\(\int x^3 (f+g x^2)^2 \log (c (d+e x^2)^p) \, dx\) [324]

Optimal result

Integrand size = 25, antiderivative size = 210 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {d (e f-d g)^2 p x^2}{2 e^3}-\frac {(e f-3 d g) (e f-d g) p \left (d+e x^2\right )^2}{8 e^4}-\frac {g (2 e f-3 d g) p \left (d+e x^2\right )^3}{18 e^4}-\frac {g^2 p \left (d+e x^2\right )^4}{32 e^4}-\frac {d^2 \left (6 e^2 f^2-8 d e f g+3 d^2 g^2\right ) p \log \left (d+e x^2\right )}{24 e^4}+\frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right ) \] Output:

1/2*d*(-d*g+e*f)^2*p*x^2/e^3-1/8*(-3*d*g+e*f)*(-d*g+e*f)*p*(e*x^2+d)^2/e^4 
-1/18*g*(-3*d*g+2*e*f)*p*(e*x^2+d)^3/e^4-1/32*g^2*p*(e*x^2+d)^4/e^4-1/24*d 
^2*(3*d^2*g^2-8*d*e*f*g+6*e^2*f^2)*p*ln(e*x^2+d)/e^4+1/4*f^2*x^4*ln(c*(e*x 
^2+d)^p)+1/3*f*g*x^6*ln(c*(e*x^2+d)^p)+1/8*g^2*x^8*ln(c*(e*x^2+d)^p)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.22 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {d f^2 p x^2}{4 e}-\frac {d^2 f g p x^2}{3 e^2}+\frac {d^3 g^2 p x^2}{8 e^3}-\frac {1}{8} f^2 p x^4+\frac {d f g p x^4}{6 e}-\frac {d^2 g^2 p x^4}{16 e^2}-\frac {1}{9} f g p x^6+\frac {d g^2 p x^6}{24 e}-\frac {1}{32} g^2 p x^8-\frac {d^2 f^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac {d^3 f g p \log \left (d+e x^2\right )}{3 e^3}-\frac {d^4 g^2 p \log \left (d+e x^2\right )}{8 e^4}+\frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right ) \] Input:

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

Output:

(d*f^2*p*x^2)/(4*e) - (d^2*f*g*p*x^2)/(3*e^2) + (d^3*g^2*p*x^2)/(8*e^3) - 
(f^2*p*x^4)/8 + (d*f*g*p*x^4)/(6*e) - (d^2*g^2*p*x^4)/(16*e^2) - (f*g*p*x^ 
6)/9 + (d*g^2*p*x^6)/(24*e) - (g^2*p*x^8)/32 - (d^2*f^2*p*Log[d + e*x^2])/ 
(4*e^2) + (d^3*f*g*p*Log[d + e*x^2])/(3*e^3) - (d^4*g^2*p*Log[d + e*x^2])/ 
(8*e^4) + (f^2*x^4*Log[c*(d + e*x^2)^p])/4 + (f*g*x^6*Log[c*(d + e*x^2)^p] 
)/3 + (g^2*x^8*Log[c*(d + e*x^2)^p])/8
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2925, 2861, 27, 1195, 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.

\(\displaystyle \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\)

\(\Big \downarrow \) 2925

\(\displaystyle \frac {1}{2} \int x^2 \left (g x^2+f\right )^2 \log \left (c \left (e x^2+d\right )^p\right )dx^2\)

\(\Big \downarrow \) 2861

\(\displaystyle \frac {1}{2} \left (-e p \int \frac {x^4 \left (3 g^2 x^4+8 f g x^2+6 f^2\right )}{12 \left (e x^2+d\right )}dx^2+\frac {1}{2} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {1}{12} e p \int \frac {x^4 \left (3 g^2 x^4+8 f g x^2+6 f^2\right )}{e x^2+d}dx^2+\frac {1}{2} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )\right )\)

\(\Big \downarrow \) 1195

\(\displaystyle \frac {1}{2} \left (-\frac {1}{12} e p \int \left (\frac {3 g^2 \left (e x^2+d\right )^3}{e^4}+\frac {4 g (2 e f-3 d g) \left (e x^2+d\right )^2}{e^4}+\frac {6 (e f-3 d g) (e f-d g) \left (e x^2+d\right )}{e^4}-\frac {12 d (d g-e f)^2}{e^4}+\frac {d^2 \left (6 e^2 f^2-8 d e g f+3 d^2 g^2\right )}{e^4 \left (e x^2+d\right )}\right )dx^2+\frac {1}{2} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{12} e p \left (\frac {d^2 \left (3 d^2 g^2-8 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{e^5}+\frac {4 g \left (d+e x^2\right )^3 (2 e f-3 d g)}{3 e^5}+\frac {3 \left (d+e x^2\right )^2 (e f-3 d g) (e f-d g)}{e^5}+\frac {3 g^2 \left (d+e x^2\right )^4}{4 e^5}-\frac {12 d x^2 (e f-d g)^2}{e^4}\right )\right )\)

