3.77 \(\int x^5 \log ^2(c (a+b x^2)^p) \, dx\)

Optimal. Leaf size=215 \[ \frac {a^3 p \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^3}-\frac {a^3 p^2 \log ^2\left (a+b x^2\right )}{6 b^3}-\frac {a^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}+\frac {a^2 p^2 x^2}{b^2}-\frac {p \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}+\frac {a p \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {p^2 \left (a+b x^2\right )^3}{27 b^3}-\frac {a p^2 \left (a+b x^2\right )^2}{4 b^3}+\frac {1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right ) \]

[Out]

a^2*p^2*x^2/b^2-1/4*a*p^2*(b*x^2+a)^2/b^3+1/27*p^2*(b*x^2+a)^3/b^3-1/6*a^3*p^2*ln(b*x^2+a)^2/b^3-a^2*p*(b*x^2+
a)*ln(c*(b*x^2+a)^p)/b^3+1/2*a*p*(b*x^2+a)^2*ln(c*(b*x^2+a)^p)/b^3-1/9*p*(b*x^2+a)^3*ln(c*(b*x^2+a)^p)/b^3+1/3
*a^3*p*ln(b*x^2+a)*ln(c*(b*x^2+a)^p)/b^3+1/6*x^6*ln(c*(b*x^2+a)^p)^2

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Rubi [A]  time = 0.30, antiderivative size = 175, normalized size of antiderivative = 0.81, number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ -\frac {1}{18} p \left (\frac {18 a^2 \left (a+b x^2\right )}{b^3}-\frac {6 a^3 \log \left (a+b x^2\right )}{b^3}-\frac {9 a \left (a+b x^2\right )^2}{b^3}+\frac {2 \left (a+b x^2\right )^3}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac {a^2 p^2 x^2}{b^2}-\frac {a^3 p^2 \log ^2\left (a+b x^2\right )}{6 b^3}+\frac {p^2 \left (a+b x^2\right )^3}{27 b^3}-\frac {a p^2 \left (a+b x^2\right )^2}{4 b^3}+\frac {1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5*Log[c*(a + b*x^2)^p]^2,x]

[Out]

(a^2*p^2*x^2)/b^2 - (a*p^2*(a + b*x^2)^2)/(4*b^3) + (p^2*(a + b*x^2)^3)/(27*b^3) - (a^3*p^2*Log[a + b*x^2]^2)/
(6*b^3) - (p*((18*a^2*(a + b*x^2))/b^3 - (9*a*(a + b*x^2)^2)/b^3 + (2*(a + b*x^2)^3)/b^3 - (6*a^3*Log[a + b*x^
2])/b^3)*Log[c*(a + b*x^2)^p])/18 + (x^6*Log[c*(a + b*x^2)^p]^2)/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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

