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

Optimal. Leaf size=253 \[ -\frac {e f^2 p}{40 d x^8}+\frac {e f (2 e f-5 d g) p}{60 d^2 x^6}-\frac {e \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p}{120 d^3 x^4}+\frac {e^2 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p}{60 d^4 x^2}+\frac {e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p \log (x)}{30 d^5}-\frac {e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p \log \left (d+e x^2\right )}{60 d^5}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6} \]

[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]
time = 0.23, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2525, 45, 2461, 12, 907} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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}{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]
time = 0.16, size = 215, normalized size = 0.85 \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[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 [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.51, size = 748, normalized size = 2.96

method	result	size
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 {-15 i \pi \,d^{5} f g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )+10 i \pi \,d^{5} g^{2} x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-15 i \pi \,d^{5} f g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}-6 i \pi \,d^{5} f^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )+10 i \pi \,d^{5} g^{2} x^{4} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}-10 i \pi \,d^{5} g^{2} x^{4} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )+15 i \pi \,d^{5} f g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}+15 i \pi \,d^{5} f g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+6 i \pi \,d^{5} f^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}+6 i \pi \,d^{5} f^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-10 i \pi \,d^{5} g^{2} x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+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}+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}-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}-6 i \pi \,d^{5} f^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+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}+30 \ln \left (c \right ) d^{5} f g \,x^{2}-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 (c \right ) d^{5} g^{2} x^{4}+12 \ln \left (c \right ) d^{5} f^{2}}{120 d^{5} x^{10}}\)	\(748\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(10*g^2*x^4+15*f*g*x^2+6*f^2)/x^10*ln((e*x^2+d)^p)-1/120*(-15*I*Pi*d^5*f*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(
I*c*(e*x^2+d)^p)*csgn(I*c)-10*I*Pi*d^5*g^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+15*I*Pi*d^5
*f*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+15*I*Pi*d^5*f*g*x^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+6*I
*Pi*d^5*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+6*I*Pi*d^5*f^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-10*I*
Pi*d^5*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^3+20*ln(e*x^2+d)*d^2*e^3*g^2*p*x^10-40*ln(x)*d^2*e^3*g^2*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+10*I*Pi*d^5*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-15*I
