}}\right )-192 e f g h^{5/2} \sqrt {f g-e h} (e+f x)^{3/2} \left (\frac {f (g+h x)}{h (e+f x)}\right )^{3/2} \arcsin \left (\frac {\sqrt {-f g+e h}}{\sqrt {h} \sqrt {e+f x}}\right )+96 e^2 h^{7/2} \sqrt {f g-e h} (e+f x)^{3/2} \left (\frac {f (g+h x)}{h (e+f x)}\right )^{3/2} \arcsin \left (\frac {\sqrt {-f g+e h}}{\sqrt {h} \sqrt {e+f x}}\right )-f^2 \sqrt {-(f g-e h)^2} (g+h x)^2 \left (8 (13 f g-12 e h+f h x)-12 (4 f g-3 e h+f h x) \log (e+f x)+9 f (g+h x) \log ^2(e+f x)\right )-9 f^3 g^2 \sqrt {-f g+e h} (g+h x) \left (4 \sqrt {f (g+h x)} \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \left (\log (e+f x)-\log \left (\frac {h (e+f x)}{-f g+e h}\right )\right )-\sqrt {f g-e h} \sqrt {\frac {f (g+h x)}{f g-e h}} \left (\log ^2\left (\frac {h (e+f x)}{-f g+e h}\right )-4 \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 \log ^2\left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )\right )+18 e f^2 g h \sqrt {-f g+e h} (g+h x) \left (4 \sqrt {f (g+h x)} \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \left (\log (e+f x)-\log \left (\frac {h (e+f x)}{-f g+e h}\right )\right )-\sqrt {f g-e h} \sqrt {\frac {f (g+h x)}{f g-e h}} \left (\log ^2\left (\frac {h (e+f x)}{-f g+e h}\right )-4 \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 \log ^2\left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )\right )-9 e^2 f h^2 \sqrt {-f g+e h} (g+h x) \left (4 \sqrt {f (g+h x)} \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \left (\log (e+f x)-\log \left (\frac {h (e+f x)}{-f g+e h}\right )\right )-\sqrt {f g-e h} \sqrt {\frac {f (g+h x)}{f g-e h}} \left (\log ^2\left (\frac {h (e+f x)}{-f g+e h}\right )-4 \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 \log ^2\left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )\right )\right )}{f^3 \sqrt {-(f g-e h)^2} (g+h x)^{3/2}}\right )}{27 h} \]
(2*((-6*b*p*q*(6*(f*g - e*h)^(3/2)*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f* g - e*h]] + Sqrt[f]*Sqrt[g + h*x]*(6*e*h - 2*f*(4*g + h*x) + 3*f*(g + h*x) *Log[e + f*x]))*(-a + b*p*q*Log[e + f*x] - b*Log[c*(d*(e + f*x)^p)^q]))/f^ (3/2) + 9*(g + h*x)^(3/2)*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p )^q])^2 - (b^2*p^2*q^2*(96*f^2*g^2*h^(3/2)*Sqrt[f*g - e*h]*(e + f*x)^(3/2) *((f*(g + h*x))/(h*(e + f*x)))^(3/2)*ArcSin[Sqrt[-(f*g) + e*h]/(Sqrt[h]*Sq rt[e + f*x])] - 192*e*f*g*h^(5/2)*Sqrt[f*g - e*h]*(e + f*x)^(3/2)*((f*(g + h*x))/(h*(e + f*x)))^(3/2)*ArcSin[Sqrt[-(f*g) + e*h]/(Sqrt[h]*Sqrt[e + f* x])] + 96*e^2*h^(7/2)*Sqrt[f*g - e*h]*(e + f*x)^(3/2)*((f*(g + h*x))/(h*(e + f*x)))^(3/2)*ArcSin[Sqrt[-(f*g) + e*h]/(Sqrt[h]*Sqrt[e + f*x])] - f^2*S qrt[-(f*g - e*h)^2]*(g + h*x)^2*(8*(13*f*g - 12*e*h + f*h*x) - 12*(4*f*g - 3*e*h + f*h*x)*Log[e + f*x] + 9*f*(g + h*x)*Log[e + f*x]^2) - 9*f^3*g^2*S qrt[-(f*g) + e*h]*(g + h*x)*(4*Sqrt[f*(g + h*x)]*ArcTanh[Sqrt[f*(g + h*x)] /Sqrt[f*g - e*h]]*(Log[e + f*x] - Log[(h*(e + f*x))/(-(f*g) + e*h)]) - Sqr t[f*g - e*h]*Sqrt[(f*(g + h*x))/(f*g - e*h)]*(Log[(h*(e + f*x))/(-(f*g) + e*h)]^2 - 4*Log[(h*(e + f*x))/(-(f*g) + e*h)]*Log[(1 + Sqrt[(f*(g + h*x))/ (f*g - e*h)])/2] + 2*Log[(1 + Sqrt[(f*(g + h*x))/(f*g - e*h)])/2]^2 - 4*Po lyLog[2, 1/2 - Sqrt[(f*(g + h*x))/(f*g - e*h)]/2])) + 18*e*f^2*g*h*Sqrt[-( f*g) + e*h]*(g + h*x)*(4*Sqrt[f*(g + h*x)]*ArcTanh[Sqrt[f*(g + h*x)]/Sqrt[ f*g - e*h]]*(Log[e + f*x] - Log[(h*(e + f*x))/(-(f*g) + e*h)]) - Sqrt[f...
Time = 4.85 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.44, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {2895, 2845, 2858, 2788, 2756, 60, 60, 73, 221, 2788, 2756, 60, 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.
| \(\displaystyle \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2dx\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b f p q \int \frac {(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{e+f x}dx}{3 h}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \int \frac {\left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x}d(e+f x)}{3 h}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\frac {h \int \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )d(e+f x)}{f}+\left (g-\frac {e h}{f}\right ) \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x}d(e+f x)\right )}{3 h}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \int \frac {\left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2}}{e+f x}d(e+f x)}{3 