\(\int \frac {(f+g x^2)^2 \log (c (d+e x^2)^p)}{x^{11}} \, dx\) [331]

Optimal result

Integrand size = 25, antiderivative size = 253 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^{11}} \, dx=-\frac {e f^2 p}{40 d x^8}+\frac {e f (2 e f-5 d g) p}{60 d^2 x^6}-\frac {e \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p}{120 d^3 x^4}+\frac {e^2 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p}{60 d^4 x^2}+\frac {e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p \log (x)}{30 d^5}-\frac {e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p \log \left (d+e x^2\right )}{60 d^5}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6} \]

[Out]

-1/40*e*f^2*p/d/x^8+1/60*e*f*(-5*d*g+2*e*f)*p/d^2/x^6-1/120*e*(10*d^2*g^2-15*d*e*f*g+6*e^2*f^2)*p/d^3/x^4+1/60
*e^2*(10*d^2*g^2-15*d*e*f*g+6*e^2*f^2)*p/d^4/x^2+1/30*e^3*(10*d^2*g^2-15*d*e*f*g+6*e^2*f^2)*p*ln(x)/d^5-1/60*e
^3*(10*d^2*g^2-15*d*e*f*g+6*e^2*f^2)*p*ln(e*x^2+d)/d^5-1/10*f^2*ln(c*(e*x^2+d)^p)/x^10-1/4*f*g*ln(c*(e*x^2+d)^
p)/x^8-1/6*g^2*ln(c*(e*x^2+d)^p)/x^6

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 253, 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} \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^{11}} \, dx=-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}+\frac {e f p (2 e f-5 d g)}{60 d^2 x^6}-\frac {e^3 p \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{60 d^5}+\frac {e^3 p \log (x) \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right )}{30 d^5}+\frac {e^2 p \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right )}{60 d^4 x^2}-\frac {e p \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right )}{120 d^3 x^4}-\frac {e f^2 p}{40 d x^8} \]

[In]

Int[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^11,x]

[Out]

-1/40*(e*f^2*p)/(d*x^8) + (e*f*(2*e*f - 5*d*g)*p)/(60*d^2*x^6) - (e*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p)/(
120*d^3*x^4) + (e^2*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p)/(60*d^4*x^2) + (e^3*(6*e^2*f^2 - 15*d*e*f*g + 10*
d^2*g^2)*p*Log[x])/(30*d^5) - (e^3*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p*Log[d + e*x^2])/(60*d^5) - (f^2*Log
[c*(d + e*x^2)^p])/(10*x^10) - (f*g*Log[c*(d + e*x^2)^p])/(4*x^8) - (g^2*Log[c*(d + e*x^2)^p])/(6*x^6)

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 \frac {(f+g x)^2 \log \left (c (d+e x)^p\right )}{x^6} \, dx,x,x^2\right ) \\ & = -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {1}{2} (e p) \text {Subst}\left (\int \frac {-6 f^2-15 f g x-10 g^2 x^2}{30 x^5 (d+e x)} \, dx,x,x^2\right ) \\ & = -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {1}{60} (e p) \text {Subst}\left (\int \frac {-6 f^2-15 f g x-10 g^2 x^2}{x^5 (d+e x)} \, dx,x,x^2\right ) \\ & = -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {1}{60} (e p) \text {Subst}\left (\int \left (-\frac {6 f^2}{d x^5}-\frac {3 f (-2 e f+5 d g)}{d^2 x^4}+\frac {-6 e^2 f^2+15 d e f g-10 d^2 g^2}{d^3 x^3}+\frac {e \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right )}{d^4 x^2}-\frac {e^2 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right )}{d^5 x}+\frac {e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right )}{d^5 (d+e x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {e f^2 p}{40 d x^8}+\frac {e f (2 e f-5 d g) p}{60 d^2 x^6}-\frac {e \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p}{120 d^3 x^4}+\frac {e^2 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p}{60 d^4 x^2}+\frac {e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p \log (x)}{30 d^5}-\frac {e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p \log \left (d+e x^2\right )}{60 d^5}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.85 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^{11}} \, dx=-\frac {d e p x^2 \left (-12 e^3 f^2 x^6+6 d e^2 f x^4 \left (f+5 g x^2\right )+d^3 \left (3 f^2+10 f g x^2+10 g^2 x^4\right )-d^2 e x^2 \left (4 f^2+15 f g x^2+20 g^2 x^4\right )\right )-4 e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p x^{10} \log (x)+2 e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p x^{10} \log \left (d+e x^2\right )+2 d^5 \left (6 f^2+15 f g x^2+10 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )}{120 d^5 x^{10}} \]

