Integrand size = 25, antiderivative size = 210 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {d (e f-d g)^2 p x^2}{2 e^3}-\frac {(e f-3 d g) (e f-d g) p \left (d+e x^2\right )^2}{8 e^4}-\frac {g (2 e f-3 d g) p \left (d+e x^2\right )^3}{18 e^4}-\frac {g^2 p \left (d+e x^2\right )^4}{32 e^4}-\frac {d^2 \left (6 e^2 f^2-8 d e f g+3 d^2 g^2\right ) p \log \left (d+e x^2\right )}{24 e^4}+\frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right ) \]
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)
Time = 0.06 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.22 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {d f^2 p x^2}{4 e}-\frac {d^2 f g p x^2}{3 e^2}+\frac {d^3 g^2 p x^2}{8 e^3}-\frac {1}{8} f^2 p x^4+\frac {d f g p x^4}{6 e}-\frac {d^2 g^2 p x^4}{16 e^2}-\frac {1}{9} f g p x^6+\frac {d g^2 p x^6}{24 e}-\frac {1}{32} g^2 p x^8-\frac {d^2 f^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac {d^3 f g p \log \left (d+e x^2\right )}{3 e^3}-\frac {d^4 g^2 p \log \left (d+e x^2\right )}{8 e^4}+\frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right ) \]
(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
Time = 0.46 (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 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{12} e p \left (\frac {d^2 \left (3 d^2 g^2-8 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{e^5}+\frac {4 g \left (d+e x^2\right )^3 (2 e f-3 d g)}{3 e^5}+\frac {3 \left (d+e x^2\right )^2 (e f-3 d g) (e f-d g)}{e^5}+\frac {3 g^2 \left (d+e x^2\right )^4}{4 e^5}-\frac {12 d x^2 (e f-d g)^2}{e^4}\right )\right )\) |
(-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
3.4.24.3.1 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[((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]
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]
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])
Time = 2.47 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00
| method | result | size |
| parts | \(\frac {g^{2} x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{8}+\frac {f g \,x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{3}+\frac {f^{2} x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4}-\frac {p e \left (-\frac {-\frac {3}{4} e^{3} g^{2} x^{8}+d \,e^{2} g^{2} x^{6}-\frac {8}{3} e^{3} f g \,x^{6}-\frac {3}{2} d^{2} e \,g^{2} x^{4}+4 d f g \,x^{4} e^{2}-3 x^{4} e^{3} f^{2}+3 x^{2} d^{3} g^{2}-8 d^{2} e f g \,x^{2}+6 d \,e^{2} f^{2} x^{2}}{2 e^{4}}+\frac {d^{2} \left (3 g^{2} d^{2}-8 d e f g +6 e^{2} f^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 e^{5}}\right )}{12}\) | \(211\) |
| 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 ) 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}-\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 {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 (c \right ) f g \,x^{6}}{3}+\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 {x^{8} g^{2} p}{32}-\frac {x^{4} f^{2} p}{8}-\frac {x^{6} f g p}{9}+\left (\frac {1}{8} g^{2} x^{8}+\frac {1}{3} f g \,x^{6}+\frac {1}{4} x^{4} f^{2}\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 )}^{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}\) | \(643\) |
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*p*e*(-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*x^4*e^3*f^2+3*x^2*d^3*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))
Time = 0.30 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.07 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {9 \, e^{4} g^{2} p x^{8} + 4 \, {\left (8 \, e^{4} f g - 3 \, d e^{3} g^{2}\right )} p x^{6} + 6 \, {\left (6 \, e^{4} f^{2} - 8 \, d e^{3} f g + 3 \, d^{2} e^{2} g^{2}\right )} p x^{4} - 12 \, {\left (6 \, d e^{3} f^{2} - 8 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} p x^{2} - 12 \, {\left (3 \, e^{4} g^{2} p x^{8} + 8 \, e^{4} f g p x^{6} + 6 \, e^{4} f^{2} p x^{4} - {\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} p\right )} \log \left (e x^{2} + d\right ) - 12 \, {\left (3 \, e^{4} g^{2} x^{8} + 8 \, e^{4} f g x^{6} + 6 \, e^{4} f^{2} x^{4}\right )} \log \left (c\right )}{288 \, e^{4}} \]
-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
Timed out. \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {1}{288} \, e p {\left (\frac {9 \, e^{3} g^{2} x^{8} + 4 \, {\left (8 \, e^{3} f g - 3 \, d e^{2} g^{2}\right )} x^{6} + 6 \, {\left (6 \, e^{3} f^{2} - 8 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} x^{4} - 12 \, {\left (6 \, d e^{2} f^{2} - 8 \, d^{2} e f g + 3 \, d^{3} g^{2}\right )} x^{2}}{e^{4}} + \frac {12 \, {\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} \log \left (e x^{2} + d\right )}{e^{5}}\right )} + \frac {1}{24} \, {\left (3 \, g^{2} x^{8} + 8 \, f g x^{6} + 6 \, f^{2} x^{4}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]
-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)
Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (194) = 388\).
Time = 0.32 (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}} \]
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
Time = 1.65 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.88 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2\,x^4}{4}+\frac {f\,g\,x^6}{3}+\frac {g^2\,x^8}{8}\right )-x^4\,\left (\frac {f^2\,p}{8}-\frac {d\,\left (\frac {2\,f\,g\,p}{3}-\frac {d\,g^2\,p}{4\,e}\right )}{4\,e}\right )-x^6\,\left (\frac {f\,g\,p}{9}-\frac {d\,g^2\,p}{24\,e}\right )-\frac {g^2\,p\,x^8}{32}-\frac {\ln \left (e\,x^2+d\right )\,\left (3\,p\,d^4\,g^2-8\,p\,d^3\,e\,f\,g+6\,p\,d^2\,e^2\,f^2\right )}{24\,e^4}+\frac {d\,x^2\,\left (\frac {f^2\,p}{2}-\frac {d\,\left (\frac {2\,f\,g\,p}{3}-\frac {d\,g^2\,p}{4\,e}\right )}{e}\right )}{2\,e} \]
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)