*Pi*d^5*f*g*x^2*csgn(I*c*(e*x^2+d)^p)^3-6*I*Pi*d^5*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+10*
I*Pi*d^5*g^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-30*ln(e*x^2+d)*d*e^4*f*g*p*x^10+60*ln(x)*d*e^4*f*
g*p*x^10-6*I*Pi*d^5*f^2*csgn(I*c*(e*x^2+d)^p)^3+12*ln(e*x^2+d)*e^5*f^2*p*x^10-24*ln(x)*e^5*f^2*p*x^10+30*ln(c)
*d^5*f*g*x^2-20*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(c)*d^5*g^2*x^4+12*ln(c)*d^5*f^2)/d^5/x^10

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 220, normalized size = 0.87 \begin {gather*} -\frac {1}{120} \, p {\left (\frac {2 \, {\left (10 \, d^{2} g^{2} e^{2} - 15 \, d f g e^{3} + 6 \, f^{2} e^{4}\right )} \log \left (x^{2} e + d\right )}{d^{5}} - \frac {2 \, {\left (10 \, d^{2} g^{2} e^{2} - 15 \, d f g e^{3} + 6 \, f^{2} e^{4}\right )} \log \left (x^{2}\right )}{d^{5}} - \frac {2 \, {\left (10 \, d^{2} g^{2} e - 15 \, d f g e^{2} + 6 \, f^{2} e^{3}\right )} x^{6} - 3 \, d^{3} f^{2} - {\left (10 \, d^{3} g^{2} - 15 \, d^{2} f g e + 6 \, d f^{2} e^{2}\right )} x^{4} - 2 \, {\left (5 \, d^{3} f g - 2 \, d^{2} f^{2} e\right )} x^{2}}{d^{4} x^{8}}\right )} e - \frac {{\left (10 \, g^{2} x^{4} + 15 \, f g x^{2} + 6 \, f^{2}\right )} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{60 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 282, normalized size = 1.11 \begin {gather*} \frac {12 \, d f^{2} p x^{8} e^{4} - 6 \, {\left (5 \, d^{2} f g p x^{8} + d^{2} f^{2} p x^{6}\right )} e^{3} + {\left (20 \, d^{3} g^{2} p x^{8} + 15 \, d^{3} f g p x^{6} + 4 \, d^{3} f^{2} p x^{4}\right )} e^{2} - {\left (10 \, d^{4} g^{2} p x^{6} + 10 \, d^{4} f g p x^{4} + 3 \, d^{4} f^{2} p x^{2}\right )} e - 2 \, {\left (10 \, d^{2} g^{2} p x^{10} e^{3} - 15 \, d f g p x^{10} e^{4} + 6 \, f^{2} p x^{10} e^{5} + 10 \, d^{5} g^{2} p x^{4} + 15 \, d^{5} f g p x^{2} + 6 \, d^{5} f^{2} p\right )} \log \left (x^{2} e + 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 ) + 4 \, {\left (10 \, d^{2} g^{2} p x^{10} e^{3} - 15 \, d f g p x^{10} e^{4} + 6 \, f^{2} p x^{10} e^{5}\right )} \log \left (x\right )}{120 \, d^{5} x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(12*d*f^2*p*x^8*e^4 - 6*(5*d^2*f*g*p*x^8 + d^2*f^2*p*x^6)*e^3 + (20*d^3*g^2*p*x^8 + 15*d^3*f*g*p*x^6 + 4
*d^3*f^2*p*x^4)*e^2 - (10*d^4*g^2*p*x^6 + 10*d^4*f*g*p*x^4 + 3*d^4*f^2*p*x^2)*e - 2*(10*d^2*g^2*p*x^10*e^3 - 1
5*d*f*g*p*x^10*e^4 + 6*f^2*p*x^10*e^5 + 10*d^5*g^2*p*x^4 + 15*d^5*f*g*p*x^2 + 6*d^5*f^2*p)*log(x^2*e + d) - 2*
(10*d^5*g^2*x^4 + 15*d^5*f*g*x^2 + 6*d^5*f^2)*log(c) + 4*(10*d^2*g^2*p*x^10*e^3 - 15*d*f*g*p*x^10*e^4 + 6*f^2*
p*x^10*e^5)*log(x))/(d^5*x^10)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1340 vs. \(2 (240) = 480\).
