\(\int x^5 (f+g x^2)^2 \log (c (d+e x^2)^p) \, dx\) [323]

Optimal result

Integrand size = 25, antiderivative size = 251 \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {d^2 (e f-d g)^2 p x^2}{2 e^4}+\frac {d (e f-2 d g) (e f-d g) p \left (d+e x^2\right )^2}{4 e^5}-\frac {\left (e^2 f^2-6 d e f g+6 d^2 g^2\right ) p \left (d+e x^2\right )^3}{18 e^5}-\frac {g (e f-2 d g) p \left (d+e x^2\right )^4}{16 e^5}-\frac {g^2 p \left (d+e x^2\right )^5}{50 e^5}+\frac {d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )}{60 e^5}+\frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right ) \]

[Out]

-1/2*d^2*(-d*g+e*f)^2*p*x^2/e^4+1/4*d*(-2*d*g+e*f)*(-d*g+e*f)*p*(e*x^2+d)^2/e^5-1/18*(6*d^2*g^2-6*d*e*f*g+e^2*
f^2)*p*(e*x^2+d)^3/e^5-1/16*g*(-2*d*g+e*f)*p*(e*x^2+d)^4/e^5-1/50*g^2*p*(e*x^2+d)^5/e^5+1/60*d^3*(6*d^2*g^2-15
*d*e*f*g+10*e^2*f^2)*p*ln(e*x^2+d)/e^5+1/6*f^2*x^6*ln(c*(e*x^2+d)^p)+1/4*f*g*x^8*ln(c*(e*x^2+d)^p)+1/10*g^2*x^
10*ln(c*(e*x^2+d)^p)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 251, 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} \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^2 p x^2 (e f-d g)^2}{2 e^4}-\frac {p \left (d+e x^2\right )^3 \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )}{18 e^5}+\frac {d^3 p \left (6 d^2 g^2-15 d e f g+10 e^2 f^2\right ) \log \left (d+e x^2\right )}{60 e^5}-\frac {g p \left (d+e x^2\right )^4 (e f-2 d g)}{16 e^5}+\frac {d p \left (d+e x^2\right )^2 (e f-2 d g) (e f-d g)}{4 e^5}-\frac {g^2 p \left (d+e x^2\right )^5}{50 e^5} \]

[In]

Int[x^5*(f + g*x^2)^2*Log[c*(d + e*x^2)^p],x]

[Out]

-1/2*(d^2*(e*f - d*g)^2*p*x^2)/e^4 + (d*(e*f - 2*d*g)*(e*f - d*g)*p*(d + e*x^2)^2)/(4*e^5) - ((e^2*f^2 - 6*d*e
*f*g + 6*d^2*g^2)*p*(d + e*x^2)^3)/(18*e^5) - (g*(e*f - 2*d*g)*p*(d + e*x^2)^4)/(16*e^5) - (g^2*p*(d + e*x^2)^
5)/(50*e^5) + (d^3*(10*e^2*f^2 - 15*d*e*f*g + 6*d^2*g^2)*p*Log[d + e*x^2])/(60*e^5) + (f^2*x^6*Log[c*(d + e*x^
2)^p])/6 + (f*g*x^8*Log[c*(d + e*x^2)^p])/4 + (g^2*x^10*Log[c*(d + e*x^2)^p])/10

