\(\int (f+g x^3)^3 \log ^2(c (d+e x^2)^p) \, dx\) [293]

Optimal result

Integrand size = 24, antiderivative size = 1221 \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx =\text {Too large to display} \] Output:

-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*l 
n(c*(e*x^2+d)^p)/e^5+1/4*d*g^3*p*(e*x^2+d)^4*ln(c*(e*x^2+d)^p)/e^5-3/2*d*f 
^2*g*(e*x^2+d)*ln(c*(e*x^2+d)^p)^2/e^2+1/5*d^5*g^3*p*ln(e*x^2+d)*ln(c*(e*x 
^2+d)^p)/e^5-1408/245*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-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+d^4*g^3*p^2*x^2/e 
^4+1408/245*d^(7/2)*f*g^2*p^2*arctan(e^(1/2)*x/d^(1/2))/e^(7/2)+8*d^(1/2)* 
f^3*p^2*arctan(e^(1/2)*x/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*e^(1/2)*x))/e^(1 
/2)+4*d^(1/2)*f^3*p*arctan(e^(1/2)*x/d^(1/2))*ln(c*(e*x^2+d)^p)/e^(1/2)+4* 
I*d^(1/2)*f^3*p^2*arctan(e^(1/2)*x/d^(1/2))^2/e^(1/2)+4*I*d^(1/2)*f^3*p^2* 
polylog(2,1-2*d^(1/2)/(d^(1/2)+I*e^(1/2)*x))/e^(1/2)-12/7*d^(7/2)*f*g^2*p* 
arctan(e^(1/2)*x/d^(1/2))*ln(c*(e*x^2+d)^p)/e^(7/2)-24/7*d^(7/2)*f*g^2*p^2 
*arctan(e^(1/2)*x/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*e^(1/2)*x))/e^(7/2)-12/ 
7*I*d^(7/2)*f*g^2*p^2*arctan(e^(1/2)*x/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*e^(1/2)*x))/e^(7/2)+12/7*d^3*f*g 
^2*p*x*ln(c*(e*x^2+d)^p)/e^3-4/7*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+f^3*x*ln(c*(e*x^2+d)^p)^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^5-12/49*f*g^2*p*x^7*ln(c*(e*x^2+d)^p)-1/25*g^3*p*(e*x^2+d)^5*ln...
 

Mathematica [A] (verified)

Time = 1.33 (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}} \] Input:

Integrate[(f + g*x^3)^3*Log[c*(d + e*x^2)^p]^2,x]
 

Output:

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))
 

Rubi [A] (verified)

Time = 3.38 (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}}\)

Input:

Int[(f + g*x^3)^3*Log[c*(d + e*x^2)^p]^2,x]
 

Output:

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*...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2921

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]))
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 21.77 (sec) , antiderivative size = 1584, normalized size of antiderivative = 1.30

method	result	size
risch	\(\text {Expression too large to display}\)	\(1584\)

Input:

int((g*x^3+f)^3*ln(c*(e*x^2+d)^p)^2,x,method=_RETURNVERBOSE)
 

Output:

137/300*d^4*g^3*p^2*x^2/e^4-9/400/e*p^2*d*g^3*x^8-1408/245*d^3*f*g^2*p^2*x 
/e^3-9/4*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+24/343*f*g^2*p^2*x^7+47/900/e^2*p^2*d^2*g^3*x^6-77/600/e^3*p^2*x 
^4*d^3*g^3+3/8*p^2*x^4*f^2*g-137/300/e^5*p^2*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+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)-4/7/e^2*p*d^2*f*g^2*x^3*ln((e*x^2+d)^p)+(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+d)^p)*csgn(I*c* 
(e*x^2+d)^p)*csgn(I*c)-I*Pi*csgn(I*c*(e*x^2+d)^p)^3+I*Pi*csgn(I*c*(e*x^2+d 
)^p)^2*csgn(I*c)+2*ln(c))*(1/10*ln((e*x^2+d)^p)*g^3*x^10+3/7*ln((e*x^2+d)^ 
p)*g^2*f*x^7+3/4*ln((e*x^2+d)^p)*f^2*g*x^4+ln((e*x^2+d)^p)*x*f^3-1/70*e*p* 
(1/e^5*(7/5*e^4*g^3*x^10-7/4*d*e^3*g^3*x^8+60/7*e^4*f*g^2*x^7+7/3*d^2*e^2* 
g^3*x^6-12*d*e^3*f*g^2*x^5-7/2*d^3*e*g^3*x^4+105/4*e^4*f^2*g*x^4+20*d^2*e^ 
2*f*g^2*x^3+7*d^4*g^3*x^2-105/2*d*f^2*g*x^2*e^3-60*x*d^3*f*g^2*e+140*x*e^4 
*f^3)-d/e^5*(1/2*(14*d^4*g^3-105*d*e^3*f^2*g)/e*ln(e*x^2+d)+(-60*d^3*e*f*g 
^2+140*e^4*f^3)/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2)))))+1/20/e*p*d*g^3*x^8* 
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+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-I*Pi*csgn(I*c*(e*x^2+ 
d)^p)^3+I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+2*ln(c))^2*(1/10*g^3*x^1...
 

