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

Optimal. Leaf size=216 \[ -\frac {e f^2 p}{24 d x^6}+\frac {e f (3 e f-8 d g) p}{48 d^2 x^4}-\frac {e \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p}{24 d^3 x^2}-\frac {e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p \log (x)}{12 d^4}+\frac {e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )}{24 d^4}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4} \]

[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

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Rubi [A]
time = 0.20, antiderivative size = 216, 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 )}{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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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}{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*}

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Mathematica [A]
time = 0.12, size = 184, normalized size = 0.85 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.50, size = 713, normalized size = 3.30

method	result	size
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 {-6 i \pi \,d^{4} 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 )+8 i \pi \,d^{4} 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}+8 i \pi \,d^{4} f g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+12 \ln \left (c \right ) d^{4} g^{2} x^{4}+6 \ln \left (c \right ) d^{4} f^{2}-3 i \pi \,d^{4} f^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+16 \ln \left (c \right ) d^{4} f g \,x^{2}-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}+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 \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}-16 d^{2} e^{2} f g p \,x^{6}+8 d^{3} e f g p \,x^{4}-8 i \pi \,d^{4} 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 )+6 i \pi \,d^{4} g^{2} x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-8 i \pi \,d^{4} f g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}-3 i \pi \,d^{4} 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 )+6 i \pi \,d^{4} 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}-6 i \pi \,d^{4} g^{2} x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+3 i \pi \,d^{4} 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}+3 i \pi \,d^{4} f^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {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}}{48 d^{4} x^{8}}\)	\(713\)

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^9,x,method=_RETURNVERBOSE)

[Out]

-1/24*(6*g^2*x^4+8*f*g*x^2+3*f^2)/x^8*ln((e*x^2+d)^p)-1/48*(-6*I*Pi*d^4*g^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(
e*x^2+d)^p)*csgn(I*c)+8*I*Pi*d^4*f*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+8*I*Pi*d^4*f*g*x^2*csgn(I
*c*(e*x^2+d)^p)^2*csgn(I*c)+12*ln(c)*d^4*g^2*x^4+6*ln(c)*d^4*f^2-3*I*Pi*d^4*f^2*csgn(I*c*(e*x^2+d)^p)^3+16*ln(
c)*d^4*f*g*x^2-6*ln(-e*x^2-d)*e^4*f^2*p*x^8+12*ln(x)*e^4*f^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+6*I*Pi*d^4*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-8*I*Pi*d^4*f*g*x^2*csgn(I
*c*(e*x^2+d)^p)^3-3*I*Pi*d^4*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+6*I*Pi*d^4*g^2*x^4*csgn(I
*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+16*ln(-e*x^2-d)*d*e^3*f*g*p*x^8-32*ln(x)*d*e^3*f*g*p*x^8-16*d^2*e^2*f*g*
p*x^6+8*d^3*e*f*g*p*x^4-8*I*Pi*d^4*f*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-6*I*Pi*d^4*g^2*
x^4*csgn(I*c*(e*x^2+d)^p)^3+3*I*Pi*d^4*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+3*I*Pi*d^4*f^2*csgn(I*c
*(e*x^2+d)^p)^2*csgn(I*c)-12*ln(-e*x^2-d)*d^2*e^2*g^2*p*x^8+24*ln(x)*d^2*e^2*g^2*p*x^8)/d^4/x^8

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Maxima [A]
time = 0.28, size = 184, normalized size = 0.85 \begin {gather*} \frac {1}{48} \, p {\left (\frac {2 \, {\left (6 \, d^{2} g^{2} e - 8 \, d f g e^{2} + 3 \, f^{2} e^{3}\right )} \log \left (x^{2} e + d\right )}{d^{4}} - \frac {2 \, {\left (6 \, d^{2} g^{2} e - 8 \, d f g e^{2} + 3 \, f^{2} e^{3}\right )} \log \left (x^{2}\right )}{d^{4}} - \frac {2 \, {\left (6 \, d^{2} g^{2} - 8 \, d f g e + 3 \, f^{2} e^{2}\right )} x^{4} + 2 \, d^{2} f^{2} + {\left (8 \, d^{2} f g - 3 \, d f^{2} e\right )} x^{2}}{d^{3} x^{6}}\right )} e - \frac {{\left (6 \, g^{2} x^{4} + 8 \, f g x^{2} + 3 \, f^{2}\right )} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{24 \, x^{8}} \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^9,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 243, normalized size = 1.12 \begin {gather*} -\frac {6 \, d f^{2} p x^{6} e^{3} - {\left (16 \, d^{2} f g p x^{6} + 3 \, d^{2} f^{2} p x^{4}\right )} e^{2} + 2 \, {\left (6 \, d^{3} g^{2} p x^{6} + 4 \, d^{3} f g p x^{4} + d^{3} f^{2} p x^{2}\right )} e - 2 \, {\left (6 \, d^{2} g^{2} p x^{8} e^{2} - 8 \, d f g p x^{8} e^{3} + 3 \, f^{2} p x^{8} e^{4} - 6 \, d^{4} g^{2} p x^{4} - 8 \, d^{4} f g p x^{2} - 3 \, d^{4} f^{2} p\right )} \log \left (x^{2} e + 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 ) + 4 \, {\left (6 \, d^{2} g^{2} p x^{8} e^{2} - 8 \, d f g p x^{8} e^{3} + 3 \, f^{2} p x^{8} e^{4}\right )} \log \left (x\right )}{48 \, d^{4} x^{8}} \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^9,x, algorithm="fricas")

