Integrand size = 24, antiderivative size = 1221 \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx =\text {Too large to display} \]
-12/7*I*d^(7/2)*f*g^2*p^2*arctan(x*e^(1/2)/d^(1/2))^2/e^(7/2)-12/7*I*d^(7/ 2)*f*g^2*p^2*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))/e^(7/2)+d^4*g^3* p^2*x^2/e^4-d^4*g^3*p*(e*x^2+d)*ln(c*(e*x^2+d)^p)/e^5+d^3*g^3*p*(e*x^2+d)^ 2*ln(c*(e*x^2+d)^p)/e^5-4/7*d^2*f*g^2*p*x^3*ln(c*(e*x^2+d)^p)/e^2+12/35*d* f*g^2*p*x^5*ln(c*(e*x^2+d)^p)/e+3*d*f^2*g*p*(e*x^2+d)*ln(c*(e*x^2+d)^p)/e^ 2-12/7*d^(7/2)*f*g^2*p*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)/e^(7/2) -24/7*d^(7/2)*f*g^2*p^2*arctan(x*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I* x*e^(1/2)))/e^(7/2)+24/343*f*g^2*p^2*x^7+1/125*g^3*p^2*(e*x^2+d)^5/e^5-4*f ^3*p*x*ln(c*(e*x^2+d)^p)+3/7*f*g^2*x^7*ln(c*(e*x^2+d)^p)^2+8*f^3*p^2*x+1/1 0*g^3*x^10*ln(c*(e*x^2+d)^p)^2+f^3*x*ln(c*(e*x^2+d)^p)^2+12/7*d^3*f*g^2*p* x*ln(c*(e*x^2+d)^p)/e^3-1/10*d^5*g^3*p^2*ln(e*x^2+d)^2/e^5-12/49*f*g^2*p*x ^7*ln(c*(e*x^2+d)^p)-1/25*g^3*p*(e*x^2+d)^5*ln(c*(e*x^2+d)^p)/e^5+3/4*f^2* g*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)^2/e^2-8*f^3*p^2*arctan(x*e^(1/2)/d^(1/2))* d^(1/2)/e^(1/2)+3/8*f^2*g*p^2*(e*x^2+d)^2/e^2-1/2*d^3*g^3*p^2*(e*x^2+d)^2/ e^5+2/9*d^2*g^3*p^2*(e*x^2+d)^3/e^5-1/16*d*g^3*p^2*(e*x^2+d)^4/e^5+4*I*f^3 *p^2*arctan(x*e^(1/2)/d^(1/2))^2*d^(1/2)/e^(1/2)+4*I*f^3*p^2*polylog(2,1-2 *d^(1/2)/(d^(1/2)+I*x*e^(1/2)))*d^(1/2)/e^(1/2)-1408/245*d^3*f*g^2*p^2*x/e ^3-3*d*f^2*g*p^2*x^2/e+568/735*d^2*f*g^2*p^2*x^3/e^2-288/1225*d*f*g^2*p^2* x^5/e+1408/245*d^(7/2)*f*g^2*p^2*arctan(x*e^(1/2)/d^(1/2))/e^(7/2)-3/4*f^2 *g*p*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)/e^2-2/3*d^2*g^3*p*(e*x^2+d)^3*ln(c*(...