[In]

Integrate[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^11,x]

[Out]

-1/120*(d*e*p*x^2*(-12*e^3*f^2*x^6 + 6*d*e^2*f*x^4*(f + 5*g*x^2) + d^3*(3*f^2 + 10*f*g*x^2 + 10*g^2*x^4) - d^2
*e*x^2*(4*f^2 + 15*f*g*x^2 + 20*g^2*x^4)) - 4*e^3*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p*x^10*Log[x] + 2*e^3*
(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p*x^10*Log[d + e*x^2] + 2*d^5*(6*f^2 + 15*f*g*x^2 + 10*g^2*x^4)*Log[c*(d
+ e*x^2)^p])/(d^5*x^10)

Maple [A] (verified)

Time = 3.25 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.91

method	result	size
parts	\(-\frac {g^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{6 x^{6}}-\frac {f g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4 x^{8}}-\frac {f^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{10 x^{10}}-\frac {p e \left (-\frac {-10 g^{2} d^{2}+15 d e f g -6 e^{2} f^{2}}{4 d^{3} x^{4}}-\frac {\left (10 g^{2} d^{2}-15 d e f g +6 e^{2} f^{2}\right ) e}{2 d^{4} x^{2}}+\frac {3 f^{2}}{4 d \,x^{8}}+\frac {f \left (5 d g -2 e f \right )}{2 d^{2} x^{6}}-\frac {\left (10 g^{2} d^{2}-15 d e f g +6 e^{2} f^{2}\right ) e^{2} \ln \left (x \right )}{d^{5}}+\frac {e^{2} \left (10 g^{2} d^{2}-15 d e f g +6 e^{2} f^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 d^{5}}\right )}{30}\)	\(230\)
parallelrisch	\(\frac {40 \ln \left (x \right ) x^{10} d^{2} e^{3} g^{2} p^{2}-60 \ln \left (x \right ) x^{10} d \,e^{4} f g \,p^{2}+24 \ln \left (x \right ) x^{10} e^{5} f^{2} p^{2}-20 x^{10} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{2} e^{3} g^{2} p +30 x^{10} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d \,e^{4} f g p -12 x^{10} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{5} f^{2} p -20 x^{10} d^{2} e^{3} g^{2} p^{2}+30 x^{10} d \,e^{4} f g \,p^{2}-12 x^{10} e^{5} f^{2} p^{2}+20 x^{8} d^{3} e^{2} g^{2} p^{2}-30 x^{8} d^{2} e^{3} f g \,p^{2}+12 x^{8} d \,e^{4} f^{2} p^{2}-10 x^{6} d^{4} e \,g^{2} p^{2}+15 x^{6} d^{3} e^{2} f g \,p^{2}-6 x^{6} d^{2} e^{3} f^{2} p^{2}-20 x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{5} g^{2} p -10 x^{4} d^{4} e f g \,p^{2}+4 x^{4} d^{3} e^{2} f^{2} p^{2}-30 x^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{5} f g p -3 x^{2} d^{4} e \,f^{2} p^{2}-12 \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{5} f^{2} p}{120 x^{10} d^{5} p}\)	\(394\)
risch	\(-\frac {\left (10 g^{2} x^{4}+15 f g \,x^{2}+6 f^{2}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{60 x^{10}}-\frac {30 \ln \left (c \right ) d^{5} f g \,x^{2}-10 i \pi \,d^{5} g^{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 )-30 \ln \left (e \,x^{2}+d \right ) d \,e^{4} f g p \,x^{10}+60 \ln \left (x \right ) d \,e^{4} f g p \,x^{10}+30 d^{2} e^{3} f g p \,x^{8}-15 d^{3} e^{2} f g p \,x^{6}+10 d^{4} e f g p \,x^{4}+12 \ln \left (e \,x^{2}+d \right ) e^{5} f^{2} p \,x^{10}-24 \ln \left (x \right ) e^{5} f^{2} p \,x^{10}+20 \ln \left (c \right ) d^{5} g^{2} x^{4}-15 i \pi \,d^{5} f g \,x^{2} \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 d^{3} e^{2} g^{2} p \,x^{8}-12 d \,e^{4} f^{2} p \,x^{8}+10 d^{4} e \,g^{2} p \,x^{6}+6 d^{2} e^{3} f^{2} p \,x^{6}-4 d^{3} e^{2} f^{2} p \,x^{4}+3 d^{4} e \,f^{2} p \,x^{2}+20 \ln \left (e \,x^{2}+d \right ) d^{2} e^{3} g^{2} p \,x^{10}-40 \ln \left (x \right ) d^{2} e^{3} g^{2} p \,x^{10}+12 \ln \left (c \right ) d^{5} f^{2}+15 i \pi \,d^{5} f g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+15 i \pi \,d^{5} f g \,x^{2} \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}+10 i \pi \,d^{5} g^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-15 i \pi \,d^{5} f g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-6 i \pi \,d^{5} f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+10 i \pi \,d^{5} g^{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}-6 i \pi \,d^{5} f^{2} \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 i \pi \,d^{5} f^{2} \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 i \pi \,d^{5} f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-10 i \pi \,d^{5} g^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{120 d^{5} x^{10}}\)	\(748\)

