3.4.23 \(\int x^5 (f+g x^2)^2 \log (c (d+e x^2)^p) \, dx\) [323]

Optimal. Leaf size=251 \[ -\frac {d^2 (e f-d g)^2 p x^2}{2 e^4}+\frac {d (e f-2 d g) (e f-d g) p \left (d+e x^2\right )^2}{4 e^5}-\frac {\left (e^2 f^2-6 d e f g+6 d^2 g^2\right ) p \left (d+e x^2\right )^3}{18 e^5}-\frac {g (e f-2 d g) p \left (d+e x^2\right )^4}{16 e^5}-\frac {g^2 p \left (d+e x^2\right )^5}{50 e^5}+\frac {d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )}{60 e^5}+\frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right ) \]

[Out]

-1/2*d^2*(-d*g+e*f)^2*p*x^2/e^4+1/4*d*(-2*d*g+e*f)*(-d*g+e*f)*p*(e*x^2+d)^2/e^5-1/18*(6*d^2*g^2-6*d*e*f*g+e^2*
f^2)*p*(e*x^2+d)^3/e^5-1/16*g*(-2*d*g+e*f)*p*(e*x^2+d)^4/e^5-1/50*g^2*p*(e*x^2+d)^5/e^5+1/60*d^3*(6*d^2*g^2-15
*d*e*f*g+10*e^2*f^2)*p*ln(e*x^2+d)/e^5+1/6*f^2*x^6*ln(c*(e*x^2+d)^p)+1/4*f*g*x^8*ln(c*(e*x^2+d)^p)+1/10*g^2*x^
10*ln(c*(e*x^2+d)^p)

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Rubi [A]
time = 0.32, antiderivative size = 251, 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 {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^2 p x^2 (e f-d g)^2}{2 e^4}-\frac {p \left (d+e x^2\right )^3 \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )}{18 e^5}+\frac {d^3 p \left (6 d^2 g^2-15 d e f g+10 e^2 f^2\right ) \log \left (d+e x^2\right )}{60 e^5}-\frac {g p \left (d+e x^2\right )^4 (e f-2 d g)}{16 e^5}+\frac {d p \left (d+e x^2\right )^2 (e f-2 d g) (e f-d g)}{4 e^5}-\frac {g^2 p \left (d+e x^2\right )^5}{50 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(f + g*x^2)^2*Log[c*(d + e*x^2)^p],x]

[Out]

-1/2*(d^2*(e*f - d*g)^2*p*x^2)/e^4 + (d*(e*f - 2*d*g)*(e*f - d*g)*p*(d + e*x^2)^2)/(4*e^5) - ((e^2*f^2 - 6*d*e
*f*g + 6*d^2*g^2)*p*(d + e*x^2)^3)/(18*e^5) - (g*(e*f - 2*d*g)*p*(d + e*x^2)^4)/(16*e^5) - (g^2*p*(d + e*x^2)^
5)/(50*e^5) + (d^3*(10*e^2*f^2 - 15*d*e*f*g + 6*d^2*g^2)*p*Log[d + e*x^2])/(60*e^5) + (f^2*x^6*Log[c*(d + e*x^
2)^p])/6 + (f*g*x^8*Log[c*(d + e*x^2)^p])/4 + (g^2*x^10*Log[c*(d + e*x^2)^p])/10