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^2 (f+g x)^2 \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{2} (e p) \text {Subst}\left (\int \frac {x^3 \left (10 f^2+15 f g x+6 g^2 x^2\right )}{30 (d+e x)} \, dx,x,x^2\right ) \\ & = \frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{60} (e p) \text {Subst}\left (\int \frac {x^3 \left (10 f^2+15 f g x+6 g^2 x^2\right )}{d+e x} \, dx,x,x^2\right ) \\ & = \frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{60} (e p) \text {Subst}\left (\int \left (\frac {30 d^2 (-e f+d g)^2}{e^5}-\frac {d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right )}{e^5 (d+e x)}+\frac {30 d (e f-2 d g) (-e f+d g) (d+e x)}{e^5}+\frac {10 \left (e^2 f^2-6 d e f g+6 d^2 g^2\right ) (d+e x)^2}{e^5}+\frac {15 g (e f-2 d g) (d+e x)^3}{e^5}+\frac {6 g^2 (d+e x)^4}{e^5}\right ) \, dx,x,x^2\right ) \\ & = -\frac {d^2 (e f-d g)^2 p x^2}{2 e^4}+\frac {d (e f-2 d g) (e f-d g) p \left (d+e x^2\right )^2}{4 e^5}-\frac {\left (e^2 f^2-6 d e f g+6 d^2 g^2\right ) p \left (d+e x^2\right )^3}{18 e^5}-\frac {g (e f-2 d g) p \left (d+e x^2\right )^4}{16 e^5}-\frac {g^2 p \left (d+e x^2\right )^5}{50 e^5}+\frac {d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )}{60 e^5}+\frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.82 \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {-e p x^2 \left (360 d^4 g^2-180 d^3 e g \left (5 f+g x^2\right )-30 d e^3 x^2 \left (10 f^2+10 f g x^2+3 g^2 x^4\right )+30 d^2 e^2 \left (20 f^2+15 f g x^2+4 g^2 x^4\right )+e^4 x^4 \left (200 f^2+225 f g x^2+72 g^2 x^4\right )\right )+60 d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )+60 e^5 x^6 \left (10 f^2+15 f g x^2+6 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3600 e^5} \]

[In]

Integrate[x^5*(f + g*x^2)^2*Log[c*(d + e*x^2)^p],x]

[Out]

(-(e*p*x^2*(360*d^4*g^2 - 180*d^3*e*g*(5*f + g*x^2) - 30*d*e^3*x^2*(10*f^2 + 10*f*g*x^2 + 3*g^2*x^4) + 30*d^2*
e^2*(20*f^2 + 15*f*g*x^2 + 4*g^2*x^4) + e^4*x^4*(200*f^2 + 225*f*g*x^2 + 72*g^2*x^4))) + 60*d^3*(10*e^2*f^2 -
15*d*e*f*g + 6*d^2*g^2)*p*Log[d + e*x^2] + 60*e^5*x^6*(10*f^2 + 15*f*g*x^2 + 6*g^2*x^4)*Log[c*(d + e*x^2)^p])/
(3600*e^5)

Maple [A] (verified)