Time = 0.76 (sec) , antiderivative size = 780, normalized size of antiderivative = 0.64 \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{4} f^2 g x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3 f^2 g \left (e p^2 x^2 \left (-6 d+e x^2\right )+2 d^2 p^2 \log \left (d+e x^2\right )+2 p \left (2 d^2+2 d e x^2-e^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )-2 d^2 \log ^2\left (c \left (d+e x^2\right )^p\right )\right )}{8 e^2}+\frac {g^3 \left (e p^2 x^2 \left (8220 d^4-2310 d^3 e x^2+940 d^2 e^2 x^4-405 d e^3 x^6+144 e^4 x^8\right )-4620 d^5 p^2 \log \left (d+e x^2\right )-60 p \left (60 d^5+60 d^4 e x^2-30 d^3 e^2 x^4+20 d^2 e^3 x^6-15 d e^4 x^8+12 e^5 x^{10}\right ) \log \left (c \left (d+e x^2\right )^p\right )+1800 d^5 \log ^2\left (c \left (d+e x^2\right )^p\right )\right )}{18000 e^5}+\frac {4 f^3 p \left (i \sqrt {d} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2+\sqrt {e} x \left (2 p-\log \left (c \left (d+e x^2\right )^p\right )\right )+\sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-2 p+2 p \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )+i \sqrt {d} p \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )\right )}{\sqrt {e}}+\frac {4 f g^2 p \left (-11025 i d^{7/2} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2-105 d^{7/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-352 p+210 p \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )+105 \log \left (c \left (d+e x^2\right )^p\right )\right )+\sqrt {e} x \left (2 p \left (-18480 d^3+2485 d^2 e x^2-756 d e^2 x^4+225 e^3 x^6\right )+105 \left (105 d^3-35 d^2 e x^2+21 d e^2 x^4-15 e^3 x^6\right ) \log \left (c \left (d+e x^2\right )^p\right )\right )-11025 i d^{7/2} p \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )\right )}{25725 e^{7/2}} \]
f^3*x*Log[c*(d + e*x^2)^p]^2 + (3*f^2*g*x^4*Log[c*(d + e*x^2)^p]^2)/4 + (3 *f*g^2*x^7*Log[c*(d + e*x^2)^p]^2)/7 + (g^3*x^10*Log[c*(d + e*x^2)^p]^2)/1 0 + (3*f^2*g*(e*p^2*x^2*(-6*d + e*x^2) + 2*d^2*p^2*Log[d + e*x^2] + 2*p*(2 *d^2 + 2*d*e*x^2 - e^2*x^4)*Log[c*(d + e*x^2)^p] - 2*d^2*Log[c*(d + e*x^2) ^p]^2))/(8*e^2) + (g^3*(e*p^2*x^2*(8220*d^4 - 2310*d^3*e*x^2 + 940*d^2*e^2 *x^4 - 405*d*e^3*x^6 + 144*e^4*x^8) - 4620*d^5*p^2*Log[d + e*x^2] - 60*p*( 60*d^5 + 60*d^4*e*x^2 - 30*d^3*e^2*x^4 + 20*d^2*e^3*x^6 - 15*d*e^4*x^8 + 1 2*e^5*x^10)*Log[c*(d + e*x^2)^p] + 1800*d^5*Log[c*(d + e*x^2)^p]^2))/(1800 0*e^5) + (4*f^3*p*(I*Sqrt[d]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[e]*x*( 2*p - Log[c*(d + e*x^2)^p]) + Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-2*p + 2*p*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + Log[c*(d + e*x^2)^p]) + I*S qrt[d]*p*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)]))/ Sqrt[e] + (4*f*g^2*p*((-11025*I)*d^(7/2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 - 105*d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-352*p + 210*p*Log[(2*Sqrt[d])/( Sqrt[d] + I*Sqrt[e]*x)] + 105*Log[c*(d + e*x^2)^p]) + Sqrt[e]*x*(2*p*(-184 80*d^3 + 2485*d^2*e*x^2 - 756*d*e^2*x^4 + 225*e^3*x^6) + 105*(105*d^3 - 35 *d^2*e*x^2 + 21*d*e^2*x^4 - 15*e^3*x^6)*Log[c*(d + e*x^2)^p]) - (11025*I)* d^(7/2)*p*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)])) /(25725*e^(7/2))
Time = 1.86 (sec) , antiderivative size = 1221, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2921, 2009}
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.