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 x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int x^2 (f+g x)^2 \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )\\ &=\frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{2} (e p) \text {Subst}\left (\int \frac {x^3 \left (10 f^2+15 f g x+6 g^2 x^2\right )}{30 (d+e x)} \, dx,x,x^2\right )\\ &=\frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{60} (e p) \text {Subst}\left (\int \frac {x^3 \left (10 f^2+15 f g x+6 g^2 x^2\right )}{d+e x} \, dx,x,x^2\right )\\ &=\frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{60} (e p) \text {Subst}\left (\int \left (\frac {30 d^2 (-e f+d g)^2}{e^5}-\frac {d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right )}{e^5 (d+e x)}+\frac {30 d (e f-2 d g) (-e f+d g) (d+e x)}{e^5}+\frac {10 \left (e^2 f^2-6 d e f g+6 d^2 g^2\right ) (d+e x)^2}{e^5}+\frac {15 g (e f-2 d g) (d+e x)^3}{e^5}+\frac {6 g^2 (d+e x)^4}{e^5}\right ) \, dx,x,x^2\right )\\ &=-\frac {d^2 (e f-d g)^2 p x^2}{2 e^4}+\frac {d (e f-2 d g) (e f-d g) p \left (d+e x^2\right )^2}{4 e^5}-\frac {\left (e^2 f^2-6 d e f g+6 d^2 g^2\right ) p \left (d+e x^2\right )^3}{18 e^5}-\frac {g (e f-2 d g) p \left (d+e x^2\right )^4}{16 e^5}-\frac {g^2 p \left (d+e x^2\right )^5}{50 e^5}+\frac {d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )}{60 e^5}+\frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 205, normalized size = 0.82 \begin {gather*} \frac {-e p x^2 \left (360 d^4 g^2-180 d^3 e g \left (5 f+g x^2\right )-30 d e^3 x^2 \left (10 f^2+10 f g x^2+3 g^2 x^4\right )+30 d^2 e^2 \left (20 f^2+15 f g x^2+4 g^2 x^4\right )+e^4 x^4 \left (200 f^2+225 f g x^2+72 g^2 x^4\right )\right )+60 d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )+60 e^5 x^6 \left (10 f^2+15 f g x^2+6 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3600 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(f + g*x^2)^2*Log[c*(d + e*x^2)^p],x]

[Out]

(-(e*p*x^2*(360*d^4*g^2 - 180*d^3*e*g*(5*f + g*x^2) - 30*d*e^3*x^2*(10*f^2 + 10*f*g*x^2 + 3*g^2*x^4) + 30*d^2*
e^2*(20*f^2 + 15*f*g*x^2 + 4*g^2*x^4) + e^4*x^4*(200*f^2 + 225*f*g*x^2 + 72*g^2*x^4))) + 60*d^3*(10*e^2*f^2 -
15*d*e*f*g + 6*d^2*g^2)*p*Log[d + e*x^2] + 60*e^5*x^6*(10*f^2 + 15*f*g*x^2 + 6*g^2*x^4)*Log[c*(d + e*x^2)^p])/
(3600*e^5)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.54, size = 687, normalized size = 2.74

method	result	size
risch	\(-\frac {i \pi f g \,x^{8} \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}+\frac {\ln \left (c \right ) g^{2} x^{10}}{10}+\frac {\ln \left (c \right ) f^{2} x^{6}}{6}+\frac {d \,g^{2} p \,x^{8}}{40 e}-\frac {d^{2} g^{2} p \,x^{6}}{30 e^{2}}+\frac {d^{3} g^{2} p \,x^{4}}{20 e^{3}}+\frac {d \,f^{2} p \,x^{4}}{12 e}-\frac {d^{4} g^{2} p \,x^{2}}{10 e^{4}}-\frac {d^{2} f^{2} p \,x^{2}}{6 e^{2}}+\frac {\ln \left (e \,x^{2}+d \right ) d^{5} g^{2} p}{10 e^{5}}+\frac {\ln \left (e \,x^{2}+d \right ) d^{3} f^{2} p}{6 e^{3}}-\frac {i \pi \,f^{2} x^{6} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{12}-\frac {i \pi \,g^{2} x^{10} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{20}-\frac {i \pi \,g^{2} x^{10} \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 )}{20}+\frac {i \pi f g \,x^{8} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{8}+\frac {i \pi f g \,x^{8} \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}-\frac {i \pi \,f^{2} x^{6} \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 )}{12}+\frac {d^{3} f g p \,x^{2}}{4 e^{3}}-\frac {\ln \left (e \,x^{2}+d \right ) d^{4} f g p}{4 e^{4}}+\frac {i \pi \,g^{2} x^{10} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{20}+\frac {i \pi \,g^{2} x^{10} \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}}{20}-\frac {i \pi f g \,x^{8} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{8}+\frac {i \pi \,f^{2} x^{6} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{12}+\frac {i \pi \,f^{2} x^{6} \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}}{12}+\frac {d f g p \,x^{6}}{12 e}-\frac {d^{2} f g p \,x^{4}}{8 e^{2}}+\left (\frac {1}{10} g^{2} x^{10}+\frac {1}{4} f g \,x^{8}+\frac {1}{6} f^{2} x^{6}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )-\frac {f g p \,x^{8}}{16}+\frac {\ln \left (c \right ) f g \,x^{8}}{4}-\frac {g^{2} p \,x^{10}}{50}-\frac {f^{2} p \,x^{6}}{18}\)	\(687\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(g*x^2+f)^2*ln(c*(e*x^2+d)^p),x,method=_RETURNVERBOSE)