Time = 3.72 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.01

method	result	size
parts	\(\frac {g^{2} x^{10} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{10}+\frac {f g \,x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4}+\frac {f^{2} x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{6}-\frac {p e \left (\frac {\frac {6}{5} e^{4} g^{2} x^{10}-\frac {3}{2} x^{8} d \,e^{3} g^{2}+\frac {15}{4} e^{4} f g \,x^{8}+2 x^{6} d^{2} e^{2} g^{2}-5 x^{6} d \,e^{3} f g +\frac {10}{3} e^{4} f^{2} x^{6}-3 x^{4} d^{3} e \,g^{2}+\frac {15}{2} x^{4} d^{2} e^{2} f g -5 x^{4} d \,e^{3} f^{2}+6 d^{4} g^{2} x^{2}-15 d^{3} e f g \,x^{2}+10 d^{2} e^{2} f^{2} x^{2}}{2 e^{5}}-\frac {d^{3} \left (6 g^{2} d^{2}-15 d e f g +10 e^{2} f^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 e^{6}}\right )}{30}\)	\(253\)
parallelrisch	\(\frac {360 x^{10} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{5} g^{2}-72 e^{5} g^{2} p \,x^{10}+900 x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{5} f g +90 d \,e^{4} g^{2} p \,x^{8}-225 e^{5} f g p \,x^{8}+600 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{5} f^{2}-120 d^{2} e^{3} g^{2} p \,x^{6}+300 d \,e^{4} f g p \,x^{6}-200 e^{5} f^{2} p \,x^{6}+180 d^{3} e^{2} g^{2} p \,x^{4}-450 d^{2} e^{3} f g p \,x^{4}+300 d \,e^{4} f^{2} p \,x^{4}-360 d^{4} e \,g^{2} p \,x^{2}+900 d^{3} e^{2} f g p \,x^{2}-600 d^{2} e^{3} f^{2} p \,x^{2}+360 \ln \left (e \,x^{2}+d \right ) d^{5} g^{2} p -900 \ln \left (e \,x^{2}+d \right ) d^{4} e f g p +600 \ln \left (e \,x^{2}+d \right ) d^{3} e^{2} f^{2} p +360 d^{5} g^{2} p -900 d^{4} e f g p +600 d^{3} e^{2} f^{2} p}{3600 e^{5}}\)	\(318\)
risch	\(\frac {d f g p \,x^{6}}{12 e}+\frac {\ln \left (c \right ) g^{2} x^{10}}{10}+\frac {\ln \left (c \right ) f^{2} x^{6}}{6}+\frac {i \pi \,g^{2} x^{10} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{20}+\frac {i \pi \,g^{2} x^{10} \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}}{20}-\frac {i \pi f g \,x^{8} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{8}+\frac {i \pi \,f^{2} x^{6} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{12}+\frac {\ln \left (c \right ) f g \,x^{8}}{4}-\frac {i \pi f g \,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 )}{8}-\frac {g^{2} p \,x^{10}}{50}-\frac {f^{2} p \,x^{6}}{18}-\frac {f g p \,x^{8}}{16}+\frac {d \,g^{2} p \,x^{8}}{40 e}-\frac {d^{2} g^{2} p \,x^{6}}{30 e^{2}}+\frac {d^{3} g^{2} p \,x^{4}}{20 e^{3}}+\frac {d \,f^{2} p \,x^{4}}{12 e}-\frac {d^{4} g^{2} p \,x^{2}}{10 e^{4}}-\frac {d^{2} f^{2} p \,x^{2}}{6 e^{2}}+\frac {\ln \left (e \,x^{2}+d \right ) d^{5} g^{2} p}{10 e^{5}}+\frac {\ln \left (e \,x^{2}+d \right ) d^{3} f^{2} p}{6 e^{3}}+\frac {i \pi \,f^{2} 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}}{12}-\frac {\ln \left (e \,x^{2}+d \right ) d^{4} f g p}{4 e^{4}}-\frac {d^{2} f g p \,x^{4}}{8 e^{2}}+\frac {d^{3} f g p \,x^{2}}{4 e^{3}}-\frac {i \pi \,g^{2} x^{10} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{20}-\frac {i \pi \,f^{2} x^{6} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{12}+\left (\frac {1}{10} g^{2} x^{10}+\frac {1}{4} f g \,x^{8}+\frac {1}{6} f^{2} x^{6}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )-\frac {i \pi \,f^{2} 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 )}{12}-\frac {i \pi \,g^{2} x^{10} \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 )}{20}+\frac {i \pi f g \,x^{8} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{8}+\frac {i \pi f g \,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}}{8}\)	\(687\)

[In]

int(x^5*(g*x^2+f)^2*ln(c*(e*x^2+d)^p),x,method=_RETURNVERBOSE)

[Out]

1/10*g^2*x^10*ln(c*(e*x^2+d)^p)+1/4*f*g*x^8*ln(c*(e*x^2+d)^p)+1/6*f^2*x^6*ln(c*(e*x^2+d)^p)-1/30*p*e*(1/2/e^5*
(6/5*e^4*g^2*x^10-3/2*x^8*d*e^3*g^2+15/4*e^4*f*g*x^8+2*x^6*d^2*e^2*g^2-5*x^6*d*e^3*f*g+10/3*e^4*f^2*x^6-3*x^4*
d^3*e*g^2+15/2*x^4*d^2*e^2*f*g-5*x^4*d*e^3*f^2+6*d^4*g^2*x^2-15*d^3*e*f*g*x^2+10*d^2*e^2*f^2*x^2)-1/2*d^3*(6*d
^2*g^2-15*d*e*f*g+10*e^2*f^2)/e^6*ln(e*x^2+d))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.04 \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {72 \, e^{5} g^{2} p x^{10} + 45 \, {\left (5 \, e^{5} f g - 2 \, d e^{4} g^{2}\right )} p x^{8} + 20 \, {\left (10 \, e^{5} f^{2} - 15 \, d e^{4} f g + 6 \, d^{2} e^{3} g^{2}\right )} p x^{6} - 30 \, {\left (10 \, d e^{4} f^{2} - 15 \, d^{2} e^{3} f g + 6 \, d^{3} e^{2} g^{2}\right )} p x^{4} + 60 \, {\left (10 \, d^{2} e^{3} f^{2} - 15 \, d^{3} e^{2} f g + 6 \, d^{4} e g^{2}\right )} p x^{2} - 60 \, {\left (6 \, e^{5} g^{2} p x^{10} + 15 \, e^{5} f g p x^{8} + 10 \, e^{5} f^{2} p x^{6} + {\left (10 \, d^{3} e^{2} f^{2} - 15 \, d^{4} e f g + 6 \, d^{5} g^{2}\right )} p\right )} \log \left (e x^{2} + d\right ) - 60 \, {\left (6 \, e^{5} g^{2} x^{10} + 15 \, e^{5} f g x^{8} + 10 \, e^{5} f^{2} x^{6}\right )} \log \left (c\right )}{3600 \, e^{5}} \]