h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x}d(e+f x)\right )}{3 h}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{e+f x}d(e+f x)+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x}d(e+f x)\right )}{3 h}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (\left (g-\frac {e h}{f}\right ) \int \frac {1}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)+2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x}d(e+f x)\right )}{3 h}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (\frac {2 f \left (g-\frac {e h}{f}\right ) \int \frac {1}{e+\frac {f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )}{h}-\frac {f g}{h}}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{h}+2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x}d(e+f x)\right )}{3 h}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x}d(e+f x)+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \left (\frac {h \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)}{f}+\left (g-\frac {e h}{f}\right ) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)\right )+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \left (\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{e+f x}d(e+f x)}{h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)\right )+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \left (\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \int \frac {1}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)+2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}\right )}{h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)\right )+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \left (\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (\frac {2 f \left (g-\frac {e h}{f}\right ) \int \frac {1}{e+\frac {f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )}{h}-\frac {f g}{h}}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{h}+2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}\right )}{h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)\right )+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \left (\left (g-\frac {e h}{f}\right ) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)+\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )}{h}\right )}{f}\right )+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
| \(\Big \downarrow \) 2790 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \left (\left (g-\frac {e h}{f}\right ) \left (-b p q \int -\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h} (e+f x)}d(e+f x)-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )+\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )}{h}\right )}{f}\right )+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \left (\left (g-\frac {e h}{f}\right ) \left (\frac {2 b \sqrt {f} p q \int \frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{e+f x}d(e+f x)}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )+\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )}{h}\right )}{f}\right )+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \left (\left (g-\frac {e h}{f}\right ) \left (\frac {4 b f^{3/2} p q \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{e h-f \left (\frac {e h}{f}-\frac {h (e+f x)}{f}\right )}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )+\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )}{h}\right )}{f}\right )+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
| \(\Big \downarrow \) 2092 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \left (\left (g-\frac {e h}{f}\right ) \left (\frac {4 b f^{3/2} p q \int \frac {\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{-f g+e h+f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )+\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )}{h}\right )}{f}\right )+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \left (\left (g-\frac {e h}{f}\right ) \left (\frac {4 b f^{3/2} p q \left (\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )^2}{2 f}-\frac {\int \frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{1-\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f g-e h}}}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{\sqrt {f} \sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}-\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{\sqrt {f g-e h}}\right )+\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )}{h}\right )}{f}\right )+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\frac {h \left (\frac {2 f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\frac {2}{3} \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2}+\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \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}}\right )\right )}{3 h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \left (\frac {h \left (\frac {2 f \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (2 \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \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}}\right )}{h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \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}}-\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 {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 )\right )\right )}{3 h}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\frac {h \left (\frac {2 f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\frac {2}{3} \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )^{3/2}+\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \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}}\right )\right )}{3 h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \left (\frac {h \left (\frac {2 f \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (2 \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \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}}\right )}{h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \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 )\right )\right )}{3 h}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {4 b p q \left (\left (g-\frac {e h}{f}\right ) \left (\left (g-\frac {e h}{f}\right ) \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 )+\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )}{h}\right )}{f}\right )+\frac {h \left (\frac {2 f \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{3 h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \left (2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}-\frac {2 \sqrt {f} \left (g-\frac {e h}{f}\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}\right )+\frac {2}{3} \left (\frac {h (e+f x)}{f}-\frac {e h}{f}+g\right )^{3/2}\right )}{3 h}\right )}{f}\right )}{3 h}\) |
(2*(g + h*x)^(3/2)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(3*h) - (4*b*p*q*(( h*((-2*b*f*p*q*((2*(g - (e*h)/f + (h*(e + f*x))/f)^(3/2))/3 + (g - (e*h)/f )*(2*Sqrt[g - (e*h)/f + (h*(e + f*x))/f] - (2*Sqrt[f]*(g - (e*h)/f)*ArcTan h[(Sqrt[f]*Sqrt[g - (e*h)/f + (h*(e + f*x))/f])/Sqrt[f*g - e*h]])/Sqrt[f*g - e*h])))/(3*h) + (2*f*(g - (e*h)/f + (h*(e + f*x))/f)^(3/2)*(a + b*Log[c *d^q*(e + f*x)^(p*q)]))/(3*h)))/f + (g - (e*h)/f)*((h*((-2*b*f*p*q*(2*Sqrt [g - (e*h)/f + (h*(e + f*x))/f] - (2*Sqrt[f]*(g - (e*h)/f)*ArcTanh[(Sqrt[f ]*Sqrt[g - (e*h)/f + (h*(e + f*x))/f])/Sqrt[f*g - e*h]])/Sqrt[f*g - e*h])) /h + (2*f*Sqrt[g - (e*h)/f + (h*(e + f*x))/f]*(a + b*Log[c*d^q*(e + f*x)^( p*q)]))/h))/f + (g - (e*h)/f)*((-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[(Sq rt[f]*Sqrt[g - (e*h)/f + (h*(e + f*x))/f])/Sqrt[f*g - e*h]]*Log[2/(1 - (Sq rt[f]*Sqrt[g - (e*h)/f + (h*(e + f*x))/f])/Sqrt[f*g - e*h])])/Sqrt[f] + (S qrt[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]))))/(3*h)
3.5.90.3.1 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/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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[d Int[(d + e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x) , x], x] + Simp[e Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /; F reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && 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]]
\[\int \sqrt {h x +g}\, {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}d x\]
\[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\int { \sqrt {h x + g} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2} \,d x } \]
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, x)
\[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\int \left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2} \sqrt {g + h x}\, dx \]
Exception generated. \[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\int { \sqrt {h x + g} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\int \sqrt {g+h\,x}\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2 \,d x \]