\(\int \frac {(f+g x^2)^2 \log (c (d+e x^2)^p)}{x^9} \, dx\) [330]

Optimal result

Integrand size = 25, antiderivative size = 216 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=-\frac {e f^2 p}{24 d x^6}+\frac {e f (3 e f-8 d g) p}{48 d^2 x^4}-\frac {e \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p}{24 d^3 x^2}-\frac {e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p \log (x)}{12 d^4}+\frac {e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )}{24 d^4}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4} \]

[Out]

-1/24*e*f^2*p/d/x^6+1/48*e*f*(-8*d*g+3*e*f)*p/d^2/x^4-1/24*e*(6*d^2*g^2-8*d*e*f*g+3*e^2*f^2)*p/d^3/x^2-1/12*e^
2*(6*d^2*g^2-8*d*e*f*g+3*e^2*f^2)*p*ln(x)/d^4+1/24*e^2*(6*d^2*g^2-8*d*e*f*g+3*e^2*f^2)*p*ln(e*x^2+d)/d^4-1/8*f
^2*ln(c*(e*x^2+d)^p)/x^8-1/3*f*g*ln(c*(e*x^2+d)^p)/x^6-1/4*g^2*ln(c*(e*x^2+d)^p)/x^4

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 216, 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} \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}+\frac {e f p (3 e f-8 d g)}{48 d^2 x^4}+\frac {e^2 p \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right ) \log \left (d+e x^2\right )}{24 d^4}-\frac {e^2 p \log (x) \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )}{12 d^4}-\frac {e p \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )}{24 d^3 x^2}-\frac {e f^2 p}{24 d x^6} \]

[In]

Int[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^9,x]

[Out]

-1/24*(e*f^2*p)/(d*x^6) + (e*f*(3*e*f - 8*d*g)*p)/(48*d^2*x^4) - (e*(3*e^2*f^2 - 8*d*e*f*g + 6*d^2*g^2)*p)/(24
*d^3*x^2) - (e^2*(3*e^2*f^2 - 8*d*e*f*g + 6*d^2*g^2)*p*Log[x])/(12*d^4) + (e^2*(3*e^2*f^2 - 8*d*e*f*g + 6*d^2*
g^2)*p*Log[d + e*x^2])/(24*d^4) - (f^2*Log[c*(d + e*x^2)^p])/(8*x^8) - (f*g*Log[c*(d + e*x^2)^p])/(3*x^6) - (g
^2*Log[c*(d + e*x^2)^p])/(4*x^4)

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^5} \, dx,x,x^2\right ) \\ & = -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {1}{2} (e p) \text {Subst}\left (\int \frac {-3 f^2-8 f g x-6 g^2 x^2}{12 x^4 (d+e x)} \, dx,x,x^2\right ) \\ & = -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {1}{24} (e p) \text {Subst}\left (\int \frac {-3 f^2-8 f g x-6 g^2 x^2}{x^4 (d+e x)} \, dx,x,x^2\right ) \\ & = -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {1}{24} (e p) \text {Subst}\left (\int \left (-\frac {3 f^2}{d x^4}-\frac {f (-3 e f+8 d g)}{d^2 x^3}+\frac {-3 e^2 f^2+8 d e f g-6 d^2 g^2}{d^3 x^2}+\frac {e \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right )}{d^4 x}-\frac {e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right )}{d^4 (d+e x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {e f^2 p}{24 d x^6}+\frac {e f (3 e f-8 d g) p}{48 d^2 x^4}-\frac {e \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p}{24 d^3 x^2}-\frac {e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p \log (x)}{12 d^4}+\frac {e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )}{24 d^4}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.85 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=-\frac {d e p x^2 \left (6 e^2 f^2 x^4-d e f x^2 \left (3 f+16 g x^2\right )+2 d^2 \left (f^2+4 f g x^2+6 g^2 x^4\right )\right )+4 e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p x^8 \log (x)-2 e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p x^8 \log \left (d+e x^2\right )+2 d^4 \left (3 f^2+8 f g x^2+6 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )}{48 d^4 x^8} \]

[In]

Integrate[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^9,x]

[Out]