[In]

int((g*x^2+f)^2*ln(c*(e*x^2+d)^p)/x^11,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.06 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^{11}} \, dx=\frac {4 \, {\left (6 \, e^{5} f^{2} - 15 \, d e^{4} f g + 10 \, d^{2} e^{3} g^{2}\right )} p x^{10} \log \left (x\right ) - 3 \, d^{4} e f^{2} p x^{2} + 2 \, {\left (6 \, d e^{4} f^{2} - 15 \, d^{2} e^{3} f g + 10 \, d^{3} e^{2} g^{2}\right )} p x^{8} - {\left (6 \, d^{2} e^{3} f^{2} - 15 \, d^{3} e^{2} f g + 10 \, d^{4} e g^{2}\right )} p x^{6} + 2 \, {\left (2 \, d^{3} e^{2} f^{2} - 5 \, d^{4} e f g\right )} p x^{4} - 2 \, {\left (10 \, d^{5} g^{2} p x^{4} + {\left (6 \, e^{5} f^{2} - 15 \, d e^{4} f g + 10 \, d^{2} e^{3} g^{2}\right )} p x^{10} + 15 \, d^{5} f g p x^{2} + 6 \, d^{5} f^{2} p\right )} \log \left (e x^{2} + d\right ) - 2 \, {\left (10 \, d^{5} g^{2} x^{4} + 15 \, d^{5} f g x^{2} + 6 \, d^{5} f^{2}\right )} \log \left (c\right )}{120 \, d^{5} x^{10}} \]

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^11,x, algorithm="fricas")

[Out]

1/120*(4*(6*e^5*f^2 - 15*d*e^4*f*g + 10*d^2*e^3*g^2)*p*x^10*log(x) - 3*d^4*e*f^2*p*x^2 + 2*(6*d*e^4*f^2 - 15*d
^2*e^3*f*g + 10*d^3*e^2*g^2)*p*x^8 - (6*d^2*e^3*f^2 - 15*d^3*e^2*f*g + 10*d^4*e*g^2)*p*x^6 + 2*(2*d^3*e^2*f^2
- 5*d^4*e*f*g)*p*x^4 - 2*(10*d^5*g^2*p*x^4 + (6*e^5*f^2 - 15*d*e^4*f*g + 10*d^2*e^3*g^2)*p*x^10 + 15*d^5*f*g*p
*x^2 + 6*d^5*f^2*p)*log(e*x^2 + d) - 2*(10*d^5*g^2*x^4 + 15*d^5*f*g*x^2 + 6*d^5*f^2)*log(c))/(d^5*x^10)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^{11}} \, dx=\text {Timed out} \]

[In]

integrate((g*x**2+f)**2*ln(c*(e*x**2+d)**p)/x**11,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.88 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^{11}} \, dx=-\frac {1}{120} \, e p {\left (\frac {2 \, {\left (6 \, e^{4} f^{2} - 15 \, d e^{3} f g + 10 \, d^{2} e^{2} g^{2}\right )} \log \left (e x^{2} + d\right )}{d^{5}} - \frac {2 \, {\left (6 \, e^{4} f^{2} - 15 \, d e^{3} f g + 10 \, d^{2} e^{2} g^{2}\right )} \log \left (x^{2}\right )}{d^{5}} - \frac {2 \, {\left (6 \, e^{3} f^{2} - 15 \, d e^{2} f g + 10 \, d^{2} e g^{2}\right )} x^{6} - 3 \, d^{3} f^{2} - {\left (6 \, d e^{2} f^{2} - 15 \, d^{2} e f g + 10 \, d^{3} g^{2}\right )} x^{4} + 2 \, {\left (2 \, d^{2} e f^{2} - 5 \, d^{3} f g\right )} x^{2}}{d^{4} x^{8}}\right )} - \frac {{\left (10 \, g^{2} x^{4} + 15 \, f g x^{2} + 6 \, f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{60 \, x^{10}} \]