time = 5.51, size = 1340, normalized size = 5.30 \begin {gather*} -\frac {{\left (20 \, {\left (x^{2} e + d\right )}^{5} d^{2} g^{2} p e^{4} \log \left (x^{2} e + d\right ) - 100 \, {\left (x^{2} e + d\right )}^{4} d^{3} g^{2} p e^{4} \log \left (x^{2} e + d\right ) + 200 \, {\left (x^{2} e + d\right )}^{3} d^{4} g^{2} p e^{4} \log \left (x^{2} e + d\right ) - 180 \, {\left (x^{2} e + d\right )}^{2} d^{5} g^{2} p e^{4} \log \left (x^{2} e + d\right ) + 60 \, {\left (x^{2} e + d\right )} d^{6} g^{2} p e^{4} \log \left (x^{2} e + d\right ) - 20 \, {\left (x^{2} e + d\right )}^{5} d^{2} g^{2} p e^{4} \log \left (x^{2} e\right ) + 100 \, {\left (x^{2} e + d\right )}^{4} d^{3} g^{2} p e^{4} \log \left (x^{2} e\right ) - 200 \, {\left (x^{2} e + d\right )}^{3} d^{4} g^{2} p e^{4} \log \left (x^{2} e\right ) + 200 \, {\left (x^{2} e + d\right )}^{2} d^{5} g^{2} p e^{4} \log \left (x^{2} e\right ) - 100 \, {\left (x^{2} e + d\right )} d^{6} g^{2} p e^{4} \log \left (x^{2} e\right ) + 20 \, d^{7} g^{2} p e^{4} \log \left (x^{2} e\right ) - 20 \, {\left (x^{2} e + d\right )}^{4} d^{3} g^{2} p e^{4} + 90 \, {\left (x^{2} e + d\right )}^{3} d^{4} g^{2} p e^{4} - 150 \, {\left (x^{2} e + d\right )}^{2} d^{5} g^{2} p e^{4} + 110 \, {\left (x^{2} e + d\right )} d^{6} g^{2} p e^{4} - 30 \, d^{7} g^{2} p e^{4} - 30 \, {\left (x^{2} e + d\right )}^{5} d f g p e^{5} \log \left (x^{2} e + d\right ) + 150 \, {\left (x^{2} e + d\right )}^{4} d^{2} f g p e^{5} \log \left (x^{2} e + d\right ) - 300 \, {\left (x^{2} e + d\right )}^{3} d^{3} f g p e^{5} \log \left (x^{2} e + d\right ) + 300 \, {\left (x^{2} e + d\right )}^{2} d^{4} f g p e^{5} \log \left (x^{2} e + d\right ) - 120 \, {\left (x^{2} e + d\right )} d^{5} f g p e^{5} \log \left (x^{2} e + d\right ) + 30 \, {\left (x^{2} e + d\right )}^{5} d f g p e^{5} \log \left (x^{2} e\right ) - 150 \, {\left (x^{2} e + d\right )}^{4} d^{2} f g p e^{5} \log \left (x^{2} e\right ) + 300 \, {\left (x^{2} e + d\right )}^{3} d^{3} f g p e^{5} \log \left (x^{2} e\right ) - 300 \, {\left (x^{2} e + d\right )}^{2} d^{4} f g p e^{5} \log \left (x^{2} e\right ) + 150 \, {\left (x^{2} e + d\right )} d^{5} f g p e^{5} \log \left (x^{2} e\right ) - 30 \, d^{6} f g p e^{5} \log \left (x^{2} e\right ) + 20 \, {\left (x^{2} e + d\right )}^{2} d^{5} g^{2} e^{4} \log \left (c\right ) - 40 \, {\left (x^{2} e + d\right )} d^{6} g^{2} e^{4} \log \left (c\right ) + 20 \, d^{7} g^{2} e^{4} \log \left (c\right ) + 30 \, {\left (x^{2} e + d\right )}^{4} d^{2} f g p e^{5} - 135 \, {\left (x^{2} e + d\right )}^{3} d^{3} f g p e^{5} + 235 \, {\left (x^{2} e + d\right )}^{2} d^{4} f g p e^{5} - 185 \, {\left (x^{2} e + d\right )} d^{5} f g p e^{5} + 55 \, d^{6} f g p e^{5} + 12 \, {\left (x^{2} e + d\right )}^{5} f^{2} p e^{6} \log \left (x^{2} e + d\right ) - 60 \, {\left (x^{2} e + d\right )}^{4} d f^{2} p e^{6} \log \left (x^{2} e + d\right ) + 120 \, {\left (x^{2} e + d\right )}^{3} d^{2} f^{2} p e^{6} \log \left (x^{2} e + d\right ) - 120 \, {\left (x^{2} e + d\right )}^{2} d^{3} f^{2} p e^{6} \log \left (x^{2} e + d\right ) + 60 \, {\left (x^{2} e + d\right )} d^{4} f^{2} p e^{6} \log \left (x^{2} e + d\right ) - 12 \, {\left (x^{2} e + d\right )}^{5} f^{2} p e^{6} \log \left (x^{2} e\right ) + 60 \, {\left (x^{2} e + d\right )}^{4} d f^{2} p e^{6} \log \left (x^{2} e\right ) - 120 \, {\left (x^{2} e + d\right )}^{3} d^{2} f^{2} p e^{6} \log \left (x^{2} e\right ) + 120 \, {\left (x^{2} e + d\right )}^{2} d^{3} f^{2} p e^{6} \log \left (x^{2} e\right ) - 60 \, {\left (x^{2} e + d\right )} d^{4} f^{2} p e^{6} \log \left (x^{2} e\right ) + 12 \, d^{5} f^{2} p e^{6} \log \left (x^{2} e\right ) + 30 \, {\left (x^{2} e + d\right )} d^{5} f g e^{5} \log \left (c\right ) - 30 \, d^{6} f g e^{5} \log \left (c\right ) - 12 \, {\left (x^{2} e + d\right )}^{4} d f^{2} p e^{6} + 54 \, {\left (x^{2} e + d\right )}^{3} d^{2} f^{2} p e^{6} - 94 \, {\left (x^{2} e + d\right )}^{2} d^{3} f^{2} p e^{6} + 77 \, {\left (x^{2} e + d\right )} d^{4} f^{2} p e^{6} - 25 \, d^{5} f^{2} p e^{6} + 12 \, d^{5} f^{2} e^{6} \log \left (c\right )\right )} e^{\left (-1\right )}}{120 \, {\left ({\left (x^{2} e + d\right )}^{5} d^{5} - 5 \, {\left (x^{2} e + d\right )}^{4} d^{6} + 10 \, {\left (x^{2} e + d\right )}^{3} d^{7} - 10 \, {\left (x^{2} e + d\right )}^{2} d^{8} + 5 \, {\left (x^{2} e + d\right )} d^{9} - d^{10}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(20*(x^2*e + d)^5*d^2*g^2*p*e^4*log(x^2*e + d) - 100*(x^2*e + d)^4*d^3*g^2*p*e^4*log(x^2*e + d) + 200*(
x^2*e + d)^3*d^4*g^2*p*e^4*log(x^2*e + d) - 180*(x^2*e + d)^2*d^5*g^2*p*e^4*log(x^2*e + d) + 60*(x^2*e + d)*d^
6*g^2*p*e^4*log(x^2*e + d) - 20*(x^2*e + d)^5*d^2*g^2*p*e^4*log(x^2*e) + 100*(x^2*e + d)^4*d^3*g^2*p*e^4*log(x
^2*e) - 200*(x^2*e + d)^3*d^4*g^2*p*e^4*log(x^2*e) + 200*(x^2*e + d)^2*d^5*g^2*p*e^4*log(x^2*e) - 100*(x^2*e +
d)*d^6*g^2*p*e^4*log(x^2*e) + 20*d^7*g^2*p*e^4*log(x^2*e) - 20*(x^2*e + d)^4*d^3*g^2*p*e^4 + 90*(x^2*e + d)^3
*d^4*g^2*p*e^4 - 150*(x^2*e + d)^2*d^5*g^2*p*e^4 + 110*(x^2*e + d)*d^6*g^2*p*e^4 - 30*d^7*g^2*p*e^4 - 30*(x^2*
e + d)^5*d*f*g*p*e^5*log(x^2*e + d) + 150*(x^2*e + d)^4*d^2*f*g*p*e^5*log(x^2*e + d) - 300*(x^2*e + d)^3*d^3*f
*g*p*e^5*log(x^2*e + d) + 300*(x^2*e + d)^2*d^4*f*g*p*e^5*log(x^2*e + d) - 120*(x^2*e + d)*d^5*f*g*p*e^5*log(x
^2*e + d) + 30*(x^2*e + d)^5*d*f*g*p*e^5*log(x^2*e) - 150*(x^2*e + d)^4*d^2*f*g*p*e^5*log(x^2*e) + 300*(x^2*e
+ d)^3*d^3*f*g*p*e^5*log(x^2*e) - 300*(x^2*e + d)^2*d^4*f*g*p*e^5*log(x^2*e) + 150*(x^2*e + d)*d^5*f*g*p*e^5*l
og(x^2*e) - 30*d^6*f*g*p*e^5*log(x^2*e) + 20*(x^2*e + d)^2*d^5*g^2*e^4*log(c) - 40*(x^2*e + d)*d^6*g^2*e^4*log
(c) + 20*d^7*g^2*e^4*log(c) + 30*(x^2*e + d)^4*d^2*f*g*p*e^5 - 135*(x^2*e + d)^3*d^3*f*g*p*e^5 + 235*(x^2*e +
d)^2*d^4*f*g*p*e^5 - 185*(x^2*e + d)*d^5*f*g*p*e^5 + 55*d^6*f*g*p*e^5 + 12*(x^2*e + d)^5*f^2*p*e^6*log(x^2*e +
d) - 60*(x^2*e + d)^4*d*f^2*p*e^6*log(x^2*e + d) + 120*(x^2*e + d)^3*d^2*f^2*p*e^6*log(x^2*e + d) - 120*(x^2*
e + d)^2*d^3*f^2*p*e^6*log(x^2*e + d) + 60*(x^2*e + d)*d^4*f^2*p*e^6*log(x^2*e + d) - 12*(x^2*e + d)^5*f^2*p*e
^6*log(x^2*e) + 60*(x^2*e + d)^4*d*f^2*p*e^6*log(x^2*e) - 120*(x^2*e + d)^3*d^2*f^2*p*e^6*log(x^2*e) + 120*(x^
2*e + d)^2*d^3*f^2*p*e^6*log(x^2*e) - 60*(x^2*e + d)*d^4*f^2*p*e^6*log(x^2*e) + 12*d^5*f^2*p*e^6*log(x^2*e) +
30*(x^2*e + d)*d^5*f*g*e^5*log(c) - 30*d^6*f*g*e^5*log(c) - 12*(x^2*e + d)^4*d*f^2*p*e^6 + 54*(x^2*e + d)^3*d^
2*f^2*p*e^6 - 94*(x^2*e + d)^2*d^3*f^2*p*e^6 + 77*(x^2*e + d)*d^4*f^2*p*e^6 - 25*d^5*f^2*p*e^6 + 12*d^5*f^2*e^
6*log(c))*e^(-1)/((x^2*e + d)^5*d^5 - 5*(x^2*e + d)^4*d^6 + 10*(x^2*e + d)^3*d^7 - 10*(x^2*e + d)^2*d^8 + 5*(x
^2*e + d)*d^9 - d^10)

________________________________________________________________________________________

Mupad [B]
time = 0.46, size = 225, normalized size = 0.89 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________