-1/48*(d*e*p*x^2*(6*e^2*f^2*x^4 - d*e*f*x^2*(3*f + 16*g*x^2) + 2*d^2*(f^2 + 4*f*g*x^2 + 6*g^2*x^4)) + 4*e^2*(3
*e^2*f^2 - 8*d*e*f*g + 6*d^2*g^2)*p*x^8*Log[x] - 2*e^2*(3*e^2*f^2 - 8*d*e*f*g + 6*d^2*g^2)*p*x^8*Log[d + e*x^2
] + 2*d^4*(3*f^2 + 8*f*g*x^2 + 6*g^2*x^4)*Log[c*(d + e*x^2)^p])/(d^4*x^8)

Maple [A] (verified)

Time = 2.03 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.89

method	result	size
parts	\(-\frac {g^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4 x^{4}}-\frac {f g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{3 x^{6}}-\frac {f^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{8 x^{8}}-\frac {p e \left (-\frac {-6 g^{2} d^{2}+8 d e f g -3 e^{2} f^{2}}{2 d^{3} x^{2}}+\frac {\left (6 g^{2} d^{2}-8 d e f g +3 e^{2} f^{2}\right ) e \ln \left (x \right )}{d^{4}}+\frac {f^{2}}{2 d \,x^{6}}+\frac {f \left (8 d g -3 e f \right )}{4 d^{2} x^{4}}-\frac {e \left (6 g^{2} d^{2}-8 d e f g +3 e^{2} f^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 d^{4}}\right )}{12}\)	\(193\)
parallelrisch	\(-\frac {24 \ln \left (x \right ) x^{8} d^{2} e^{2} g^{2} p^{2}-32 \ln \left (x \right ) x^{8} d \,e^{3} f g \,p^{2}+12 \ln \left (x \right ) x^{8} e^{4} f^{2} p^{2}-12 x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{2} e^{2} g^{2} p +16 x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d \,e^{3} f g p -6 x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{4} f^{2} p -12 x^{8} d^{2} e^{2} g^{2} p^{2}+16 x^{8} d \,e^{3} f g \,p^{2}-6 x^{8} e^{4} f^{2} p^{2}+12 x^{6} d^{3} e \,g^{2} p^{2}-16 x^{6} d^{2} e^{2} f g \,p^{2}+6 x^{6} d \,e^{3} f^{2} p^{2}+12 x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{4} g^{2} p +8 x^{4} d^{3} e f g \,p^{2}-3 x^{4} d^{2} e^{2} f^{2} p^{2}+16 x^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{4} f g p +2 x^{2} d^{3} e \,f^{2} p^{2}+6 \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{4} f^{2} p}{48 x^{8} p \,d^{4}}\)	\(344\)
risch	\(-\frac {\left (6 g^{2} x^{4}+8 f g \,x^{2}+3 f^{2}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{24 x^{8}}+\frac {-16 \ln \left (-e \,x^{2}-d \right ) d \,e^{3} f g p \,x^{8}+32 \ln \left (x \right ) d \,e^{3} f g p \,x^{8}-6 \ln \left (c \right ) d^{4} f^{2}+6 i \pi \,d^{4} 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 )+12 \ln \left (-e \,x^{2}-d \right ) d^{2} e^{2} g^{2} p \,x^{8}-24 \ln \left (x \right ) d^{2} e^{2} g^{2} p \,x^{8}-12 d^{3} e \,g^{2} p \,x^{6}-6 d \,e^{3} f^{2} p \,x^{6}+3 d^{2} e^{2} f^{2} p \,x^{4}-2 d^{3} e \,f^{2} p \,x^{2}+16 d^{2} e^{2} f g p \,x^{6}-8 d^{3} e f g p \,x^{4}+6 \ln \left (-e \,x^{2}-d \right ) e^{4} f^{2} p \,x^{8}-12 \ln \left (x \right ) e^{4} f^{2} p \,x^{8}-16 \ln \left (c \right ) d^{4} f g \,x^{2}+8 i \pi \,d^{4} 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 )-8 i \pi \,d^{4} 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}-8 i \pi \,d^{4} f g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-12 \ln \left (c \right ) d^{4} g^{2} x^{4}+3 i \pi \,d^{4} 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 )+8 i \pi \,d^{4} f g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-6 i \pi \,d^{4} 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^{4} g^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+3 i \pi \,d^{4} f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+6 i \pi \,d^{4} g^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-3 i \pi \,d^{4} 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}-3 i \pi \,d^{4} f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{48 d^{4} x^{8}}\)	\(713\)