[Out]

1/10*ln(c)*g^2*x^10+1/6*ln(c)*f^2*x^6+1/40/e*d*g^2*p*x^8-1/30/e^2*d^2*g^2*p*x^6+1/20/e^3*d^3*g^2*p*x^4+1/12/e*
d*f^2*p*x^4-1/10/e^4*d^4*g^2*p*x^2-1/6/e^2*d^2*f^2*p*x^2+1/10/e^5*ln(e*x^2+d)*d^5*g^2*p+1/6/e^3*ln(e*x^2+d)*d^
3*f^2*p-1/20*I*Pi*g^2*x^10*csgn(I*c*(e*x^2+d)^p)^3-1/12*I*Pi*f^2*x^6*csgn(I*c*(e*x^2+d)^p)^3-1/20*I*Pi*g^2*x^1
0*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+1/8*I*Pi*f*g*x^8*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+1/8*I
*Pi*f*g*x^8*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-1/12*I*Pi*f^2*x^6*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+
d)^p)*csgn(I*c)+1/4/e^3*d^3*f*g*p*x^2-1/4/e^4*ln(e*x^2+d)*d^4*f*g*p+1/20*I*Pi*g^2*x^10*csgn(I*c*(e*x^2+d)^p)^2
*csgn(I*c)+1/20*I*Pi*g^2*x^10*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-1/8*I*Pi*f*g*x^8*csgn(I*c*(e*x^2+d)^
p)^3+1/12*I*Pi*f^2*x^6*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+1/12*I*Pi*f^2*x^6*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2
+d)^p)^2+1/12/e*d*f*g*p*x^6-1/8/e^2*d^2*f*g*p*x^4+(1/10*g^2*x^10+1/4*f*g*x^8+1/6*f^2*x^6)*ln((e*x^2+d)^p)-1/16
*f*g*p*x^8+1/4*ln(c)*f*g*x^8-1/50*g^2*p*x^10-1/18*f^2*p*x^6-1/8*I*Pi*f*g*x^8*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x
^2+d)^p)*csgn(I*c)

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Maxima [A]
time = 0.28, size = 218, normalized size = 0.87 \begin {gather*} \frac {1}{3600} \, {\left (60 \, {\left (6 \, d^{5} g^{2} - 15 \, d^{4} f g e + 10 \, d^{3} f^{2} e^{2}\right )} e^{\left (-6\right )} \log \left (x^{2} e + d\right ) - {\left (72 \, g^{2} x^{10} e^{4} - 45 \, {\left (2 \, d g^{2} e^{3} - 5 \, f g e^{4}\right )} x^{8} + 20 \, {\left (6 \, d^{2} g^{2} e^{2} - 15 \, d f g e^{3} + 10 \, f^{2} e^{4}\right )} x^{6} - 30 \, {\left (6 \, d^{3} g^{2} e - 15 \, d^{2} f g e^{2} + 10 \, d f^{2} e^{3}\right )} x^{4} + 60 \, {\left (6 \, d^{4} g^{2} - 15 \, d^{3} f g e + 10 \, d^{2} f^{2} e^{2}\right )} x^{2}\right )} e^{\left (-5\right )}\right )} p e + \frac {1}{60} \, {\left (6 \, g^{2} x^{10} + 15 \, f g x^{8} + 10 \, f^{2} x^{6}\right )} \log \left ({\left (x^{2} e + d\right )}^{p} c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(g*x^2+f)^2*log(c*(e*x^2+d)^p),x, algorithm="maxima")

[Out]

1/3600*(60*(6*d^5*g^2 - 15*d^4*f*g*e + 10*d^3*f^2*e^2)*e^(-6)*log(x^2*e + d) - (72*g^2*x^10*e^4 - 45*(2*d*g^2*
e^3 - 5*f*g*e^4)*x^8 + 20*(6*d^2*g^2*e^2 - 15*d*f*g*e^3 + 10*f^2*e^4)*x^6 - 30*(6*d^3*g^2*e - 15*d^2*f*g*e^2 +
10*d*f^2*e^3)*x^4 + 60*(6*d^4*g^2 - 15*d^3*f*g*e + 10*d^2*f^2*e^2)*x^2)*e^(-5))*p*e + 1/60*(6*g^2*x^10 + 15*f
*g*x^8 + 10*f^2*x^6)*log((x^2*e + d)^p*c)

