\(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^2}{\sqrt {g+h x}} \, dx\) [510]

Optimal result

Integrand size = 30, antiderivative size = 447 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt {g+h x}} \, dx=\frac {16 b^2 p^2 q^2 \sqrt {g+h x}}{h}-\frac {16 b^2 \sqrt {f g-e h} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{\sqrt {f} h}-\frac {8 b^2 \sqrt {f g-e h} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{\sqrt {f} h}-\frac {8 b p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac {8 b \sqrt {f g-e h} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {f} h}+\frac {2 \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac {16 b^2 \sqrt {f g-e h} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{\sqrt {f} h}+\frac {8 b^2 \sqrt {f g-e h} p^2 q^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{\sqrt {f} h} \] Output:

16*b^2*p^2*q^2*(h*x+g)^(1/2)/h-16*b^2*(-e*h+f*g)^(1/2)*p^2*q^2*arctanh(f^( 
1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))/f^(1/2)/h-8*b^2*(-e*h+f*g)^(1/2)*p^2* 
q^2*arctanh(f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))^2/f^(1/2)/h-8*b*p*q*(h 
*x+g)^(1/2)*(a+b*ln(c*(d*(f*x+e)^p)^q))/h+8*b*(-e*h+f*g)^(1/2)*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^(1/ 
2)/h+2*(h*x+g)^(1/2)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/h+16*b^2*(-e*h+f*g)^(1/ 
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^(1/2)/h+8*b^2*(-e*h+f*g)^(1/2)*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
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1407\) vs. \(2(447)=894\).

Time = 13.09 (sec) , antiderivative size = 1407, normalized size of antiderivative = 3.15 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt {g+h x}} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])^2/Sqrt[g + h*x],x]
 

Output:

(2*b*p*q*((4*Sqrt[f]*Sqrt[f*g - e*h]*ArcTanh[(Sqrt[f]*Sqrt[(f*g - e*h + h* 
(e + f*x))/f])/Sqrt[f*g - e*h]])/h + (2*f*Sqrt[(f*g - e*h + h*(e + f*x))/f 
]*(-2 + Log[e + f*x]))/h)*(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]) + Lo 
g[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*x)^p]))/Log[d*(e + f*x)^p])])))/f + (2* 
Sqrt[g + h*x]*(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*x)^p]))/Log[d*(e + f*x)^p])]))^2)/h + (b^2*p^2*q^2* 
((-16*f^2*g*Sqrt[e + f*x]*Sqrt[1 + (f*g - e*h)/(h*(e + f*x))]*Sqrt[(f*g - 
e*h + h*(e + f*x))/f]*ArcSin[Sqrt[-(f*g) + e*h]/(Sqrt[h]*Sqrt[e + f*x])])/ 
(Sqrt[h]*Sqrt[-(f*g) + e*h]*(f*g - e*h + h*(e + f*x))) + (16*e*f*Sqrt[h]*S 
qrt[e + f*x]*Sqrt[1 + (f*g - e*h)/(h*(e + f*x))]*Sqrt[(f*g - e*h + h*(e + 
f*x))/f]*ArcSin[Sqrt[-(f*g) + e*h]/(Sqrt[h]*Sqrt[e + f*x])])/(Sqrt[-(f*g) 
+ e*h]*(f*g - e*h + h*(e + f*x))) + (2*f*Sqrt[(f*g - e*h + h*(e + f*x))/f] 
*(8 - 4*Log[e + f*x] + Log[e + f*x]^2))/h + (2*e*f*Sqrt[(f*g - e*h + h*(e 
+ f*x))/f]*((-4*ArcTanh[Sqrt[f*g - e*h + h*(e + f*x)]/Sqrt[f*g - e*h]]*...
 

Rubi [A] (warning: unable to verify)

Time = 6.34 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.28, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2895, 2845, 2858, 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.