[In]

int((g*x^2+f)^2*ln(c*(e*x^2+d)^p)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/4*g^2*ln(c*(e*x^2+d)^p)/x^4-1/3*f*g*ln(c*(e*x^2+d)^p)/x^6-1/8*f^2*ln(c*(e*x^2+d)^p)/x^8-1/12*p*e*(-1/2*(-6*
d^2*g^2+8*d*e*f*g-3*e^2*f^2)/d^3/x^2+(6*d^2*g^2-8*d*e*f*g+3*e^2*f^2)/d^4*e*ln(x)+1/2*f^2/d/x^6+1/4*f*(8*d*g-3*
e*f)/d^2/x^4-1/2*e*(6*d^2*g^2-8*d*e*f*g+3*e^2*f^2)/d^4*ln(e*x^2+d))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.06 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=-\frac {4 \, {\left (3 \, e^{4} f^{2} - 8 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} p x^{8} \log \left (x\right ) + 2 \, d^{3} e f^{2} p x^{2} + 2 \, {\left (3 \, d e^{3} f^{2} - 8 \, d^{2} e^{2} f g + 6 \, d^{3} e g^{2}\right )} p x^{6} - {\left (3 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g\right )} p x^{4} + 2 \, {\left (6 \, d^{4} g^{2} p x^{4} - {\left (3 \, e^{4} f^{2} - 8 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} p x^{8} + 8 \, d^{4} f g p x^{2} + 3 \, d^{4} f^{2} p\right )} \log \left (e x^{2} + d\right ) + 2 \, {\left (6 \, d^{4} g^{2} x^{4} + 8 \, d^{4} f g x^{2} + 3 \, d^{4} f^{2}\right )} \log \left (c\right )}{48 \, d^{4} x^{8}} \]

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^9,x, algorithm="fricas")

[Out]

-1/48*(4*(3*e^4*f^2 - 8*d*e^3*f*g + 6*d^2*e^2*g^2)*p*x^8*log(x) + 2*d^3*e*f^2*p*x^2 + 2*(3*d*e^3*f^2 - 8*d^2*e
^2*f*g + 6*d^3*e*g^2)*p*x^6 - (3*d^2*e^2*f^2 - 8*d^3*e*f*g)*p*x^4 + 2*(6*d^4*g^2*p*x^4 - (3*e^4*f^2 - 8*d*e^3*
f*g + 6*d^2*e^2*g^2)*p*x^8 + 8*d^4*f*g*p*x^2 + 3*d^4*f^2*p)*log(e*x^2 + d) + 2*(6*d^4*g^2*x^4 + 8*d^4*f*g*x^2
+ 3*d^4*f^2)*log(c))/(d^4*x^8)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=\text {Timed out} \]

[In]

integrate((g*x**2+f)**2*ln(c*(e*x**2+d)**p)/x**9,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.85 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=\frac {1}{48} \, e p {\left (\frac {2 \, {\left (3 \, e^{3} f^{2} - 8 \, d e^{2} f g + 6 \, d^{2} e g^{2}\right )} \log \left (e x^{2} + d\right )}{d^{4}} - \frac {2 \, {\left (3 \, e^{3} f^{2} - 8 \, d e^{2} f g + 6 \, d^{2} e g^{2}\right )} \log \left (x^{2}\right )}{d^{4}} - \frac {2 \, {\left (3 \, e^{2} f^{2} - 8 \, d e f g + 6 \, d^{2} g^{2}\right )} x^{4} + 2 \, d^{2} f^{2} - {\left (3 \, d e f^{2} - 8 \, d^{2} f g\right )} x^{2}}{d^{3} x^{6}}\right )} - \frac {{\left (6 \, g^{2} x^{4} + 8 \, f g x^{2} + 3 \, f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{24 \, x^{8}} \]