\begin {align*} \int x^5 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac {1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{3} (b p) \operatorname {Subst}\left (\int \frac {x^3 \log \left (c (a+b x)^p\right )}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{3} p \operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \log \left (c x^p\right )}{x} \, dx,x,a+b x^2\right )\\ &=-\frac {1}{18} p \left (\frac {18 a^2 \left (a+b x^2\right )}{b^3}-\frac {9 a \left (a+b x^2\right )^2}{b^3}+\frac {2 \left (a+b x^2\right )^3}{b^3}-\frac {6 a^3 \log \left (a+b x^2\right )}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac {1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} p^2 \operatorname {Subst}\left (\int \frac {18 a^2 x-9 a x^2+2 x^3-6 a^3 \log (x)}{6 b^3 x} \, dx,x,a+b x^2\right )\\ &=-\frac {1}{18} p \left (\frac {18 a^2 \left (a+b x^2\right )}{b^3}-\frac {9 a \left (a+b x^2\right )^2}{b^3}+\frac {2 \left (a+b x^2\right )^3}{b^3}-\frac {6 a^3 \log \left (a+b x^2\right )}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac {1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {p^2 \operatorname {Subst}\left (\int \frac {18 a^2 x-9 a x^2+2 x^3-6 a^3 \log (x)}{x} \, dx,x,a+b x^2\right )}{18 b^3}\\ &=-\frac {1}{18} p \left (\frac {18 a^2 \left (a+b x^2\right )}{b^3}-\frac {9 a \left (a+b x^2\right )^2}{b^3}+\frac {2 \left (a+b x^2\right )^3}{b^3}-\frac {6 a^3 \log \left (a+b x^2\right )}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac {1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {p^2 \operatorname {Subst}\left (\int \left (18 a^2-9 a x+2 x^2-\frac {6 a^3 \log (x)}{x}\right ) \, dx,x,a+b x^2\right )}{18 b^3}\\ &=\frac {a^2 p^2 x^2}{b^2}-\frac {a p^2 \left (a+b x^2\right )^2}{4 b^3}+\frac {p^2 \left (a+b x^2\right )^3}{27 b^3}-\frac {1}{18} p \left (\frac {18 a^2 \left (a+b x^2\right )}{b^3}-\frac {9 a \left (a+b x^2\right )^2}{b^3}+\frac {2 \left (a+b x^2\right )^3}{b^3}-\frac {6 a^3 \log \left (a+b x^2\right )}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac {1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {\left (a^3 p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x^2\right )}{3 b^3}\\ &=\frac {a^2 p^2 x^2}{b^2}-\frac {a p^2 \left (a+b x^2\right )^2}{4 b^3}+\frac {p^2 \left (a+b x^2\right )^3}{27 b^3}-\frac {a^3 p^2 \log ^2\left (a+b x^2\right )}{6 b^3}-\frac {1}{18} p \left (\frac {18 a^2 \left (a+b x^2\right )}{b^3}-\frac {9 a \left (a+b x^2\right )^2}{b^3}+\frac {2 \left (a+b x^2\right )^3}{b^3}-\frac {6 a^3 \log \left (a+b x^2\right )}{b^3}\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac {1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 200, normalized size = 0.93 \[ \frac {a^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac {a^3 p \log \left (c \left (a+b x^2\right )^p\right )}{3 b^3}-\frac {5 a^3 p^2 \log \left (a+b x^2\right )}{18 b^3}-\frac {a^2 p x^2 \log \left (c \left (a+b x^2\right )^p\right )}{3 b^2}+\frac {11 a^2 p^2 x^2}{18 b^2}+\frac {1}{6} x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )-\frac {1}{9} p x^6 \log \left (c \left (a+b x^2\right )^p\right )+\frac {a p x^4 \log \left (c \left (a+b x^2\right )^p\right )}{6 b}-\frac {5 a p^2 x^4}{36 b}+\frac {p^2 x^6}{27} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Log[c*(a + b*x^2)^p]^2,x]

[Out]

(11*a^2*p^2*x^2)/(18*b^2) - (5*a*p^2*x^4)/(36*b) + (p^2*x^6)/27 - (5*a^3*p^2*Log[a + b*x^2])/(18*b^3) - (a^3*p
*Log[c*(a + b*x^2)^p])/(3*b^3) - (a^2*p*x^2*Log[c*(a + b*x^2)^p])/(3*b^2) + (a*p*x^4*Log[c*(a + b*x^2)^p])/(6*
b) - (p*x^6*Log[c*(a + b*x^2)^p])/9 + (a^3*Log[c*(a + b*x^2)^p]^2)/(6*b^3) + (x^6*Log[c*(a + b*x^2)^p]^2)/6

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fricas [A]  time = 0.46, size = 189, normalized size = 0.88 \[ \frac {4 \, b^{3} p^{2} x^{6} + 18 \, b^{3} x^{6} \log \relax (c)^{2} - 15 \, a b^{2} p^{2} x^{4} + 66 \, a^{2} b p^{2} x^{2} + 18 \, {\left (b^{3} p^{2} x^{6} + a^{3} p^{2}\right )} \log \left (b x^{2} + a\right )^{2} - 6 \, {\left (2 \, b^{3} p^{2} x^{6} - 3 \, a b^{2} p^{2} x^{4} + 6 \, a^{2} b p^{2} x^{2} + 11 \, a^{3} p^{2} - 6 \, {\left (b^{3} p x^{6} + a^{3} p\right )} \log \relax (c)\right )} \log \left (b x^{2} + a\right ) - 6 \, {\left (2 \, b^{3} p x^{6} - 3 \, a b^{2} p x^{4} + 6 \, a^{2} b p x^{2}\right )} \log \relax (c)}{108 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*log(c*(b*x^2+a)^p)^2,x, algorithm="fricas")