| \(\displaystyle \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx\) |
| \(\Big \downarrow \) 2921 |
| \(\displaystyle \int \left (f^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+3 f^2 g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+3 f g^2 x^6 \log ^2\left (c \left (d+e x^2\right )^p\right )+g^3 x^9 \log ^2\left (c \left (d+e x^2\right )^p\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{10} g^3 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^{10}+\frac {24}{343} f g^2 p^2 x^7+\frac {3}{7} f g^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^7-\frac {12}{49} f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^7-\frac {288 d f g^2 p^2 x^5}{1225 e}+\frac {12 d f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^5}{35 e}+\frac {568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac {4 d^2 f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^3}{7 e^2}+\frac {d^4 g^3 p^2 x^2}{e^4}-\frac {3 d f^2 g p^2 x^2}{e}+8 f^3 p^2 x-\frac {1408 d^3 f g^2 p^2 x}{245 e^3}+f^3 \log ^2\left (c \left (e x^2+d\right )^p\right ) x-4 f^3 p \log \left (c \left (e x^2+d\right )^p\right ) x+\frac {12 d^3 f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x}{7 e^3}+\frac {g^3 p^2 \left (e x^2+d\right )^5}{125 e^5}-\frac {d g^3 p^2 \left (e x^2+d\right )^4}{16 e^5}+\frac {2 d^2 g^3 p^2 \left (e x^2+d\right )^3}{9 e^5}-\frac {d^3 g^3 p^2 \left (e x^2+d\right )^2}{2 e^5}+\frac {3 f^2 g p^2 \left (e x^2+d\right )^2}{8 e^2}+\frac {4 i \sqrt {d} f^3 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}-\frac {d^5 g^3 p^2 \log ^2\left (e x^2+d\right )}{10 e^5}+\frac {3 f^2 g \left (e x^2+d\right )^2 \log ^2\left (c \left (e x^2+d\right )^p\right )}{4 e^2}-\frac {3 d f^2 g \left (e x^2+d\right ) \log ^2\left (c \left (e x^2+d\right )^p\right )}{2 e^2}-\frac {8 \sqrt {d} f^3 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} f g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{245 e^{7/2}}+\frac {8 \sqrt {d} f^3 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {24 d^{7/2} f g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{7 e^{7/2}}-\frac {g^3 p \left (e x^2+d\right )^5 \log \left (c \left (e x^2+d\right )^p\right )}{25 e^5}+\frac {d g^3 p \left (e x^2+d\right )^4 \log \left (c \left (e x^2+d\right )^p\right )}{4 e^5}-\frac {2 d^2 g^3 p \left (e x^2+d\right )^3 \log \left (c \left (e x^2+d\right )^p\right )}{3 e^5}+\frac {d^3 g^3 p \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{e^5}-\frac {3 f^2 g p \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{4 e^2}-\frac {d^4 g^3 p \left (e x^2+d\right ) \log \left (c \left (e x^2+d\right )^p\right )}{e^5}+\frac {3 d f^2 g p \left (e x^2+d\right ) \log \left (c \left (e x^2+d\right )^p\right )}{e^2}+\frac {4 \sqrt {d} f^3 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{\sqrt {e}}-\frac {12 d^{7/2} f g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{7 e^{7/2}}+\frac {d^5 g^3 p \log \left (e x^2+d\right ) \log \left (c \left (e x^2+d\right )^p\right )}{5 e^5}+\frac {4 i \sqrt {d} f^3 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{7 e^{7/2}}\) |
8*f^3*p^2*x - (1408*d^3*f*g^2*p^2*x)/(245*e^3) - (3*d*f^2*g*p^2*x^2)/e + ( d^4*g^3*p^2*x^2)/e^4 + (568*d^2*f*g^2*p^2*x^3)/(735*e^2) - (288*d*f*g^2*p^ 2*x^5)/(1225*e) + (24*f*g^2*p^2*x^7)/343 + (3*f^2*g*p^2*(d + e*x^2)^2)/(8* e^2) - (d^3*g^3*p^2*(d + e*x^2)^2)/(2*e^5) + (2*d^2*g^3*p^2*(d + e*x^2)^3) /(9*e^5) - (d*g^3*p^2*(d + e*x^2)^4)/(16*e^5) + (g^3*p^2*(d + e*x^2)^5)/(1 25*e^5) - (8*Sqrt[d]*f^3*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (1408* d^(7/2)*f*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(245*e^(7/2)) + ((4*I)*Sqrt [d]*f^3*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] - (((12*I)/7)*d^(7/2)*f *g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e^(7/2) + (8*Sqrt[d]*f^3*p^2*ArcTa n[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (24*d^(7/2)*f*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d ] + I*Sqrt[e]*x)])/(7*e^(7/2)) - (d^5*g^3*p^2*Log[d + e*x^2]^2)/(10*e^5) - 4*f^3*p*x*Log[c*(d + e*x^2)^p] + (12*d^3*f*g^2*p*x*Log[c*(d + e*x^2)^p])/ (7*e^3) - (4*d^2*f*g^2*p*x^3*Log[c*(d + e*x^2)^p])/(7*e^2) + (12*d*f*g^2*p *x^5*Log[c*(d + e*x^2)^p])/(35*e) - (12*f*g^2*p*x^7*Log[c*(d + e*x^2)^p])/ 49 + (3*d*f^2*g*p*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 - (d^4*g^3*p*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^5 - (3*f^2*g*p*(d + e*x^2)^2*Log[c*(d + e*x ^2)^p])/(4*e^2) + (d^3*g^3*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/e^5 - (2* d^2*g^3*p*(d + e*x^2)^3*Log[c*(d + e*x^2)^p])/(3*e^5) + (d*g^3*p*(d + e*x^ 2)^4*Log[c*(d + e*x^2)^p])/(4*e^5) - (g^3*p*(d + e*x^2)^5*Log[c*(d + e*...