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^9,x, algorithm="maxima")

[Out]

1/48*e*p*(2*(3*e^3*f^2 - 8*d*e^2*f*g + 6*d^2*e*g^2)*log(e*x^2 + d)/d^4 - 2*(3*e^3*f^2 - 8*d*e^2*f*g + 6*d^2*e*
g^2)*log(x^2)/d^4 - (2*(3*e^2*f^2 - 8*d*e*f*g + 6*d^2*g^2)*x^4 + 2*d^2*f^2 - (3*d*e*f^2 - 8*d^2*f*g)*x^2)/(d^3
*x^6)) - 1/24*(6*g^2*x^4 + 8*f*g*x^2 + 3*f^2)*log((e*x^2 + d)^p*c)/x^8

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (200) = 400\).

Time = 0.32 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.80 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=-\frac {\frac {2 \, {\left (3 \, e^{5} f^{2} p + 8 \, {\left (e x^{2} + d\right )} e^{4} f g p - 8 \, d e^{4} f g p + 6 \, {\left (e x^{2} + d\right )}^{2} e^{3} g^{2} p - 12 \, {\left (e x^{2} + d\right )} d e^{3} g^{2} p + 6 \, d^{2} e^{3} g^{2} p\right )} \log \left (e x^{2} + d\right )}{{\left (e x^{2} + d\right )}^{4} - 4 \, {\left (e x^{2} + d\right )}^{3} d + 6 \, {\left (e x^{2} + d\right )}^{2} d^{2} - 4 \, {\left (e x^{2} + d\right )} d^{3} + d^{4}} + \frac {6 \, {\left (e x^{2} + d\right )}^{3} e^{5} f^{2} p - 21 \, {\left (e x^{2} + d\right )}^{2} d e^{5} f^{2} p + 26 \, {\left (e x^{2} + d\right )} d^{2} e^{5} f^{2} p - 11 \, d^{3} e^{5} f^{2} p - 16 \, {\left (e x^{2} + d\right )}^{3} d e^{4} f g p + 56 \, {\left (e x^{2} + d\right )}^{2} d^{2} e^{4} f g p - 64 \, {\left (e x^{2} + d\right )} d^{3} e^{4} f g p + 24 \, d^{4} e^{4} f g p + 12 \, {\left (e x^{2} + d\right )}^{3} d^{2} e^{3} g^{2} p - 36 \, {\left (e x^{2} + d\right )}^{2} d^{3} e^{3} g^{2} p + 36 \, {\left (e x^{2} + d\right )} d^{4} e^{3} g^{2} p - 12 \, d^{5} e^{3} g^{2} p + 6 \, d^{3} e^{5} f^{2} \log \left (c\right ) + 16 \, {\left (e x^{2} + d\right )} d^{3} e^{4} f g \log \left (c\right ) - 16 \, d^{4} e^{4} f g \log \left (c\right ) + 12 \, {\left (e x^{2} + d\right )}^{2} d^{3} e^{3} g^{2} \log \left (c\right ) - 24 \, {\left (e x^{2} + d\right )} d^{4} e^{3} g^{2} \log \left (c\right ) + 12 \, d^{5} e^{3} g^{2} \log \left (c\right )}{{\left (e x^{2} + d\right )}^{4} d^{3} - 4 \, {\left (e x^{2} + d\right )}^{3} d^{4} + 6 \, {\left (e x^{2} + d\right )}^{2} d^{5} - 4 \, {\left (e x^{2} + d\right )} d^{6} + d^{7}} - \frac {2 \, {\left (3 \, e^{5} f^{2} p - 8 \, d e^{4} f g p + 6 \, d^{2} e^{3} g^{2} p\right )} \log \left (e x^{2} + d\right )}{d^{4}} + \frac {2 \, {\left (3 \, e^{5} f^{2} p - 8 \, d e^{4} f g p + 6 \, d^{2} e^{3} g^{2} p\right )} \log \left (e x^{2}\right )}{d^{4}}}{48 \, e} \]

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^9,x, algorithm="giac")

[Out]

