\(\int \frac {(f+g x^2)^2 \log (c (d+e x^2)^p)}{x^7} \, dx\) [329]

Optimal result

Integrand size = 25, antiderivative size = 130 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {e f^2 p}{12 d x^4}+\frac {e f (e f-3 d g) p}{6 d^2 x^2}+\frac {e \left (e^2 f^2-3 d e f g+3 d^2 g^2\right ) p \log (x)}{3 d^3}-\frac {(e f-d g)^3 p \log \left (d+e x^2\right )}{6 d^3 f}-\frac {\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{6 f x^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 130, 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, 37, 2461, 12, 90} \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{6 f x^6}-\frac {p (e f-d g)^3 \log \left (d+e x^2\right )}{6 d^3 f}+\frac {e f p (e f-3 d g)}{6 d^2 x^2}+\frac {e p \log (x) \left (3 d^2 g^2-3 d e f g+e^2 f^2\right )}{3 d^3}-\frac {e f^2 p}{12 d x^4} \]

[In]

Int[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^7,x]

[Out]

-1/12*(e*f^2*p)/(d*x^4) + (e*f*(e*f - 3*d*g)*p)/(6*d^2*x^2) + (e*(e^2*f^2 - 3*d*e*f*g + 3*d^2*g^2)*p*Log[x])/(
3*d^3) - ((e*f - d*g)^3*p*Log[d + e*x^2])/(6*d^3*f) - ((f + g*x^2)^3*Log[c*(d + e*x^2)^p])/(6*f*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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.08 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {d e f p x^2 \left (-2 e f x^2+d \left (f+6 g x^2\right )\right )-4 e \left (e^2 f^2-3 d e f g+3 d^2 g^2\right ) p x^6 \log (x)+2 e \left (e^2 f^2-3 d e f g+3 d^2 g^2\right ) p x^6 \log \left (d+e x^2\right )+2 d^3 \left (f^2+3 f g x^2+3 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )}{12 d^3 x^6} \]

[In]

Integrate[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^7,x]

[Out]

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

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.22

method	result	size
parts	\(-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) g^{2}}{2 x^{2}}-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f g}{2 x^{4}}-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f^{2}}{6 x^{6}}-\frac {p e \left (\frac {\left (-3 g^{2} d^{2}+3 d e f g -e^{2} f^{2}\right ) \ln \left (x \right )}{d^{3}}+\frac {f^{2}}{4 d \,x^{4}}+\frac {f \left (3 d g -e f \right )}{2 d^{2} x^{2}}+\frac {\left (3 g^{2} d^{2}-3 d e f g +e^{2} f^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 d^{3}}\right )}{3}\)	\(159\)
parallelrisch	\(\frac {12 \ln \left (x \right ) x^{6} d^{2} e^{2} g^{2} p^{2}-12 \ln \left (x \right ) x^{6} d \,e^{3} f g \,p^{2}+4 \ln \left (x \right ) x^{6} e^{4} f^{2} p^{2}-6 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{2} e^{2} g^{2} p +6 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d \,e^{3} f g p -2 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{4} f^{2} p -6 x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{3} e \,g^{2} p -6 x^{4} d^{2} e^{2} f g \,p^{2}+2 x^{4} d \,e^{3} f^{2} p^{2}-6 x^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{3} e f g p -x^{2} d^{2} e^{2} f^{2} p^{2}-2 \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{3} e \,f^{2} p}{12 x^{6} d^{3} e p}\)	\(261\)
risch	\(-\frac {\left (3 g^{2} x^{4}+3 f g \,x^{2}+f^{2}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{6 x^{6}}+\frac {-3 i \pi \,d^{3} f g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-i \pi \,d^{3} f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+3 i \pi \,d^{3} 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 )-3 i \pi \,d^{3} g^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+12 \ln \left (x \right ) d^{2} e \,g^{2} p \,x^{6}-12 \ln \left (x \right ) d \,e^{2} f g p \,x^{6}+4 \ln \left (x \right ) e^{3} f^{2} p \,x^{6}-6 \ln \left (e \,x^{2}+d \right ) d^{2} e \,g^{2} p \,x^{6}+6 \ln \left (e \,x^{2}+d \right ) d \,e^{2} f g p \,x^{6}-2 \ln \left (e \,x^{2}+d \right ) e^{3} f^{2} p \,x^{6}+i \pi \,d^{3} 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 )-3 i \pi \,d^{3} 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}-i \pi \,d^{3} 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^{3} 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}+3 i \pi \,d^{3} 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 )+3 i \pi \,d^{3} f g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+3 i \pi \,d^{3} g^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+i \pi \,d^{3} f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-6 \ln \left (c \right ) d^{3} g^{2} x^{4}-6 d^{2} e f g p \,x^{4}+2 d \,e^{2} f^{2} p \,x^{4}-6 \ln \left (c \right ) d^{3} f g \,x^{2}-d^{2} e \,f^{2} p \,x^{2}-2 \ln \left (c \right ) d^{3} f^{2}}{12 d^{3} x^{6}}\)	\(656\)