[Out]

1/108*(4*b^3*p^2*x^6 + 18*b^3*x^6*log(c)^2 - 15*a*b^2*p^2*x^4 + 66*a^2*b*p^2*x^2 + 18*(b^3*p^2*x^6 + a^3*p^2)*
log(b*x^2 + a)^2 - 6*(2*b^3*p^2*x^6 - 3*a*b^2*p^2*x^4 + 6*a^2*b*p^2*x^2 + 11*a^3*p^2 - 6*(b^3*p*x^6 + a^3*p)*l
og(c))*log(b*x^2 + a) - 6*(2*b^3*p*x^6 - 3*a*b^2*p*x^4 + 6*a^2*b*p*x^2)*log(c))/b^3

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giac [A]  time = 0.19, size = 325, normalized size = 1.51 \[ \frac {18 \, b x^{6} \log \relax (c)^{2} + {\left (\frac {18 \, {\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )^{2}}{b^{2}} - \frac {54 \, {\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )^{2}}{b^{2}} + \frac {54 \, {\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )^{2}}{b^{2}} - \frac {12 \, {\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )}{b^{2}} + \frac {54 \, {\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )}{b^{2}} - \frac {108 \, {\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )}{b^{2}} + \frac {4 \, {\left (b x^{2} + a\right )}^{3}}{b^{2}} - \frac {27 \, {\left (b x^{2} + a\right )}^{2} a}{b^{2}} + \frac {108 \, {\left (b x^{2} + a\right )} a^{2}}{b^{2}}\right )} p^{2} + 6 \, {\left (\frac {6 \, {\left (b x^{2} + a\right )}^{3} \log \left (b x^{2} + a\right )}{b^{2}} - \frac {18 \, {\left (b x^{2} + a\right )}^{2} a \log \left (b x^{2} + a\right )}{b^{2}} + \frac {18 \, {\left (b x^{2} + a\right )} a^{2} \log \left (b x^{2} + a\right )}{b^{2}} - \frac {2 \, {\left (b x^{2} + a\right )}^{3}}{b^{2}} + \frac {9 \, {\left (b x^{2} + a\right )}^{2} a}{b^{2}} - \frac {18 \, {\left (b x^{2} + a\right )} a^{2}}{b^{2}}\right )} p \log \relax (c)}{108 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*log(c*(b*x^2+a)^p)^2,x, algorithm="giac")

[Out]

1/108*(18*b*x^6*log(c)^2 + (18*(b*x^2 + a)^3*log(b*x^2 + a)^2/b^2 - 54*(b*x^2 + a)^2*a*log(b*x^2 + a)^2/b^2 +
54*(b*x^2 + a)*a^2*log(b*x^2 + a)^2/b^2 - 12*(b*x^2 + a)^3*log(b*x^2 + a)/b^2 + 54*(b*x^2 + a)^2*a*log(b*x^2 +
a)/b^2 - 108*(b*x^2 + a)*a^2*log(b*x^2 + a)/b^2 + 4*(b*x^2 + a)^3/b^2 - 27*(b*x^2 + a)^2*a/b^2 + 108*(b*x^2 +
a)*a^2/b^2)*p^2 + 6*(6*(b*x^2 + a)^3*log(b*x^2 + a)/b^2 - 18*(b*x^2 + a)^2*a*log(b*x^2 + a)/b^2 + 18*(b*x^2 +
a)*a^2*log(b*x^2 + a)/b^2 - 2*(b*x^2 + a)^3/b^2 + 9*(b*x^2 + a)^2*a/b^2 - 18*(b*x^2 + a)*a^2/b^2)*p*log(c))/b

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maple [C]  time = 0.52, size = 1436, normalized size = 6.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*ln(c*(b*x^2+a)^p)^2,x)

[Out]