[In]

integrate(x^5*(g*x^2+f)^2*log(c*(e*x^2+d)^p),x, algorithm="fricas")

[Out]

-1/3600*(72*e^5*g^2*p*x^10 + 45*(5*e^5*f*g - 2*d*e^4*g^2)*p*x^8 + 20*(10*e^5*f^2 - 15*d*e^4*f*g + 6*d^2*e^3*g^
2)*p*x^6 - 30*(10*d*e^4*f^2 - 15*d^2*e^3*f*g + 6*d^3*e^2*g^2)*p*x^4 + 60*(10*d^2*e^3*f^2 - 15*d^3*e^2*f*g + 6*
d^4*e*g^2)*p*x^2 - 60*(6*e^5*g^2*p*x^10 + 15*e^5*f*g*p*x^8 + 10*e^5*f^2*p*x^6 + (10*d^3*e^2*f^2 - 15*d^4*e*f*g
+ 6*d^5*g^2)*p)*log(e*x^2 + d) - 60*(6*e^5*g^2*x^10 + 15*e^5*f*g*x^8 + 10*e^5*f^2*x^6)*log(c))/e^5

Sympy [F(-1)]

Timed out. \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Timed out} \]

[In]

integrate(x**5*(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 = 223, normalized size of antiderivative = 0.89 \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {1}{3600} \, e p {\left (\frac {72 \, e^{4} g^{2} x^{10} + 45 \, {\left (5 \, e^{4} f g - 2 \, d e^{3} g^{2}\right )} x^{8} + 20 \, {\left (10 \, e^{4} f^{2} - 15 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} x^{6} - 30 \, {\left (10 \, d e^{3} f^{2} - 15 \, d^{2} e^{2} f g + 6 \, d^{3} e g^{2}\right )} x^{4} + 60 \, {\left (10 \, d^{2} e^{2} f^{2} - 15 \, d^{3} e f g + 6 \, d^{4} g^{2}\right )} x^{2}}{e^{5}} - \frac {60 \, {\left (10 \, d^{3} e^{2} f^{2} - 15 \, d^{4} e f g + 6 \, d^{5} g^{2}\right )} \log \left (e x^{2} + d\right )}{e^{6}}\right )} + \frac {1}{60} \, {\left (6 \, g^{2} x^{10} + 15 \, f g x^{8} + 10 \, f^{2} x^{6}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]

[In]

integrate(x^5*(g*x^2+f)^2*log(c*(e*x^2+d)^p),x, algorithm="maxima")

[Out]

-1/3600*e*p*((72*e^4*g^2*x^10 + 45*(5*e^4*f*g - 2*d*e^3*g^2)*x^8 + 20*(10*e^4*f^2 - 15*d*e^3*f*g + 6*d^2*e^2*g
^2)*x^6 - 30*(10*d*e^3*f^2 - 15*d^2*e^2*f*g + 6*d^3*e*g^2)*x^4 + 60*(10*d^2*e^2*f^2 - 15*d^3*e*f*g + 6*d^4*g^2
)*x^2)/e^5 - 60*(10*d^3*e^2*f^2 - 15*d^4*e*f*g + 6*d^5*g^2)*log(e*x^2 + d)/e^6) + 1/60*(6*g^2*x^10 + 15*f*g*x^
8 + 10*f^2*x^6)*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. 753 vs. \(2 (233) = 466\).