[In]

int((g*x^2+f)^2*ln(c*(e*x^2+d)^p)/x^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.41 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\frac {4 \, {\left (e^{3} f^{2} - 3 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} p x^{6} \log \left (x\right ) - d^{2} e f^{2} p x^{2} + 2 \, {\left (d e^{2} f^{2} - 3 \, d^{2} e f g\right )} p x^{4} - 2 \, {\left (3 \, d^{3} g^{2} p x^{4} + 3 \, d^{3} f g p x^{2} + {\left (e^{3} f^{2} - 3 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} p x^{6} + d^{3} f^{2} p\right )} \log \left (e x^{2} + d\right ) - 2 \, {\left (3 \, d^{3} g^{2} x^{4} + 3 \, d^{3} f g x^{2} + d^{3} f^{2}\right )} \log \left (c\right )}{12 \, d^{3} x^{6}} \]

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^7,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\text {Timed out} \]

[In]

integrate((g*x**2+f)**2*ln(c*(e*x**2+d)**p)/x**7,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.05 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {1}{12} \, e p {\left (\frac {2 \, {\left (e^{2} f^{2} - 3 \, d e f g + 3 \, d^{2} g^{2}\right )} \log \left (e x^{2} + d\right )}{d^{3}} - \frac {2 \, {\left (e^{2} f^{2} - 3 \, d e f g + 3 \, d^{2} g^{2}\right )} \log \left (x^{2}\right )}{d^{3}} + \frac {d f^{2} - 2 \, {\left (e f^{2} - 3 \, d f g\right )} x^{2}}{d^{2} x^{4}}\right )} - \frac {{\left (3 \, g^{2} x^{4} + 3 \, f g x^{2} + f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{6 \, x^{6}} \]

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^7,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (120) = 240\).

Time = 0.32 (sec) , antiderivative size = 464, normalized size of antiderivative = 3.57 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {\frac {2 \, {\left (e^{4} f^{2} p + 3 \, {\left (e x^{2} + d\right )} e^{3} f g p - 3 \, d e^{3} f g p + 3 \, {\left (e x^{2} + d\right )}^{2} e^{2} g^{2} p - 6 \, {\left (e x^{2} + d\right )} d e^{2} g^{2} p + 3 \, d^{2} e^{2} g^{2} p\right )} \log \left (e x^{2} + d\right )}{{\left (e x^{2} + d\right )}^{3} - 3 \, {\left (e x^{2} + d\right )}^{2} d + 3 \, {\left (e x^{2} + d\right )} d^{2} - d^{3}} - \frac {2 \, {\left (e x^{2} + d\right )}^{2} e^{4} f^{2} p - 5 \, {\left (e x^{2} + d\right )} d e^{4} f^{2} p + 3 \, d^{2} e^{4} f^{2} p - 6 \, {\left (e x^{2} + d\right )}^{2} d e^{3} f g p + 12 \, {\left (e x^{2} + d\right )} d^{2} e^{3} f g p - 6 \, d^{3} e^{3} f g p - 2 \, d^{2} e^{4} f^{2} \log \left (c\right ) - 6 \, {\left (e x^{2} + d\right )} d^{2} e^{3} f g \log \left (c\right ) + 6 \, d^{3} e^{3} f g \log \left (c\right ) - 6 \, {\left (e x^{2} + d\right )}^{2} d^{2} e^{2} g^{2} \log \left (c\right ) + 12 \, {\left (e x^{2} + d\right )} d^{3} e^{2} g^{2} \log \left (c\right ) - 6 \, d^{4} e^{2} g^{2} \log \left (c\right )}{{\left (e x^{2} + d\right )}^{3} d^{2} - 3 \, {\left (e x^{2} + d\right )}^{2} d^{3} + 3 \, {\left (e x^{2} + d\right )} d^{4} - d^{5}} + \frac {2 \, {\left (e^{4} f^{2} p - 3 \, d e^{3} f g p + 3 \, d^{2} e^{2} g^{2} p\right )} \log \left (e x^{2} + d\right )}{d^{3}} - \frac {2 \, {\left (e^{4} f^{2} p - 3 \, d e^{3} f g p + 3 \, d^{2} e^{2} g^{2} p\right )} \log \left (e x^{2}\right )}{d^{3}}}{12 \, e} \]

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^7,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.66 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.16 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\frac {\ln \left (x\right )\,\left (3\,p\,d^2\,e\,g^2-3\,p\,d\,e^2\,f\,g+p\,e^3\,f^2\right )}{3\,d^3}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2}{6}+\frac {f\,g\,x^2}{2}+\frac {g^2\,x^4}{2}\right )}{x^6}-\frac {\ln \left (e\,x^2+d\right )\,\left (3\,p\,d^2\,e\,g^2-3\,p\,d\,e^2\,f\,g+p\,e^3\,f^2\right )}{6\,d^3}-\frac {\frac {e\,f^2\,p}{4\,d}+\frac {e\,f\,p\,x^2\,\left (3\,d\,g-e\,f\right )}{2\,d^2}}{3\,x^4} \]

[In]

int((log(c*(d + e*x^2)^p)*(f + g*x^2)^2)/x^7,x)

[Out]

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