Input:

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

Output:

(-1/12*(e*p*((-12*d*(e*f - d*g)^2*x^2)/e^4 + (3*(e*f - 3*d*g)*(e*f - d*g)* 
(d + e*x^2)^2)/e^5 + (4*g*(2*e*f - 3*d*g)*(d + e*x^2)^3)/(3*e^5) + (3*g^2* 
(d + e*x^2)^4)/(4*e^5) + (d^2*(6*e^2*f^2 - 8*d*e*f*g + 3*d^2*g^2)*Log[d + 
e*x^2])/e^5)) + (f^2*x^4*Log[c*(d + e*x^2)^p])/2 + (2*f*g*x^6*Log[c*(d + e 
*x^2)^p])/3 + (g^2*x^8*Log[c*(d + e*x^2)^p])/4)/2
 

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 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

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

rule 2861

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*(x_)^(m_.)*((f_) + 
 (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(f + g*x^r)^q, 
 x]}, Simp[(a + b*Log[c*(d + e*x)^n])   u, x] - Simp[b*e*n   Int[SimplifyIn 
tegrand[u/(d + e*x), x], x], x] /; InverseFunctionFreeQ[u, x]] /; FreeQ[{a, 
 b, c, d, e, f, g, m, n, q, r}, x] && IntegerQ[m] && IntegerQ[q] && Integer 
Q[r]
 

rule 2925

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Si 
mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], 
x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer 
Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 
] || IGtQ[q, 0])
 

Maple [A] (verified)

Time = 4.33 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00

method	result	size
parts	\(\frac {g^{2} x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{8}+\frac {f g \,x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{3}+\frac {f^{2} x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4}-\frac {e p \left (-\frac {-\frac {3}{4} e^{3} g^{2} x^{8}+d \,e^{2} g^{2} x^{6}-\frac {8}{3} e^{3} f g \,x^{6}-\frac {3}{2} d^{2} e \,g^{2} x^{4}+4 d f g \,x^{4} e^{2}-3 e^{3} f^{2} x^{4}+3 d^{3} x^{2} g^{2}-8 d^{2} e f g \,x^{2}+6 d \,e^{2} f^{2} x^{2}}{2 e^{4}}+\frac {d^{2} \left (3 d^{2} g^{2}-8 f g e d +6 f^{2} e^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 e^{5}}\right )}{12}\)	\(211\)
parallelrisch	\(-\frac {-36 x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{4} g^{2}+9 x^{8} e^{4} g^{2} p -96 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{4} f g -12 x^{6} d \,e^{3} g^{2} p +32 x^{6} e^{4} f g p -72 x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{4} f^{2}+18 x^{4} d^{2} e^{2} g^{2} p -48 x^{4} d \,e^{3} f g p +36 x^{4} e^{4} f^{2} p -36 x^{2} d^{3} e \,g^{2} p +96 x^{2} d^{2} e^{2} f g p -72 x^{2} d \,e^{3} f^{2} p +36 \ln \left (e \,x^{2}+d \right ) d^{4} g^{2} p -96 \ln \left (e \,x^{2}+d \right ) d^{3} e f g p +72 \ln \left (e \,x^{2}+d \right ) d^{2} e^{2} f^{2} p +36 d^{4} g^{2} p -96 d^{3} e f g p +72 d^{2} e^{2} f^{2} p}{288 e^{4}}\)	\(274\)
risch	\(\frac {\ln \left (c \right ) f g \,x^{6}}{3}-\frac {x^{6} f g p}{9}-\frac {x^{8} g^{2} p}{32}-\frac {x^{4} f^{2} p}{8}+\frac {x^{6} d \,g^{2} p}{24 e}-\frac {x^{4} d^{2} g^{2} p}{16 e^{2}}+\frac {x^{2} d^{3} g^{2} p}{8 e^{3}}+\frac {x^{2} d \,f^{2} p}{4 e}-\frac {\ln \left (e \,x^{2}+d \right ) d^{4} g^{2} p}{8 e^{4}}-\frac {\ln \left (e \,x^{2}+d \right ) d^{2} f^{2} p}{4 e^{2}}-\frac {i \pi \,g^{2} x^{8} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{16}-\frac {i \pi \,f^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{8}+\frac {x^{4} d f g p}{6 e}-\frac {x^{2} d^{2} f g p}{3 e^{2}}+\frac {\ln \left (e \,x^{2}+d \right ) d^{3} f g p}{3 e^{3}}-\frac {i \pi f g \,x^{6} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{6}+\frac {\ln \left (c \right ) g^{2} x^{8}}{8}+\frac {\ln \left (c \right ) f^{2} x^{4}}{4}+\frac {i \pi f g \,x^{6} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{6}-\frac {i \pi \,f^{2} x^{4} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{8}+\left (\frac {1}{8} g^{2} x^{8}+\frac {1}{3} g f \,x^{6}+\frac {1}{4} f^{2} x^{4}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )-\frac {i \pi \,g^{2} x^{8} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{16}+\frac {i \pi f g \,x^{6} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{6}+\frac {i \pi \,g^{2} x^{8} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{16}+\frac {i \pi \,g^{2} x^{8} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{16}-\frac {i \pi f g \,x^{6} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{6}+\frac {i \pi \,f^{2} x^{4} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{8}+\frac {i \pi \,f^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{8}\)	\(643\)