Time = 0.33 (sec) , antiderivative size = 753, normalized size of antiderivative = 3.00 \[ \int x^5 \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 )}^{3} f^{2} p \log \left (e x^{2} + d\right )}{6 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{2} d f^{2} p \log \left (e x^{2} + d\right )}{2 \, e^{3}} + \frac {{\left (e x^{2} + d\right )}^{4} f g p \log \left (e x^{2} + d\right )}{4 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{3} d f g p \log \left (e x^{2} + d\right )}{e^{4}} + \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} f g p \log \left (e x^{2} + d\right )}{2 \, e^{4}} + \frac {{\left (e x^{2} + d\right )}^{5} g^{2} p \log \left (e x^{2} + d\right )}{10 \, e^{5}} - \frac {{\left (e x^{2} + d\right )}^{4} d g^{2} p \log \left (e x^{2} + d\right )}{2 \, e^{5}} + \frac {{\left (e x^{2} + d\right )}^{3} d^{2} g^{2} p \log \left (e x^{2} + d\right )}{e^{5}} - \frac {{\left (e x^{2} + d\right )}^{2} d^{3} g^{2} p \log \left (e x^{2} + d\right )}{e^{5}} - \frac {{\left (e x^{2} + d\right )}^{3} f^{2} p}{18 \, e^{3}} + \frac {{\left (e x^{2} + d\right )}^{2} d f^{2} p}{4 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{4} f g p}{16 \, e^{4}} + \frac {{\left (e x^{2} + d\right )}^{3} d f g p}{3 \, e^{4}} - \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} f g p}{4 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{5} g^{2} p}{50 \, e^{5}} + \frac {{\left (e x^{2} + d\right )}^{4} d g^{2} p}{8 \, e^{5}} - \frac {{\left (e x^{2} + d\right )}^{3} d^{2} g^{2} p}{3 \, e^{5}} + \frac {{\left (e x^{2} + d\right )}^{2} d^{3} g^{2} p}{2 \, e^{5}} + \frac {{\left (e x^{2} + d\right )}^{3} f^{2} \log \left (c\right )}{6 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{2} d f^{2} \log \left (c\right )}{2 \, e^{3}} + \frac {{\left (e x^{2} + d\right )}^{4} f g \log \left (c\right )}{4 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{3} d f g \log \left (c\right )}{e^{4}} + \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} f g \log \left (c\right )}{2 \, e^{4}} + \frac {{\left (e x^{2} + d\right )}^{5} g^{2} \log \left (c\right )}{10 \, e^{5}} - \frac {{\left (e x^{2} + d\right )}^{4} d g^{2} \log \left (c\right )}{2 \, e^{5}} + \frac {{\left (e x^{2} + d\right )}^{3} d^{2} g^{2} \log \left (c\right )}{e^{5}} - \frac {{\left (e x^{2} + d\right )}^{2} d^{3} g^{2} \log \left (c\right )}{e^{5}} - \frac {{\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d^{2} 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^{3} e f g p + {\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d^{4} g^{2} p - {\left (e x^{2} + d\right )} d^{2} e^{2} f^{2} \log \left (c\right ) + 2 \, {\left (e x^{2} + d\right )} d^{3} e f g \log \left (c\right ) - {\left (e x^{2} + d\right )} d^{4} g^{2} \log \left (c\right )}{2 \, e^{5}} \]

[In]

integrate(x^5*(g*x^2+f)^2*log(c*(e*x^2+d)^p),x, algorithm="giac")

[Out]

