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

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2525, 45, 2461, 12, 907} \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^2 p \left (3 d^2 g^2-8 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{24 e^4}-\frac {g p \left (d+e x^2\right )^3 (2 e f-3 d g)}{18 e^4}-\frac {p \left (d+e x^2\right )^2 (e f-3 d g) (e f-d g)}{8 e^4}-\frac {g^2 p \left (d+e x^2\right )^4}{32 e^4}+\frac {d p x^2 (e f-d g)^2}{2 e^3} \]

[In]

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

[Out]

(d*(e*f - d*g)^2*p*x^2)/(2*e^3) - ((e*f - 3*d*g)*(e*f - d*g)*p*(d + e*x^2)^2)/(8*e^4) - (g*(2*e*f - 3*d*g)*p*(
d + e*x^2)^3)/(18*e^4) - (g^2*p*(d + e*x^2)^4)/(32*e^4) - (d^2*(6*e^2*f^2 - 8*d*e*f*g + 3*d^2*g^2)*p*Log[d + e
*x^2])/(24*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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 907

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}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2461

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]}, Dist[a + b*Log[c*(d + e*x)^n], u, x] - Dist[b*e*n, Int[SimplifyI
ntegrand[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] && IntegerQ[r]

Rule 2525

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(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] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x (f+g x)^2 \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right ) \\ & = \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 )-\frac {1}{2} (e p) \text {Subst}\left (\int \frac {x^2 \left (6 f^2+8 f g x+3 g^2 x^2\right )}{12 (d+e x)} \, dx,x,x^2\right ) \\ & = \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 )-\frac {1}{24} (e p) \text {Subst}\left (\int \frac {x^2 \left (6 f^2+8 f g x+3 g^2 x^2\right )}{d+e x} \, dx,x,x^2\right ) \\ & = \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 )-\frac {1}{24} (e p) \text {Subst}\left (\int \left (-\frac {12 d (-e f+d g)^2}{e^4}+\frac {d^2 \left (6 e^2 f^2-8 d e f g+3 d^2 g^2\right )}{e^4 (d+e x)}+\frac {6 (e f-3 d g) (e f-d g) (d+e x)}{e^4}+\frac {4 g (2 e f-3 d g) (d+e x)^2}{e^4}+\frac {3 g^2 (d+e x)^3}{e^4}\right ) \, dx,x,x^2\right ) \\ & = \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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.22 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {d f^2 p x^2}{4 e}-\frac {d^2 f g p x^2}{3 e^2}+\frac {d^3 g^2 p x^2}{8 e^3}-\frac {1}{8} f^2 p x^4+\frac {d f g p x^4}{6 e}-\frac {d^2 g^2 p x^4}{16 e^2}-\frac {1}{9} f g p x^6+\frac {d g^2 p x^6}{24 e}-\frac {1}{32} g^2 p x^8-\frac {d^2 f^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac {d^3 f g p \log \left (d+e x^2\right )}{3 e^3}-\frac {d^4 g^2 p \log \left (d+e x^2\right )}{8 e^4}+\frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right ) \]

[In]

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

[Out]

(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

Maple [A] (verified)

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}\)	\(643\)

[In]

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

[Out]

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))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.07 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {9 \, e^{4} g^{2} p x^{8} + 4 \, {\left (8 \, e^{4} f g - 3 \, d e^{3} g^{2}\right )} p x^{6} + 6 \, {\left (6 \, e^{4} f^{2} - 8 \, d e^{3} f g + 3 \, d^{2} e^{2} g^{2}\right )} p x^{4} - 12 \, {\left (6 \, d e^{3} f^{2} - 8 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} p x^{2} - 12 \, {\left (3 \, e^{4} g^{2} p x^{8} + 8 \, e^{4} f g p x^{6} + 6 \, e^{4} f^{2} p x^{4} - {\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} p\right )} \log \left (e x^{2} + d\right ) - 12 \, {\left (3 \, e^{4} g^{2} x^{8} + 8 \, e^{4} f g x^{6} + 6 \, e^{4} f^{2} x^{4}\right )} \log \left (c\right )}{288 \, e^{4}} \]

[In]

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

[Out]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {1}{288} \, e p {\left (\frac {9 \, e^{3} g^{2} x^{8} + 4 \, {\left (8 \, e^{3} f g - 3 \, d e^{2} g^{2}\right )} x^{6} + 6 \, {\left (6 \, e^{3} f^{2} - 8 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} x^{4} - 12 \, {\left (6 \, d e^{2} f^{2} - 8 \, d^{2} e f g + 3 \, d^{3} g^{2}\right )} x^{2}}{e^{4}} + \frac {12 \, {\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} \log \left (e x^{2} + d\right )}{e^{5}}\right )} + \frac {1}{24} \, {\left (3 \, g^{2} x^{8} + 8 \, f g x^{6} + 6 \, f^{2} x^{4}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]

[In]

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

[Out]

-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 = 0.32 (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}} \]

[In]

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

[Out]

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*log(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 = 1.65 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.88 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2\,x^4}{4}+\frac {f\,g\,x^6}{3}+\frac {g^2\,x^8}{8}\right )-x^4\,\left (\frac {f^2\,p}{8}-\frac {d\,\left (\frac {2\,f\,g\,p}{3}-\frac {d\,g^2\,p}{4\,e}\right )}{4\,e}\right )-x^6\,\left (\frac {f\,g\,p}{9}-\frac {d\,g^2\,p}{24\,e}\right )-\frac {g^2\,p\,x^8}{32}-\frac {\ln \left (e\,x^2+d\right )\,\left (3\,p\,d^4\,g^2-8\,p\,d^3\,e\,f\,g+6\,p\,d^2\,e^2\,f^2\right )}{24\,e^4}+\frac {d\,x^2\,\left (\frac {f^2\,p}{2}-\frac {d\,\left (\frac {2\,f\,g\,p}{3}-\frac {d\,g^2\,p}{4\,e}\right )}{e}\right )}{2\,e} \]

[In]

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

[Out]

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)