Input:

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

Output:

1/8*g^2*x^8*ln(c*(e*x^2+d)^p)+1/3*f*g*x^6*ln(c*(e*x^2+d)^p)+1/4*f^2*x^4*ln 
(c*(e*x^2+d)^p)-1/12*e*p*(-1/2/e^4*(-3/4*e^3*g^2*x^8+d*e^2*g^2*x^6-8/3*e^3 
*f*g*x^6-3/2*d^2*e*g^2*x^4+4*d*f*g*x^4*e^2-3*e^3*f^2*x^4+3*d^3*x^2*g^2-8*d 
^2*e*f*g*x^2+6*d*e^2*f^2*x^2)+1/2*d^2*(3*d^2*g^2-8*d*e*f*g+6*e^2*f^2)/e^5* 
ln(e*x^2+d))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.07 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {9 \, e^{4} g^{2} p x^{8} + 4 \, {\left (8 \, e^{4} f g - 3 \, d e^{3} g^{2}\right )} p x^{6} + 6 \, {\left (6 \, e^{4} f^{2} - 8 \, d e^{3} f g + 3 \, d^{2} e^{2} g^{2}\right )} p x^{4} - 12 \, {\left (6 \, d e^{3} f^{2} - 8 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} p x^{2} - 12 \, {\left (3 \, e^{4} g^{2} p x^{8} + 8 \, e^{4} f g p x^{6} + 6 \, e^{4} f^{2} p x^{4} - {\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} p\right )} \log \left (e x^{2} + d\right ) - 12 \, {\left (3 \, e^{4} g^{2} x^{8} + 8 \, e^{4} f g x^{6} + 6 \, e^{4} f^{2} x^{4}\right )} \log \left (c\right )}{288 \, e^{4}} \] Input:

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

Output:

-1/288*(9*e^4*g^2*p*x^8 + 4*(8*e^4*f*g - 3*d*e^3*g^2)*p*x^6 + 6*(6*e^4*f^2 
 - 8*d*e^3*f*g + 3*d^2*e^2*g^2)*p*x^4 - 12*(6*d*e^3*f^2 - 8*d^2*e^2*f*g + 
3*d^3*e*g^2)*p*x^2 - 12*(3*e^4*g^2*p*x^8 + 8*e^4*f*g*p*x^6 + 6*e^4*f^2*p*x 
^4 - (6*d^2*e^2*f^2 - 8*d^3*e*f*g + 3*d^4*g^2)*p)*log(e*x^2 + d) - 12*(3*e 
^4*g^2*x^8 + 8*e^4*f*g*x^6 + 6*e^4*f^2*x^4)*log(c))/e^4
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {1}{288} \, e p {\left (\frac {9 \, e^{3} g^{2} x^{8} + 4 \, {\left (8 \, e^{3} f g - 3 \, d e^{2} g^{2}\right )} x^{6} + 6 \, {\left (6 \, e^{3} f^{2} - 8 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} x^{4} - 12 \, {\left (6 \, d e^{2} f^{2} - 8 \, d^{2} e f g + 3 \, d^{3} g^{2}\right )} x^{2}}{e^{4}} + \frac {12 \, {\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} \log \left (e x^{2} + d\right )}{e^{5}}\right )} + \frac {1}{24} \, {\left (3 \, g^{2} x^{8} + 8 \, f g x^{6} + 6 \, f^{2} x^{4}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \] Input:

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

Output:

-1/288*e*p*((9*e^3*g^2*x^8 + 4*(8*e^3*f*g - 3*d*e^2*g^2)*x^6 + 6*(6*e^3*f^ 
2 - 8*d*e^2*f*g + 3*d^2*e*g^2)*x^4 - 12*(6*d*e^2*f^2 - 8*d^2*e*f*g + 3*d^3 
*g^2)*x^2)/e^4 + 12*(6*d^2*e^2*f^2 - 8*d^3*e*f*g + 3*d^4*g^2)*log(e*x^2 + 
d)/e^5) + 1/24*(3*g^2*x^8 + 8*f*g*x^6 + 6*f^2*x^4)*log((e*x^2 + d)^p*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (194) = 388\).

Time = 0.13 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.59 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {{\left (e x^{2} + d\right )}^{2} f^{2} p \log \left (e x^{2} + d\right )}{4 \, e^{2}} + \frac {{\left (e x^{2} + d\right )}^{3} f g p \log \left (e x^{2} + d\right )}{3 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{2} d f g p \log \left (e x^{2} + d\right )}{e^{3}} + \frac {{\left (e x^{2} + d\right )}^{4} g^{2} p \log \left (e x^{2} + d\right )}{8 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{3} d g^{2} p \log \left (e x^{2} + d\right )}{2 \, e^{4}} + \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} g^{2} p \log \left (e x^{2} + d\right )}{4 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{2} f^{2} p}{8 \, e^{2}} - \frac {{\left (e x^{2} + d\right )}^{3} f g p}{9 \, e^{3}} + \frac {{\left (e x^{2} + d\right )}^{2} d f g p}{2 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{4} g^{2} p}{32 \, e^{4}} + \frac {{\left (e x^{2} + d\right )}^{3} d g^{2} p}{6 \, e^{4}} - \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} g^{2} p}{8 \, e^{4}} + \frac {{\left (e x^{2} + d\right )}^{2} f^{2} \log \left (c\right )}{4 \, e^{2}} + \frac {{\left (e x^{2} + d\right )}^{3} f g \log \left (c\right )}{3 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{2} d f g \log \left (c\right )}{e^{3}} + \frac {{\left (e x^{2} + d\right )}^{4} g^{2} \log \left (c\right )}{8 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{3} d g^{2} \log \left (c\right )}{2 \, e^{4}} + \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} g^{2} \log \left (c\right )}{4 \, e^{4}} + \frac {{\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d e^{2} f^{2} p - 2 \, {\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d^{2} e f g p + {\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d^{3} g^{2} p - {\left (e x^{2} + d\right )} d e^{2} f^{2} \log \left (c\right ) + 2 \, {\left (e x^{2} + d\right )} d^{2} e f g \log \left (c\right ) - {\left (e x^{2} + d\right )} d^{3} g^{2} \log \left (c\right )}{2 \, e^{4}} \] Input:

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

Output:

1/4*(e*x^2 + d)^2*f^2*p*log(e*x^2 + d)/e^2 + 1/3*(e*x^2 + d)^3*f*g*p*log(e 
*x^2 + d)/e^3 - (e*x^2 + d)^2*d*f*g*p*log(e*x^2 + d)/e^3 + 1/8*(e*x^2 + d) 
^4*g^2*p*log(e*x^2 + d)/e^4 - 1/2*(e*x^2 + d)^3*d*g^2*p*log(e*x^2 + d)/e^4 
 + 3/4*(e*x^2 + d)^2*d^2*g^2*p*log(e*x^2 + d)/e^4 - 1/8*(e*x^2 + d)^2*f^2* 
p/e^2 - 1/9*(e*x^2 + d)^3*f*g*p/e^3 + 1/2*(e*x^2 + d)^2*d*f*g*p/e^3 - 1/32 
*(e*x^2 + d)^4*g^2*p/e^4 + 1/6*(e*x^2 + d)^3*d*g^2*p/e^4 - 3/8*(e*x^2 + d) 
^2*d^2*g^2*p/e^4 + 1/4*(e*x^2 + d)^2*f^2*log(c)/e^2 + 1/3*(e*x^2 + d)^3*f* 
g*log(c)/e^3 - (e*x^2 + d)^2*d*f*g*log(c)/e^3 + 1/8*(e*x^2 + d)^4*g^2*log( 
c)/e^4 - 1/2*(e*x^2 + d)^3*d*g^2*log(c)/e^4 + 3/4*(e*x^2 + d)^2*d^2*g^2*lo 
g(c)/e^4 + 1/2*((e*x^2 - (e*x^2 + d)*log(e*x^2 + d) + d)*d*e^2*f^2*p - 2*( 
e*x^2 - (e*x^2 + d)*log(e*x^2 + d) + d)*d^2*e*f*g*p + (e*x^2 - (e*x^2 + d) 
*log(e*x^2 + d) + d)*d^3*g^2*p - (e*x^2 + d)*d*e^2*f^2*log(c) + 2*(e*x^2 + 
 d)*d^2*e*f*g*log(c) - (e*x^2 + d)*d^3*g^2*log(c))/e^4
 