1/6*(e*x^2 + d)^3*f^2*p*log(e*x^2 + d)/e^3 - 1/2*(e*x^2 + d)^2*d*f^2*p*log(e*x^2 + d)/e^3 + 1/4*(e*x^2 + d)^4*
f*g*p*log(e*x^2 + d)/e^4 - (e*x^2 + d)^3*d*f*g*p*log(e*x^2 + d)/e^4 + 3/2*(e*x^2 + d)^2*d^2*f*g*p*log(e*x^2 +
d)/e^4 + 1/10*(e*x^2 + d)^5*g^2*p*log(e*x^2 + d)/e^5 - 1/2*(e*x^2 + d)^4*d*g^2*p*log(e*x^2 + d)/e^5 + (e*x^2 +
d)^3*d^2*g^2*p*log(e*x^2 + d)/e^5 - (e*x^2 + d)^2*d^3*g^2*p*log(e*x^2 + d)/e^5 - 1/18*(e*x^2 + d)^3*f^2*p/e^3
+ 1/4*(e*x^2 + d)^2*d*f^2*p/e^3 - 1/16*(e*x^2 + d)^4*f*g*p/e^4 + 1/3*(e*x^2 + d)^3*d*f*g*p/e^4 - 3/4*(e*x^2 +
d)^2*d^2*f*g*p/e^4 - 1/50*(e*x^2 + d)^5*g^2*p/e^5 + 1/8*(e*x^2 + d)^4*d*g^2*p/e^5 - 1/3*(e*x^2 + d)^3*d^2*g^2
*p/e^5 + 1/2*(e*x^2 + d)^2*d^3*g^2*p/e^5 + 1/6*(e*x^2 + d)^3*f^2*log(c)/e^3 - 1/2*(e*x^2 + d)^2*d*f^2*log(c)/e
^3 + 1/4*(e*x^2 + d)^4*f*g*log(c)/e^4 - (e*x^2 + d)^3*d*f*g*log(c)/e^4 + 3/2*(e*x^2 + d)^2*d^2*f*g*log(c)/e^4
+ 1/10*(e*x^2 + d)^5*g^2*log(c)/e^5 - 1/2*(e*x^2 + d)^4*d*g^2*log(c)/e^5 + (e*x^2 + d)^3*d^2*g^2*log(c)/e^5 -
(e*x^2 + d)^2*d^3*g^2*log(c)/e^5 - 1/2*((e*x^2 - (e*x^2 + d)*log(e*x^2 + d) + d)*d^2*e^2*f^2*p - 2*(e*x^2 - (e
*x^2 + d)*log(e*x^2 + d) + d)*d^3*e*f*g*p + (e*x^2 - (e*x^2 + d)*log(e*x^2 + d) + d)*d^4*g^2*p - (e*x^2 + d)*d
^2*e^2*f^2*log(c) + 2*(e*x^2 + d)*d^3*e*f*g*log(c) - (e*x^2 + d)*d^4*g^2*log(c))/e^5

Mupad [B] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.89 \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2\,x^6}{6}+\frac {f\,g\,x^8}{4}+\frac {g^2\,x^{10}}{10}\right )-x^6\,\left (\frac {f^2\,p}{18}-\frac {d\,\left (\frac {f\,g\,p}{2}-\frac {d\,g^2\,p}{5\,e}\right )}{6\,e}\right )-x^8\,\left (\frac {f\,g\,p}{16}-\frac {d\,g^2\,p}{40\,e}\right )-\frac {g^2\,p\,x^{10}}{50}+\frac {\ln \left (e\,x^2+d\right )\,\left (6\,p\,d^5\,g^2-15\,p\,d^4\,e\,f\,g+10\,p\,d^3\,e^2\,f^2\right )}{60\,e^5}+\frac {d\,x^4\,\left (\frac {f^2\,p}{3}-\frac {d\,\left (\frac {f\,g\,p}{2}-\frac {d\,g^2\,p}{5\,e}\right )}{e}\right )}{4\,e}-\frac {d^2\,x^2\,\left (\frac {f^2\,p}{3}-\frac {d\,\left (\frac {f\,g\,p}{2}-\frac {d\,g^2\,p}{5\,e}\right )}{e}\right )}{2\,e^2} \]

[In]

int(x^5*log(c*(d + e*x^2)^p)*(f + g*x^2)^2,x)

[Out]

log(c*(d + e*x^2)^p)*((f^2*x^6)/6 + (g^2*x^10)/10 + (f*g*x^8)/4) - x^6*((f^2*p)/18 - (d*((f*g*p)/2 - (d*g^2*p)
/(5*e)))/(6*e)) - x^8*((f*g*p)/16 - (d*g^2*p)/(40*e)) - (g^2*p*x^10)/50 + (log(d + e*x^2)*(6*d^5*g^2*p + 10*d^
3*e^2*f^2*p - 15*d^4*e*f*g*p))/(60*e^5) + (d*x^4*((f^2*p)/3 - (d*((f*g*p)/2 - (d*g^2*p)/(5*e)))/e))/(4*e) - (d
^2*x^2*((f^2*p)/3 - (d*((f*g*p)/2 - (d*g^2*p)/(5*e)))/e))/(2*e^2)