3.3.93.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> With[{t = ExpandIntegrand[(a + b*Log[ c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && Integ erQ[r] && IntegerQ[s] && (EqQ[q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 10.28 (sec) , antiderivative size = 1584, normalized size of antiderivative = 1.30
47/900*p^2/e^2*d^2*g^3*x^6-77/600*p^2/e^3*x^4*d^3*g^3+3/8*p^2*x^4*f^2*g-13 7/300*p^2/e^5*d^5*ln(e*x^2+d)*g^3-12/49*p*f*g^2*x^7*ln((e*x^2+d)^p)-3/4*p* f^2*g*x^4*ln((e*x^2+d)^p)-8*p^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^3- 12/7*p/e^3*d^4/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f*g^2*ln((e*x^2+d)^p)+1 2/7*p^2/e^3*d^4/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f*g^2*ln(e*x^2+d)-9/40 0*p^2/e*d*g^3*x^8+1/10*ln((e*x^2+d)^p)^2*g^3*x^10+ln((e*x^2+d)^p)^2*x*f^3+ 3/7*ln((e*x^2+d)^p)^2*g^2*f*x^7+3/4*ln((e*x^2+d)^p)^2*f^2*g*x^4+1/125*p^2* g^3*x^10-1/25*p*g^3*x^10*ln((e*x^2+d)^p)-4*p*x*f^3*ln((e*x^2+d)^p)+12/35*p /e*d*f*g^2*x^5*ln((e*x^2+d)^p)-4/7*p/e^2*d^2*f*g^2*x^3*ln((e*x^2+d)^p)+3/2 *p/e*d*f^2*g*x^2*ln((e*x^2+d)^p)+12/7*p/e^3*x*d^3*f*g^2*ln((e*x^2+d)^p)-3/ 2*p/e^2*d^2*ln(e*x^2+d)*f^2*g*ln((e*x^2+d)^p)+1408/245*p^2/e^3*f*g^2*d^4/( d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-1/35*p^2*e*Sum(-1/2*(ln(x-_alpha)*ln(e* x^2+d)-2*e*(1/4/_alpha/e*ln(x-_alpha)^2+1/2*_alpha/d*ln(x-_alpha)*ln(1/2*( x+_alpha)/_alpha)+1/2*_alpha/d*dilog(1/2*(x+_alpha)/_alpha)))*d*(14*_alpha *d^4*g^3-105*_alpha*d*e^3*f^2*g-60*d^3*e*f*g^2+140*e^4*f^3)/e^6/_alpha,_al pha=RootOf(_Z^2*e+d))-4*p^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^3*ln(e *x^2+d)+4*p*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^3*ln((e*x^2+d)^p)+1/20 *p/e*d*g^3*x^8*ln((e*x^2+d)^p)-1/15*p/e^2*d^2*g^3*x^6*ln((e*x^2+d)^p)+1/10 *p/e^3*d^3*g^3*x^4*ln((e*x^2+d)^p)-1/5*p/e^4*d^4*g^3*x^2*ln((e*x^2+d)^p)+1 /4*(I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-I*Pi*csgn(I*(e*x^2...
\[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]
\[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f + g x^{3}\right )^{3} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \]
Exception generated. \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Exception raised: ValueError} \]
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>0)', see `assume?` for more de tails)Is e
\[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]
Timed out. \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (g\,x^3+f\right )}^3 \,d x \]