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Fricas [A]
time = 0.36, size = 252, normalized size = 1.00 \begin {gather*} -\frac {1}{3600} \, {\left (360 \, d^{4} g^{2} p x^{2} e - 60 \, {\left (6 \, g^{2} x^{10} + 15 \, f g x^{8} + 10 \, f^{2} x^{6}\right )} e^{5} \log \left (c\right ) + {\left (72 \, g^{2} p x^{10} + 225 \, f g p x^{8} + 200 \, f^{2} p x^{6}\right )} e^{5} - 30 \, {\left (3 \, d g^{2} p x^{8} + 10 \, d f g p x^{6} + 10 \, d f^{2} p x^{4}\right )} e^{4} + 30 \, {\left (4 \, d^{2} g^{2} p x^{6} + 15 \, d^{2} f g p x^{4} + 20 \, d^{2} f^{2} p x^{2}\right )} e^{3} - 180 \, {\left (d^{3} g^{2} p x^{4} + 5 \, d^{3} f g p x^{2}\right )} e^{2} - 60 \, {\left (6 \, d^{5} g^{2} p - 15 \, d^{4} f g p e + 10 \, d^{3} f^{2} p e^{2} + {\left (6 \, g^{2} p x^{10} + 15 \, f g p x^{8} + 10 \, f^{2} p x^{6}\right )} e^{5}\right )} \log \left (x^{2} e + d\right )\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(g*x^2+f)^2*log(c*(e*x^2+d)^p),x, algorithm="fricas")

[Out]

-1/3600*(360*d^4*g^2*p*x^2*e - 60*(6*g^2*x^10 + 15*f*g*x^8 + 10*f^2*x^6)*e^5*log(c) + (72*g^2*p*x^10 + 225*f*g
*p*x^8 + 200*f^2*p*x^6)*e^5 - 30*(3*d*g^2*p*x^8 + 10*d*f*g*p*x^6 + 10*d*f^2*p*x^4)*e^4 + 30*(4*d^2*g^2*p*x^6 +
15*d^2*f*g*p*x^4 + 20*d^2*f^2*p*x^2)*e^3 - 180*(d^3*g^2*p*x^4 + 5*d^3*f*g*p*x^2)*e^2 - 60*(6*d^5*g^2*p - 15*d
^4*f*g*p*e + 10*d^3*f^2*p*e^2 + (6*g^2*p*x^10 + 15*f*g*p*x^8 + 10*f^2*p*x^6)*e^5)*log(x^2*e + d))*e^(-5)