Mupad [B] (verification not implemented)

Time = 25.77 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.88 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2\,x^4}{4}+\frac {f\,g\,x^6}{3}+\frac {g^2\,x^8}{8}\right )-x^4\,\left (\frac {f^2\,p}{8}-\frac {d\,\left (\frac {2\,f\,g\,p}{3}-\frac {d\,g^2\,p}{4\,e}\right )}{4\,e}\right )-x^6\,\left (\frac {f\,g\,p}{9}-\frac {d\,g^2\,p}{24\,e}\right )-\frac {g^2\,p\,x^8}{32}-\frac {\ln \left (e\,x^2+d\right )\,\left (3\,p\,d^4\,g^2-8\,p\,d^3\,e\,f\,g+6\,p\,d^2\,e^2\,f^2\right )}{24\,e^4}+\frac {d\,x^2\,\left (\frac {f^2\,p}{2}-\frac {d\,\left (\frac {2\,f\,g\,p}{3}-\frac {d\,g^2\,p}{4\,e}\right )}{e}\right )}{2\,e} \] Input:

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

Output:

log(c*(d + e*x^2)^p)*((f^2*x^4)/4 + (g^2*x^8)/8 + (f*g*x^6)/3) - x^4*((f^2 
*p)/8 - (d*((2*f*g*p)/3 - (d*g^2*p)/(4*e)))/(4*e)) - x^6*((f*g*p)/9 - (d*g 
^2*p)/(24*e)) - (g^2*p*x^8)/32 - (log(d + e*x^2)*(3*d^4*g^2*p + 6*d^2*e^2* 
f^2*p - 8*d^3*e*f*g*p))/(24*e^4) + (d*x^2*((f^2*p)/2 - (d*((2*f*g*p)/3 - ( 
d*g^2*p)/(4*e)))/e))/(2*e)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.20 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {-36 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{4} g^{2}+96 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{3} e f g -72 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{2} e^{2} f^{2}+72 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{4} f^{2} x^{4}+96 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{4} f g \,x^{6}+36 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{4} g^{2} x^{8}+36 d^{3} e \,g^{2} p \,x^{2}-96 d^{2} e^{2} f g p \,x^{2}-18 d^{2} e^{2} g^{2} p \,x^{4}+72 d \,e^{3} f^{2} p \,x^{2}+48 d \,e^{3} f g p \,x^{4}+12 d \,e^{3} g^{2} p \,x^{6}-36 e^{4} f^{2} p \,x^{4}-32 e^{4} f g p \,x^{6}-9 e^{4} g^{2} p \,x^{8}}{288 e^{4}} \] Input:

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

Output:

( - 36*log((d + e*x**2)**p*c)*d**4*g**2 + 96*log((d + e*x**2)**p*c)*d**3*e 
*f*g - 72*log((d + e*x**2)**p*c)*d**2*e**2*f**2 + 72*log((d + e*x**2)**p*c 
)*e**4*f**2*x**4 + 96*log((d + e*x**2)**p*c)*e**4*f*g*x**6 + 36*log((d + e 
*x**2)**p*c)*e**4*g**2*x**8 + 36*d**3*e*g**2*p*x**2 - 96*d**2*e**2*f*g*p*x 
**2 - 18*d**2*e**2*g**2*p*x**4 + 72*d*e**3*f**2*p*x**2 + 48*d*e**3*f*g*p*x 
**4 + 12*d*e**3*g**2*p*x**6 - 36*e**4*f**2*p*x**4 - 32*e**4*f*g*p*x**6 - 9 
*e**4*g**2*p*x**8)/(288*e**4)