Fricas [F]

\[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \] Input:

integrate((g*x^3+f)^3*log(c*(e*x^2+d)^p)^2,x, algorithm="fricas")
 

Output:

integral((g^3*x^9 + 3*f*g^2*x^6 + 3*f^2*g*x^3 + f^3)*log((e*x^2 + d)^p*c)^ 
2, x)
 

Sympy [F]

\[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f + g x^{3}\right )^{3} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \] Input:

integrate((g*x**3+f)**3*ln(c*(e*x**2+d)**p)**2,x)
 

Output:

Integral((f + g*x**3)**3*log(c*(d + e*x**2)**p)**2, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Exception raised: ValueError} \] Input:

integrate((g*x^3+f)^3*log(c*(e*x^2+d)^p)^2,x, algorithm="maxima")
 

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>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \] Input:

integrate((g*x^3+f)^3*log(c*(e*x^2+d)^p)^2,x, algorithm="giac")
 

Output:

integrate((g*x^3 + f)^3*log((e*x^2 + d)^p*c)^2, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (g\,x^3+f\right )}^3 \,d x \] Input:

int(log(c*(d + e*x^2)^p)^2*(f + g*x^3)^3,x)
 

Output:

int(log(c*(d + e*x^2)^p)^2*(f + g*x^3)^3, x)
 

Reduce [F]

\[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx =\text {Too large to display} \] Input:

int((g*x^3+f)^3*log(c*(e*x^2+d)^p)^2,x)
 

Output:

(35481600*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*d**3*e*f*g**2*p**2 
 - 49392000*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*e**4*f**3*p**2 - 
 10584000*int(log((d + e*x**2)**p*c)/(d + e*x**2),x)*d**4*e**2*f*g**2*p + 
24696000*int(log((d + e*x**2)**p*c)/(d + e*x**2),x)*d*e**5*f**3*p + 617400 
*log((d + e*x**2)**p*c)**2*d**5*g**3 - 4630500*log((d + e*x**2)**p*c)**2*d 
**2*e**3*f**2*g + 6174000*log((d + e*x**2)**p*c)**2*e**5*f**3*x + 4630500* 
log((d + e*x**2)**p*c)**2*e**5*f**2*g*x**4 + 2646000*log((d + e*x**2)**p*c 
)**2*e**5*f*g**2*x**7 + 617400*log((d + e*x**2)**p*c)**2*e**5*g**3*x**10 - 
 2819460*log((d + e*x**2)**p*c)*d**5*g**3*p - 1234800*log((d + e*x**2)**p* 
c)*d**4*e*g**3*p*x**2 + 10584000*log((d + e*x**2)**p*c)*d**3*e**2*f*g**2*p 
*x + 617400*log((d + e*x**2)**p*c)*d**3*e**2*g**3*p*x**4 + 13891500*log((d 
 + e*x**2)**p*c)*d**2*e**3*f**2*g*p - 3528000*log((d + e*x**2)**p*c)*d**2* 
e**3*f*g**2*p*x**3 - 411600*log((d + e*x**2)**p*c)*d**2*e**3*g**3*p*x**6 + 
 9261000*log((d + e*x**2)**p*c)*d*e**4*f**2*g*p*x**2 + 2116800*log((d + e* 
x**2)**p*c)*d*e**4*f*g**2*p*x**5 + 308700*log((d + e*x**2)**p*c)*d*e**4*g* 
*3*p*x**8 - 24696000*log((d + e*x**2)**p*c)*e**5*f**3*p*x - 4630500*log((d 
 + e*x**2)**p*c)*e**5*f**2*g*p*x**4 - 1512000*log((d + e*x**2)**p*c)*e**5* 
f*g**2*p*x**7 - 246960*log((d + e*x**2)**p*c)*e**5*g**3*p*x**10 + 2819460* 
d**4*e*g**3*p**2*x**2 - 35481600*d**3*e**2*f*g**2*p**2*x - 792330*d**3*e** 
2*g**3*p**2*x**4 + 4771200*d**2*e**3*f*g**2*p**2*x**3 + 322420*d**2*e**...