[Out]

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

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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**9,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1089 vs. \(2 (206) = 412\).
time = 3.47, size = 1089, normalized size = 5.04 \begin {gather*} \frac {{\left (12 \, {\left (x^{2} e + d\right )}^{4} d^{2} g^{2} p e^{3} \log \left (x^{2} e + d\right ) - 48 \, {\left (x^{2} e + d\right )}^{3} d^{3} g^{2} p e^{3} \log \left (x^{2} e + d\right ) + 60 \, {\left (x^{2} e + d\right )}^{2} d^{4} g^{2} p e^{3} \log \left (x^{2} e + d\right ) - 24 \, {\left (x^{2} e + d\right )} d^{5} g^{2} p e^{3} \log \left (x^{2} e + d\right ) - 12 \, {\left (x^{2} e + d\right )}^{4} d^{2} g^{2} p e^{3} \log \left (x^{2} e\right ) + 48 \, {\left (x^{2} e + d\right )}^{3} d^{3} g^{2} p e^{3} \log \left (x^{2} e\right ) - 72 \, {\left (x^{2} e + d\right )}^{2} d^{4} g^{2} p e^{3} \log \left (x^{2} e\right ) + 48 \, {\left (x^{2} e + d\right )} d^{5} g^{2} p e^{3} \log \left (x^{2} e\right ) - 12 \, d^{6} g^{2} p e^{3} \log \left (x^{2} e\right ) - 12 \, {\left (x^{2} e + d\right )}^{3} d^{3} g^{2} p e^{3} + 36 \, {\left (x^{2} e + d\right )}^{2} d^{4} g^{2} p e^{3} - 36 \, {\left (x^{2} e + d\right )} d^{5} g^{2} p e^{3} + 12 \, d^{6} g^{2} p e^{3} - 16 \, {\left (x^{2} e + d\right )}^{4} d f g p e^{4} \log \left (x^{2} e + d\right ) + 64 \, {\left (x^{2} e + d\right )}^{3} d^{2} f g p e^{4} \log \left (x^{2} e + d\right ) - 96 \, {\left (x^{2} e + d\right )}^{2} d^{3} f g p e^{4} \log \left (x^{2} e + d\right ) + 48 \, {\left (x^{2} e + d\right )} d^{4} f g p e^{4} \log \left (x^{2} e + d\right ) + 16 \, {\left (x^{2} e + d\right )}^{4} d f g p e^{4} \log \left (x^{2} e\right ) - 64 \, {\left (x^{2} e + d\right )}^{3} d^{2} f g p e^{4} \log \left (x^{2} e\right ) + 96 \, {\left (x^{2} e + d\right )}^{2} d^{3} f g p e^{4} \log \left (x^{2} e\right ) - 64 \, {\left (x^{2} e + d\right )} d^{4} f g p e^{4} \log \left (x^{2} e\right ) + 16 \, d^{5} f g p e^{4} \log \left (x^{2} e\right ) - 12 \, {\left (x^{2} e + d\right )}^{2} d^{4} g^{2} e^{3} \log \left (c\right ) + 24 \, {\left (x^{2} e + d\right )} d^{5} g^{2} e^{3} \log \left (c\right ) - 12 \, d^{6} g^{2} e^{3} \log \left (c\right ) + 16 \, {\left (x^{2} e + d\right )}^{3} d^{2} f g p e^{4} - 56 \, {\left (x^{2} e + d\right )}^{2} d^{3} f g p e^{4} + 64 \, {\left (x^{2} e + d\right )} d^{4} f g p e^{4} - 24 \, d^{5} f g p e^{4} + 6 \, {\left (x^{2} e + d\right )}^{4} f^{2} p e^{5} \log \left (x^{2} e + d\right ) - 24 \, {\left (x^{2} e + d\right )}^{3} d f^{2} p e^{5} \log \left (x^{2} e + d\right ) + 36 \, {\left (x^{2} e + d\right )}^{2} d^{2} f^{2} p e^{5} \log \left (x^{2} e + d\right ) - 24 \, {\left (x^{2} e + d\right )} d^{3} f^{2} p e^{5} \log \left (x^{2} e + d\right ) - 6 \, {\left (x^{2} e + d\right )}^{4} f^{2} p e^{5} \log \left (x^{2} e\right ) + 24 \, {\left (x^{2} e + d\right )}^{3} d f^{2} p e^{5} \log \left (x^{2} e\right ) - 36 \, {\left (x^{2} e + d\right )}^{2} d^{2} f^{2} p e^{5} \log \left (x^{2} e\right ) + 24 \, {\left (x^{2} e + d\right )} d^{3} f^{2} p e^{5} \log \left (x^{2} e\right ) - 6 \, d^{4} f^{2} p e^{5} \log \left (x^{2} e\right ) - 16 \, {\left (x^{2} e + d\right )} d^{4} f g e^{4} \log \left (c\right ) + 16 \, d^{5} f g e^{4} \log \left (c\right ) - 6 \, {\left (x^{2} e + d\right )}^{3} d f^{2} p e^{5} + 21 \, {\left (x^{2} e + d\right )}^{2} d^{2} f^{2} p e^{5} - 26 \, {\left (x^{2} e + d\right )} d^{3} f^{2} p e^{5} + 11 \, d^{4} f^{2} p e^{5} - 6 \, d^{4} f^{2} e^{5} \log \left (c\right )\right )} e^{\left (-1\right )}}{48 \, {\left ({\left (x^{2} e + d\right )}^{4} d^{4} - 4 \, {\left (x^{2} e + d\right )}^{3} d^{5} + 6 \, {\left (x^{2} e + d\right )}^{2} d^{6} - 4 \, {\left (x^{2} e + d\right )} d^{7} + d^{8}\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^9,x, algorithm="giac")