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^11,x, algorithm="maxima")

[Out]

-1/120*e*p*(2*(6*e^4*f^2 - 15*d*e^3*f*g + 10*d^2*e^2*g^2)*log(e*x^2 + d)/d^5 - 2*(6*e^4*f^2 - 15*d*e^3*f*g + 1
0*d^2*e^2*g^2)*log(x^2)/d^5 - (2*(6*e^3*f^2 - 15*d*e^2*f*g + 10*d^2*e*g^2)*x^6 - 3*d^3*f^2 - (6*d*e^2*f^2 - 15
*d^2*e*f*g + 10*d^3*g^2)*x^4 + 2*(2*d^2*e*f^2 - 5*d^3*f*g)*x^2)/(d^4*x^8)) - 1/60*(10*g^2*x^4 + 15*f*g*x^2 + 6
*f^2)*log((e*x^2 + d)^p*c)/x^10

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 699 vs. \(2 (235) = 470\).

Time = 0.34 (sec) , antiderivative size = 699, normalized size of antiderivative = 2.76 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^{11}} \, dx=-\frac {\frac {2 \, {\left (6 \, e^{6} f^{2} p + 15 \, {\left (e x^{2} + d\right )} e^{5} f g p - 15 \, d e^{5} f g p + 10 \, {\left (e x^{2} + d\right )}^{2} e^{4} g^{2} p - 20 \, {\left (e x^{2} + d\right )} d e^{4} g^{2} p + 10 \, d^{2} e^{4} g^{2} p\right )} \log \left (e x^{2} + d\right )}{{\left (e x^{2} + d\right )}^{5} - 5 \, {\left (e x^{2} + d\right )}^{4} d + 10 \, {\left (e x^{2} + d\right )}^{3} d^{2} - 10 \, {\left (e x^{2} + d\right )}^{2} d^{3} + 5 \, {\left (e x^{2} + d\right )} d^{4} - d^{5}} - \frac {12 \, {\left (e x^{2} + d\right )}^{4} e^{6} f^{2} p - 54 \, {\left (e x^{2} + d\right )}^{3} d e^{6} f^{2} p + 94 \, {\left (e x^{2} + d\right )}^{2} d^{2} e^{6} f^{2} p - 77 \, {\left (e x^{2} + d\right )} d^{3} e^{6} f^{2} p + 25 \, d^{4} e^{6} f^{2} p - 30 \, {\left (e x^{2} + d\right )}^{4} d e^{5} f g p + 135 \, {\left (e x^{2} + d\right )}^{3} d^{2} e^{5} f g p - 235 \, {\left (e x^{2} + d\right )}^{2} d^{3} e^{5} f g p + 185 \, {\left (e x^{2} + d\right )} d^{4} e^{5} f g p - 55 \, d^{5} e^{5} f g p + 20 \, {\left (e x^{2} + d\right )}^{4} d^{2} e^{4} g^{2} p - 90 \, {\left (e x^{2} + d\right )}^{3} d^{3} e^{4} g^{2} p + 150 \, {\left (e x^{2} + d\right )}^{2} d^{4} e^{4} g^{2} p - 110 \, {\left (e x^{2} + d\right )} d^{5} e^{4} g^{2} p + 30 \, d^{6} e^{4} g^{2} p - 12 \, d^{4} e^{6} f^{2} \log \left (c\right ) - 30 \, {\left (e x^{2} + d\right )} d^{4} e^{5} f g \log \left (c\right ) + 30 \, d^{5} e^{5} f g \log \left (c\right ) - 20 \, {\left (e x^{2} + d\right )}^{2} d^{4} e^{4} g^{2} \log \left (c\right ) + 40 \, {\left (e x^{2} + d\right )} d^{5} e^{4} g^{2} \log \left (c\right ) - 20 \, d^{6} e^{4} g^{2} \log \left (c\right )}{{\left (e x^{2} + d\right )}^{5} d^{4} - 5 \, {\left (e x^{2} + d\right )}^{4} d^{5} + 10 \, {\left (e x^{2} + d\right )}^{3} d^{6} - 10 \, {\left (e x^{2} + d\right )}^{2} d^{7} + 5 \, {\left (e x^{2} + d\right )} d^{8} - d^{9}} + \frac {2 \, {\left (6 \, e^{6} f^{2} p - 15 \, d e^{5} f g p + 10 \, d^{2} e^{4} g^{2} p\right )} \log \left (e x^{2} + d\right )}{d^{5}} - \frac {2 \, {\left (6 \, e^{6} f^{2} p - 15 \, d e^{5} f g p + 10 \, d^{2} e^{4} g^{2} p\right )} \log \left (e x^{2}\right )}{d^{5}}}{120 \, e} \]