\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt {g+h x}} \, dx\)

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2845

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 2788

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}-\frac {4 b p q \left (\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (\left (g-\frac {e h}{f}\right ) \int \frac {1}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)+2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}\right )}{h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)\right )}{h}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}-\frac {4 b p q \left (\frac {h \left (\frac {2 f \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {2 b f p q \left (\frac {2 f \left (g-\frac {e h}{f}\right ) \int \frac {1}{e+\frac {f \left (g-\frac {e h}{f}+\frac {h (e+f x)}{f}\right )}{h}-\frac {f g}{h}}d\sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}{h}+2 \sqrt {\frac {h (e+f x)}{f}-\frac {e h}{f}+g}\right )}{h}\right )}{f}+\left (g-\frac {e h}{f}\right ) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt {g-\frac {e h}{f}+\frac {h (e+f x)}{f}}}d(e+f x)\right )}{h}\)

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 2790

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 2092

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])^2/Sqrt[g + h*x],x]
 

Output:

(2*Sqrt[g + h*x]*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/h - (4*b*p*q*((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]])/S 
qrt[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]*Sqr 
t[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
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

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]
 

rule 73

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]
 

rule 221

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]
 

rule 2092

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

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2756

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]))
 

rule 2788

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]
 

rule 2790

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.)) 
/(x_), x_Symbol] :> With[{u = IntHide[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*L 
og[c*x^n]), x] - Simp[b*n   Int[1/x   u, x], x]] /; FreeQ[{a, b, c, d, e, n 
, r}, x] && IntegerQ[q - 1/2]
 

rule 2845

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

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 2858

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

rule 2895

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]]
 

rule 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 
2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 
, 0]
 

rule 6546

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}}{\sqrt {h x +g}}d x\]

Input:

int((a+b*ln(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(1/2),x)
 

Output:

int((a+b*ln(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(1/2),x)
 

Fricas [F]

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

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(1/2),x, algorithm="frica 
s")
 

Output:

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*x + g), x)
 

Sympy [F]

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

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))**2/(h*x+g)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d*(e + f*x)**p)**q))**2/sqrt(g + h*x), x)
 

Maxima [F(-2)]

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

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(1/2),x, algorithm="maxim 
a")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e*h-f*g>0)', see `assume?` for m 
ore detail
 

Giac [F]

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

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(1/2),x, algorithm="giac" 
)
 

Output:

integrate((b*log(((f*x + e)^p*d)^q*c) + a)^2/sqrt(h*x + g), x)
 

Mupad [F(-1)]

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

int((a + b*log(c*(d*(e + f*x)^p)^q))^2/(g + h*x)^(1/2),x)
 

Output:

int((a + b*log(c*(d*(e + f*x)^p)^q))^2/(g + h*x)^(1/2), x)
 

Reduce [F]

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

int((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(1/2),x)
 

Output:

(2*(4*sqrt(f)*sqrt(e*h - f*g)*atan((sqrt(g + h*x)*f)/(sqrt(f)*sqrt(e*h - f 
*g)))*a*b*e*h*p*q - 8*sqrt(f)*sqrt(e*h - f*g)*atan((sqrt(g + h*x)*f)/(sqrt 
(f)*sqrt(e*h - f*g)))*b**2*f*g*p**2*q**2 + sqrt(g + h*x)*log(d**q*(e + f*x 
)**(p*q)*c)**2*b**2*e*f*h + 2*sqrt(g + h*x)*log(d**q*(e + f*x)**(p*q)*c)*a 
*b*e*f*h - 4*sqrt(g + h*x)*log(d**q*(e + f*x)**(p*q)*c)*b**2*f**2*g*p*q + 
sqrt(g + h*x)*a**2*e*f*h - 4*sqrt(g + h*x)*a*b*e*f*h*p*q + 8*sqrt(g + h*x) 
*b**2*f**2*g*p**2*q**2 - 2*int((sqrt(g + h*x)*log(d**q*(e + f*x)**(p*q)*c) 
*x)/(e*g + e*h*x + f*g*x + f*h*x**2),x)*b**2*e*f**2*h**2*p*q + 2*int((sqrt 
(g + h*x)*log(d**q*(e + f*x)**(p*q)*c)*x)/(e*g + e*h*x + f*g*x + f*h*x**2) 
,x)*b**2*f**3*g*h*p*q))/(e*f*h**2)