[Out]

1/48*(12*(x^2*e + d)^4*d^2*g^2*p*e^3*log(x^2*e + d) - 48*(x^2*e + d)^3*d^3*g^2*p*e^3*log(x^2*e + d) + 60*(x^2*
e + d)^2*d^4*g^2*p*e^3*log(x^2*e + d) - 24*(x^2*e + d)*d^5*g^2*p*e^3*log(x^2*e + d) - 12*(x^2*e + d)^4*d^2*g^2
*p*e^3*log(x^2*e) + 48*(x^2*e + d)^3*d^3*g^2*p*e^3*log(x^2*e) - 72*(x^2*e + d)^2*d^4*g^2*p*e^3*log(x^2*e) + 48
*(x^2*e + d)*d^5*g^2*p*e^3*log(x^2*e) - 12*d^6*g^2*p*e^3*log(x^2*e) - 12*(x^2*e + d)^3*d^3*g^2*p*e^3 + 36*(x^2
*e + d)^2*d^4*g^2*p*e^3 - 36*(x^2*e + d)*d^5*g^2*p*e^3 + 12*d^6*g^2*p*e^3 - 16*(x^2*e + d)^4*d*f*g*p*e^4*log(x
^2*e + d) + 64*(x^2*e + d)^3*d^2*f*g*p*e^4*log(x^2*e + d) - 96*(x^2*e + d)^2*d^3*f*g*p*e^4*log(x^2*e + d) + 48
*(x^2*e + d)*d^4*f*g*p*e^4*log(x^2*e + d) + 16*(x^2*e + d)^4*d*f*g*p*e^4*log(x^2*e) - 64*(x^2*e + d)^3*d^2*f*g
*p*e^4*log(x^2*e) + 96*(x^2*e + d)^2*d^3*f*g*p*e^4*log(x^2*e) - 64*(x^2*e + d)*d^4*f*g*p*e^4*log(x^2*e) + 16*d
^5*f*g*p*e^4*log(x^2*e) - 12*(x^2*e + d)^2*d^4*g^2*e^3*log(c) + 24*(x^2*e + d)*d^5*g^2*e^3*log(c) - 12*d^6*g^2
*e^3*log(c) + 16*(x^2*e + d)^3*d^2*f*g*p*e^4 - 56*(x^2*e + d)^2*d^3*f*g*p*e^4 + 64*(x^2*e + d)*d^4*f*g*p*e^4 -
24*d^5*f*g*p*e^4 + 6*(x^2*e + d)^4*f^2*p*e^5*log(x^2*e + d) - 24*(x^2*e + d)^3*d*f^2*p*e^5*log(x^2*e + d) + 3
6*(x^2*e + d)^2*d^2*f^2*p*e^5*log(x^2*e + d) - 24*(x^2*e + d)*d^3*f^2*p*e^5*log(x^2*e + d) - 6*(x^2*e + d)^4*f
^2*p*e^5*log(x^2*e) + 24*(x^2*e + d)^3*d*f^2*p*e^5*log(x^2*e) - 36*(x^2*e + d)^2*d^2*f^2*p*e^5*log(x^2*e) + 24
*(x^2*e + d)*d^3*f^2*p*e^5*log(x^2*e) - 6*d^4*f^2*p*e^5*log(x^2*e) - 16*(x^2*e + d)*d^4*f*g*e^4*log(c) + 16*d^
5*f*g*e^4*log(c) - 6*(x^2*e + d)^3*d*f^2*p*e^5 + 21*(x^2*e + d)^2*d^2*f^2*p*e^5 - 26*(x^2*e + d)*d^3*f^2*p*e^5
+ 11*d^4*f^2*p*e^5 - 6*d^4*f^2*e^5*log(c))*e^(-1)/((x^2*e + d)^4*d^4 - 4*(x^2*e + d)^3*d^5 + 6*(x^2*e + d)^2*
d^6 - 4*(x^2*e + d)*d^7 + d^8)

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Mupad [B]
time = 0.43, size = 190, normalized size = 0.88 \begin {gather*} \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} \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^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)

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