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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(x**5*(g*x**2+f)**2*ln(c*(e*x**2+d)**p),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (241) = 482\).
time = 6.93, size = 773, normalized size = 3.08 \begin {gather*} \frac {1}{10} \, {\left (x^{2} e + d\right )}^{5} g^{2} p e^{\left (-5\right )} \log \left (x^{2} e + d\right ) - \frac {1}{2} \, {\left (x^{2} e + d\right )}^{4} d g^{2} p e^{\left (-5\right )} \log \left (x^{2} e + d\right ) + {\left (x^{2} e + d\right )}^{3} d^{2} g^{2} p e^{\left (-5\right )} \log \left (x^{2} e + d\right ) - {\left (x^{2} e + d\right )}^{2} d^{3} g^{2} p e^{\left (-5\right )} \log \left (x^{2} e + d\right ) - \frac {1}{50} \, {\left (x^{2} e + d\right )}^{5} g^{2} p e^{\left (-5\right )} + \frac {1}{8} \, {\left (x^{2} e + d\right )}^{4} d g^{2} p e^{\left (-5\right )} - \frac {1}{3} \, {\left (x^{2} e + d\right )}^{3} d^{2} g^{2} p e^{\left (-5\right )} + \frac {1}{2} \, {\left (x^{2} e + d\right )}^{2} d^{3} g^{2} p e^{\left (-5\right )} + \frac {1}{4} \, {\left (x^{2} e + d\right )}^{4} f g p e^{\left (-4\right )} \log \left (x^{2} e + d\right ) - {\left (x^{2} e + d\right )}^{3} d f g p e^{\left (-4\right )} \log \left (x^{2} e + d\right ) + \frac {3}{2} \, {\left (x^{2} e + d\right )}^{2} d^{2} f g p e^{\left (-4\right )} \log \left (x^{2} e + d\right ) + \frac {1}{10} \, {\left (x^{2} e + d\right )}^{5} g^{2} e^{\left (-5\right )} \log \left (c\right ) - \frac {1}{2} \, {\left (x^{2} e + d\right )}^{4} d g^{2} e^{\left (-5\right )} \log \left (c\right ) + {\left (x^{2} e + d\right )}^{3} d^{2} g^{2} e^{\left (-5\right )} \log \left (c\right ) - {\left (x^{2} e + d\right )}^{2} d^{3} g^{2} e^{\left (-5\right )} \log \left (c\right ) - \frac {1}{16} \, {\left (x^{2} e + d\right )}^{4} f g p e^{\left (-4\right )} + \frac {1}{3} \, {\left (x^{2} e + d\right )}^{3} d f g p e^{\left (-4\right )} - \frac {3}{4} \, {\left (x^{2} e + d\right )}^{2} d^{2} f g p e^{\left (-4\right )} + \frac {1}{6} \, {\left (x^{2} e + d\right )}^{3} f^{2} p e^{\left (-3\right )} \log \left (x^{2} e + d\right ) - \frac {1}{2} \, {\left (x^{2} e + d\right )}^{2} d f^{2} p e^{\left (-3\right )} \log \left (x^{2} e + d\right ) + \frac {1}{4} \, {\left (x^{2} e + d\right )}^{4} f g e^{\left (-4\right )} \log \left (c\right ) - {\left (x^{2} e + d\right )}^{3} d f g e^{\left (-4\right )} \log \left (c\right ) + \frac {3}{2} \, {\left (x^{2} e + d\right )}^{2} d^{2} f g e^{\left (-4\right )} \log \left (c\right ) - \frac {1}{18} \, {\left (x^{2} e + d\right )}^{3} f^{2} p e^{\left (-3\right )} + \frac {1}{4} \, {\left (x^{2} e + d\right )}^{2} d f^{2} p e^{\left (-3\right )} + \frac {1}{6} \, {\left (x^{2} e + d\right )}^{3} f^{2} e^{\left (-3\right )} \log \left (c\right ) - \frac {1}{2} \, {\left (x^{2} e + d\right )}^{2} d f^{2} e^{\left (-3\right )} \log \left (c\right ) - \frac {1}{2} \, {\left ({\left (x^{2} e - {\left (x^{2} e + d\right )} \log \left (x^{2} e + d\right ) + d\right )} d^{4} g^{2} p - 2 \, {\left (x^{2} e - {\left (x^{2} e + d\right )} \log \left (x^{2} e + d\right ) + d\right )} d^{3} f g p e - {\left (x^{2} e + d\right )} d^{4} g^{2} \log \left (c\right ) + 2 \, {\left (x^{2} e + d\right )} d^{3} f g e \log \left (c\right ) + {\left (x^{2} e - {\left (x^{2} e + d\right )} \log \left (x^{2} e + d\right ) + d\right )} d^{2} f^{2} p e^{2} - {\left (x^{2} e + d\right )} d^{2} f^{2} e^{2} \log \left (c\right )\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(g*x^2+f)^2*log(c*(e*x^2+d)^p),x, algorithm="giac")

[Out]