-1/48*(2*(3*e^5*f^2*p + 8*(e*x^2 + d)*e^4*f*g*p - 8*d*e^4*f*g*p + 6*(e*x^2 + d)^2*e^3*g^2*p - 12*(e*x^2 + d)*d
*e^3*g^2*p + 6*d^2*e^3*g^2*p)*log(e*x^2 + d)/((e*x^2 + d)^4 - 4*(e*x^2 + d)^3*d + 6*(e*x^2 + d)^2*d^2 - 4*(e*x
^2 + d)*d^3 + d^4) + (6*(e*x^2 + d)^3*e^5*f^2*p - 21*(e*x^2 + d)^2*d*e^5*f^2*p + 26*(e*x^2 + d)*d^2*e^5*f^2*p
- 11*d^3*e^5*f^2*p - 16*(e*x^2 + d)^3*d*e^4*f*g*p + 56*(e*x^2 + d)^2*d^2*e^4*f*g*p - 64*(e*x^2 + d)*d^3*e^4*f*
g*p + 24*d^4*e^4*f*g*p + 12*(e*x^2 + d)^3*d^2*e^3*g^2*p - 36*(e*x^2 + d)^2*d^3*e^3*g^2*p + 36*(e*x^2 + d)*d^4*
e^3*g^2*p - 12*d^5*e^3*g^2*p + 6*d^3*e^5*f^2*log(c) + 16*(e*x^2 + d)*d^3*e^4*f*g*log(c) - 16*d^4*e^4*f*g*log(c
) + 12*(e*x^2 + d)^2*d^3*e^3*g^2*log(c) - 24*(e*x^2 + d)*d^4*e^3*g^2*log(c) + 12*d^5*e^3*g^2*log(c))/((e*x^2 +
d)^4*d^3 - 4*(e*x^2 + d)^3*d^4 + 6*(e*x^2 + d)^2*d^5 - 4*(e*x^2 + d)*d^6 + d^7) - 2*(3*e^5*f^2*p - 8*d*e^4*f*
g*p + 6*d^2*e^3*g^2*p)*log(e*x^2 + d)/d^4 + 2*(3*e^5*f^2*p - 8*d*e^4*f*g*p + 6*d^2*e^3*g^2*p)*log(e*x^2)/d^4)/
e

Mupad [B] (verification not implemented)

Time = 1.69 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.88 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx=\frac {\ln \left (e\,x^2+d\right )\,\left (6\,p\,d^2\,e^2\,g^2-8\,p\,d\,e^3\,f\,g+3\,p\,e^4\,f^2\right )}{24\,d^4}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2}{8}+\frac {f\,g\,x^2}{3}+\frac {g^2\,x^4}{4}\right )}{x^8}-\frac {\frac {e\,f^2\,p}{2\,d}+\frac {e\,p\,x^4\,\left (6\,d^2\,g^2-8\,d\,e\,f\,g+3\,e^2\,f^2\right )}{2\,d^3}+\frac {e\,f\,p\,x^2\,\left (8\,d\,g-3\,e\,f\right )}{4\,d^2}}{12\,x^6}-\frac {\ln \left (x\right )\,\left (6\,p\,d^2\,e^2\,g^2-8\,p\,d\,e^3\,f\,g+3\,p\,e^4\,f^2\right )}{12\,d^4} \]

[In]

int((log(c*(d + e*x^2)^p)*(f + g*x^2)^2)/x^9,x)

[Out]

(log(d + e*x^2)*(3*e^4*f^2*p + 6*d^2*e^2*g^2*p - 8*d*e^3*f*g*p))/(24*d^4) - (log(c*(d + e*x^2)^p)*(f^2/8 + (g^
2*x^4)/4 + (f*g*x^2)/3))/x^8 - ((e*f^2*p)/(2*d) + (e*p*x^4*(6*d^2*g^2 + 3*e^2*f^2 - 8*d*e*f*g))/(2*d^3) + (e*f
*p*x^2*(8*d*g - 3*e*f))/(4*d^2))/(12*x^6) - (log(x)*(3*e^4*f^2*p + 6*d^2*e^2*g^2*p - 8*d*e^3*f*g*p))/(12*d^4)