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^11,x, algorithm="giac")

[Out]

-1/120*(2*(6*e^6*f^2*p + 15*(e*x^2 + d)*e^5*f*g*p - 15*d*e^5*f*g*p + 10*(e*x^2 + d)^2*e^4*g^2*p - 20*(e*x^2 +
d)*d*e^4*g^2*p + 10*d^2*e^4*g^2*p)*log(e*x^2 + d)/((e*x^2 + d)^5 - 5*(e*x^2 + d)^4*d + 10*(e*x^2 + d)^3*d^2 -
10*(e*x^2 + d)^2*d^3 + 5*(e*x^2 + d)*d^4 - d^5) - (12*(e*x^2 + d)^4*e^6*f^2*p - 54*(e*x^2 + d)^3*d*e^6*f^2*p +
94*(e*x^2 + d)^2*d^2*e^6*f^2*p - 77*(e*x^2 + d)*d^3*e^6*f^2*p + 25*d^4*e^6*f^2*p - 30*(e*x^2 + d)^4*d*e^5*f*g
*p + 135*(e*x^2 + d)^3*d^2*e^5*f*g*p - 235*(e*x^2 + d)^2*d^3*e^5*f*g*p + 185*(e*x^2 + d)*d^4*e^5*f*g*p - 55*d^
5*e^5*f*g*p + 20*(e*x^2 + d)^4*d^2*e^4*g^2*p - 90*(e*x^2 + d)^3*d^3*e^4*g^2*p + 150*(e*x^2 + d)^2*d^4*e^4*g^2*
p - 110*(e*x^2 + d)*d^5*e^4*g^2*p + 30*d^6*e^4*g^2*p - 12*d^4*e^6*f^2*log(c) - 30*(e*x^2 + d)*d^4*e^5*f*g*log(
c) + 30*d^5*e^5*f*g*log(c) - 20*(e*x^2 + d)^2*d^4*e^4*g^2*log(c) + 40*(e*x^2 + d)*d^5*e^4*g^2*log(c) - 20*d^6*
e^4*g^2*log(c))/((e*x^2 + d)^5*d^4 - 5*(e*x^2 + d)^4*d^5 + 10*(e*x^2 + d)^3*d^6 - 10*(e*x^2 + d)^2*d^7 + 5*(e*
x^2 + d)*d^8 - d^9) + 2*(6*e^6*f^2*p - 15*d*e^5*f*g*p + 10*d^2*e^4*g^2*p)*log(e*x^2 + d)/d^5 - 2*(6*e^6*f^2*p
- 15*d*e^5*f*g*p + 10*d^2*e^4*g^2*p)*log(e*x^2)/d^5)/e

Mupad [B] (verification not implemented)

Time = 1.79 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.89 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^{11}} \, dx=\frac {\ln \left (x\right )\,\left (10\,p\,d^2\,e^3\,g^2-15\,p\,d\,e^4\,f\,g+6\,p\,e^5\,f^2\right )}{30\,d^5}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2}{10}+\frac {f\,g\,x^2}{4}+\frac {g^2\,x^4}{6}\right )}{x^{10}}-\frac {\ln \left (e\,x^2+d\right )\,\left (10\,p\,d^2\,e^3\,g^2-15\,p\,d\,e^4\,f\,g+6\,p\,e^5\,f^2\right )}{60\,d^5}-\frac {\frac {3\,e\,f^2\,p}{4\,d}-\frac {e^2\,p\,x^6\,\left (10\,d^2\,g^2-15\,d\,e\,f\,g+6\,e^2\,f^2\right )}{2\,d^4}+\frac {e\,p\,x^4\,\left (10\,d^2\,g^2-15\,d\,e\,f\,g+6\,e^2\,f^2\right )}{4\,d^3}+\frac {e\,f\,p\,x^2\,\left (5\,d\,g-2\,e\,f\right )}{2\,d^2}}{30\,x^8} \]

[In]

int((log(c*(d + e*x^2)^p)*(f + g*x^2)^2)/x^11,x)

[Out]

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