1/10*(x^2*e + d)^5*g^2*p*e^(-5)*log(x^2*e + d) - 1/2*(x^2*e + d)^4*d*g^2*p*e^(-5)*log(x^2*e + d) + (x^2*e + d)
^3*d^2*g^2*p*e^(-5)*log(x^2*e + d) - (x^2*e + d)^2*d^3*g^2*p*e^(-5)*log(x^2*e + d) - 1/50*(x^2*e + d)^5*g^2*p*
e^(-5) + 1/8*(x^2*e + d)^4*d*g^2*p*e^(-5) - 1/3*(x^2*e + d)^3*d^2*g^2*p*e^(-5) + 1/2*(x^2*e + d)^2*d^3*g^2*p*e
^(-5) + 1/4*(x^2*e + d)^4*f*g*p*e^(-4)*log(x^2*e + d) - (x^2*e + d)^3*d*f*g*p*e^(-4)*log(x^2*e + d) + 3/2*(x^2
*e + d)^2*d^2*f*g*p*e^(-4)*log(x^2*e + d) + 1/10*(x^2*e + d)^5*g^2*e^(-5)*log(c) - 1/2*(x^2*e + d)^4*d*g^2*e^(
-5)*log(c) + (x^2*e + d)^3*d^2*g^2*e^(-5)*log(c) - (x^2*e + d)^2*d^3*g^2*e^(-5)*log(c) - 1/16*(x^2*e + d)^4*f*
g*p*e^(-4) + 1/3*(x^2*e + d)^3*d*f*g*p*e^(-4) - 3/4*(x^2*e + d)^2*d^2*f*g*p*e^(-4) + 1/6*(x^2*e + d)^3*f^2*p*e
^(-3)*log(x^2*e + d) - 1/2*(x^2*e + d)^2*d*f^2*p*e^(-3)*log(x^2*e + d) + 1/4*(x^2*e + d)^4*f*g*e^(-4)*log(c) -
(x^2*e + d)^3*d*f*g*e^(-4)*log(c) + 3/2*(x^2*e + d)^2*d^2*f*g*e^(-4)*log(c) - 1/18*(x^2*e + d)^3*f^2*p*e^(-3)
+ 1/4*(x^2*e + d)^2*d*f^2*p*e^(-3) + 1/6*(x^2*e + d)^3*f^2*e^(-3)*log(c) - 1/2*(x^2*e + d)^2*d*f^2*e^(-3)*log
(c) - 1/2*((x^2*e - (x^2*e + d)*log(x^2*e + d) + d)*d^4*g^2*p - 2*(x^2*e - (x^2*e + d)*log(x^2*e + d) + d)*d^3
*f*g*p*e - (x^2*e + d)*d^4*g^2*log(c) + 2*(x^2*e + d)*d^3*f*g*e*log(c) + (x^2*e - (x^2*e + d)*log(x^2*e + d) +
d)*d^2*f^2*p*e^2 - (x^2*e + d)*d^2*f^2*e^2*log(c))*e^(-5)

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Mupad [B]
time = 0.36, size = 224, normalized size = 0.89 \begin {gather*} \ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2\,x^6}{6}+\frac {f\,g\,x^8}{4}+\frac {g^2\,x^{10}}{10}\right )-x^6\,\left (\frac {f^2\,p}{18}-\frac {d\,\left (\frac {f\,g\,p}{2}-\frac {d\,g^2\,p}{5\,e}\right )}{6\,e}\right )-x^8\,\left (\frac {f\,g\,p}{16}-\frac {d\,g^2\,p}{40\,e}\right )-\frac {g^2\,p\,x^{10}}{50}+\frac {\ln \left (e\,x^2+d\right )\,\left (6\,p\,d^5\,g^2-15\,p\,d^4\,e\,f\,g+10\,p\,d^3\,e^2\,f^2\right )}{60\,e^5}+\frac {d\,x^4\,\left (\frac {f^2\,p}{3}-\frac {d\,\left (\frac {f\,g\,p}{2}-\frac {d\,g^2\,p}{5\,e}\right )}{e}\right )}{4\,e}-\frac {d^2\,x^2\,\left (\frac {f^2\,p}{3}-\frac {d\,\left (\frac {f\,g\,p}{2}-\frac {d\,g^2\,p}{5\,e}\right )}{e}\right )}{2\,e^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*log(c*(d + e*x^2)^p)*(f + g*x^2)^2,x)

[Out]

log(c*(d + e*x^2)^p)*((f^2*x^6)/6 + (g^2*x^10)/10 + (f*g*x^8)/4) - x^6*((f^2*p)/18 - (d*((f*g*p)/2 - (d*g^2*p)
/(5*e)))/(6*e)) - x^8*((f*g*p)/16 - (d*g^2*p)/(40*e)) - (g^2*p*x^10)/50 + (log(d + e*x^2)*(6*d^5*g^2*p + 10*d^
3*e^2*f^2*p - 15*d^4*e*f*g*p))/(60*e^5) + (d*x^4*((f^2*p)/3 - (d*((f*g*p)/2 - (d*g^2*p)/(5*e)))/e))/(4*e) - (d
^2*x^2*((f^2*p)/3 - (d*((f*g*p)/2 - (d*g^2*p)/(5*e)))/e))/(2*e^2)

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