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

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

[Out]

-(d^2*(e*f - d*g)^2*p*x^2)/(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

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Rubi [A]  time = 0.470699, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2475, 43, 2414, 12, 893} \[ \frac{1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac{p \left (d+e x^2\right )^3 \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )}{18 e^5}+\frac{d^3 p \left (6 d^2 g^2-15 d e f g+10 e^2 f^2\right ) \log \left (d+e x^2\right )}{60 e^5}-\frac{d^2 p x^2 (e f-d g)^2}{2 e^4}-\frac{g p \left (d+e x^2\right )^4 (e f-2 d g)}{16 e^5}+\frac{d p \left (d+e x^2\right )^2 (e f-2 d g) (e f-d g)}{4 e^5}-\frac{g^2 p \left (d+e x^2\right )^5}{50 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^2*(e*f - d*g)^2*p*x^2)/(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 2475

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])

Rule 43

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 2414

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 893

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]))

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} \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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*}

Mathematica [A]  time = 0.171382, size = 205, normalized size = 0.82 \[ \frac{60 e^5 x^6 \left (10 f^2+15 f g x^2+6 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )-e p x^2 \left (30 d^2 e^2 \left (20 f^2+15 f g x^2+4 g^2 x^4\right )-180 d^3 e g \left (5 f+g x^2\right )+360 d^4 g^2-30 d e^3 x^2 \left (10 f^2+10 f g x^2+3 g^2 x^4\right )+e^4 x^4 \left (200 f^2+225 f g x^2+72 g^2 x^4\right )\right )+60 d^3 p \left (6 d^2 g^2-15 d e f g+10 e^2 f^2\right ) \log \left (d+e x^2\right )}{3600 e^5} \]

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]  time = 0.585, size = 687, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

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

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Maxima [A]  time = 1.02508, size = 301, normalized size = 1.2 \begin{align*} -\frac{1}{3600} \, e p{\left (\frac{72 \, e^{4} g^{2} x^{10} + 45 \,{\left (5 \, e^{4} f g - 2 \, d e^{3} g^{2}\right )} x^{8} + 20 \,{\left (10 \, e^{4} f^{2} - 15 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} x^{6} - 30 \,{\left (10 \, d e^{3} f^{2} - 15 \, d^{2} e^{2} f g + 6 \, d^{3} e g^{2}\right )} x^{4} + 60 \,{\left (10 \, d^{2} e^{2} f^{2} - 15 \, d^{3} e f g + 6 \, d^{4} g^{2}\right )} x^{2}}{e^{5}} - \frac{60 \,{\left (10 \, d^{3} e^{2} f^{2} - 15 \, d^{4} e f g + 6 \, d^{5} g^{2}\right )} \log \left (e x^{2} + d\right )}{e^{6}}\right )} + \frac{1}{60} \,{\left (6 \, g^{2} x^{10} + 15 \, f g x^{8} + 10 \, f^{2} x^{6}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \end{align*}

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

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Fricas [A]  time = 2.08771, size = 582, normalized size = 2.32 \begin{align*} -\frac{72 \, e^{5} g^{2} p x^{10} + 45 \,{\left (5 \, e^{5} f g - 2 \, d e^{4} g^{2}\right )} p x^{8} + 20 \,{\left (10 \, e^{5} f^{2} - 15 \, d e^{4} f g + 6 \, d^{2} e^{3} g^{2}\right )} p x^{6} - 30 \,{\left (10 \, d e^{4} f^{2} - 15 \, d^{2} e^{3} f g + 6 \, d^{3} e^{2} g^{2}\right )} p x^{4} + 60 \,{\left (10 \, d^{2} e^{3} f^{2} - 15 \, d^{3} e^{2} f g + 6 \, d^{4} e g^{2}\right )} p x^{2} - 60 \,{\left (6 \, e^{5} g^{2} p x^{10} + 15 \, e^{5} f g p x^{8} + 10 \, e^{5} f^{2} p x^{6} +{\left (10 \, d^{3} e^{2} f^{2} - 15 \, d^{4} e f g + 6 \, d^{5} g^{2}\right )} p\right )} \log \left (e x^{2} + d\right ) - 60 \,{\left (6 \, e^{5} g^{2} x^{10} + 15 \, e^{5} f g x^{8} + 10 \, e^{5} f^{2} x^{6}\right )} \log \left (c\right )}{3600 \, e^{5}} \end{align*}

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

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

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]  time = 1.3058, size = 721, normalized size = 2.87 \begin{align*} \text{result too large to display} \end{align*}

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]

1/3600*(360*g^2*x^10*e*log(c) + 900*f*g*x^8*e*log(c) + 600*f^2*x^6*e*log(c) + 100*(6*(x^2*e + d)^3*e^(-2)*log(
x^2*e + d) - 18*(x^2*e + d)^2*d*e^(-2)*log(x^2*e + d) + 18*(x^2*e + d)*d^2*e^(-2)*log(x^2*e + d) - 2*(x^2*e +
d)^3*e^(-2) + 9*(x^2*e + d)^2*d*e^(-2) - 18*(x^2*e + d)*d^2*e^(-2))*f^2*p + 75*(12*(x^2*e + d)^4*e^(-3)*log(x^
2*e + d) - 48*(x^2*e + d)^3*d*e^(-3)*log(x^2*e + d) + 72*(x^2*e + d)^2*d^2*e^(-3)*log(x^2*e + d) - 48*(x^2*e +
d)*d^3*e^(-3)*log(x^2*e + d) - 3*(x^2*e + d)^4*e^(-3) + 16*(x^2*e + d)^3*d*e^(-3) - 36*(x^2*e + d)^2*d^2*e^(-
3) + 48*(x^2*e + d)*d^3*e^(-3))*f*g*p + 6*(60*(x^2*e + d)^5*e^(-4)*log(x^2*e + d) - 300*(x^2*e + d)^4*d*e^(-4)
*log(x^2*e + d) + 600*(x^2*e + d)^3*d^2*e^(-4)*log(x^2*e + d) - 600*(x^2*e + d)^2*d^3*e^(-4)*log(x^2*e + d) +
300*(x^2*e + d)*d^4*e^(-4)*log(x^2*e + d) - 12*(x^2*e + d)^5*e^(-4) + 75*(x^2*e + d)^4*d*e^(-4) - 200*(x^2*e +
d)^3*d^2*e^(-4) + 300*(x^2*e + d)^2*d^3*e^(-4) - 300*(x^2*e + d)*d^4*e^(-4))*g^2*p)*e^(-1)