1/18*(3*I*Pi*b^3*x^6*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-3*I*Pi*b^3*x^6*csgn(I*(b*x^2+a)^p)*csgn(I*c*(
b*x^2+a)^p)*csgn(I*c)-3*I*Pi*b^3*x^6*csgn(I*c*(b*x^2+a)^p)^3+3*I*Pi*b^3*x^6*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+
6*ln(c)*b^3*x^6-2*b^3*p*x^6+3*a*b^2*p*x^4-6*a^2*b*p*x^2+6*a^3*p*ln(b*x^2+a))/b^3*ln((b*x^2+a)^p)-1/6*a^3*p^2*l
n(b*x^2+a)^2/b^3-1/9*ln(c)*p*x^6-1/24*Pi^2*x^6*csgn(I*c*(b*x^2+a)^p)^6-11/18*a^3*p^2/b^3*ln(b*x^2+a)+1/27*p^2*
x^6+11/18*a^2*p^2*x^2/b^2+1/6*ln(c)^2*x^6-5/36/b*a*p^2*x^4-1/24*Pi^2*x^6*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)^2+1
/12*Pi^2*x^6*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)-1/24*Pi^2*x^6*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^4+1/1
2*Pi^2*x^6*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^5+1/6*x^6*ln((b*x^2+a)^p)^2+1/6/b*ln(c)*a*p*x^4-1/24*Pi^2
*x^6*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)^2+1/12*Pi^2*x^6*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x
^2+a)^p)^3*csgn(I*c)^2+1/12*Pi^2*x^6*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)-1/6*Pi^2*x^6*csgn
(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)-1/3/b^2*ln(c)*a^2*p*x^2+1/3/b^3*ln(c)*ln(b*x^2+a)*a^3*p-1/6*
I*ln(c)*Pi*x^6*csgn(I*c*(b*x^2+a)^p)^3+1/18*I*Pi*p*x^6*csgn(I*c*(b*x^2+a)^p)^3+1/6*I*ln(c)*Pi*x^6*csgn(I*c*(b*
x^2+a)^p)^2*csgn(I*c)+1/6*I*ln(c)*Pi*x^6*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-1/18*I*Pi*p*x^6*csgn(I*c*
(b*x^2+a)^p)^2*csgn(I*c)-1/18*I*Pi*p*x^6*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-1/12*I/b*Pi*a*p*x^4*csgn(
I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+1/6*I/b^2*Pi*a^2*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p
)*csgn(I*c)-1/6*I/b^3*Pi*ln(b*x^2+a)*a^3*p*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-1/12*I/b*Pi*a*p
*x^4*csgn(I*c*(b*x^2+a)^p)^3+1/6*I/b^2*Pi*a^2*p*x^2*csgn(I*c*(b*x^2+a)^p)^3-1/6*I/b^3*Pi*ln(b*x^2+a)*a^3*p*csg
n(I*c*(b*x^2+a)^p)^3-1/6*I*ln(c)*Pi*x^6*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+1/18*I*Pi*p*x^6*cs
gn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+1/12*I/b*Pi*a*p*x^4*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+1/12*I
/b*Pi*a*p*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-1/6*I/b^2*Pi*a^2*p*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(
I*c)-1/6*I/b^2*Pi*a^2*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+1/6*I/b^3*Pi*ln(b*x^2+a)*a^3*p*csgn(I*
c*(b*x^2+a)^p)^2*csgn(I*c)+1/6*I/b^3*Pi*ln(b*x^2+a)*a^3*p*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2

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maxima [A]  time = 0.70, size = 145, normalized size = 0.67 \[ \frac {1}{6} \, x^{6} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} + \frac {1}{18} \, b p {\left (\frac {6 \, a^{3} \log \left (b x^{2} + a\right )}{b^{4}} - \frac {2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{b^{3}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) + \frac {{\left (4 \, b^{3} x^{6} - 15 \, a b^{2} x^{4} + 66 \, a^{2} b x^{2} - 18 \, a^{3} \log \left (b x^{2} + a\right )^{2} - 66 \, a^{3} \log \left (b x^{2} + a\right )\right )} p^{2}}{108 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*log(c*(b*x^2+a)^p)^2,x, algorithm="maxima")

[Out]

1/6*x^6*log((b*x^2 + a)^p*c)^2 + 1/18*b*p*(6*a^3*log(b*x^2 + a)/b^4 - (2*b^2*x^6 - 3*a*b*x^4 + 6*a^2*x^2)/b^3)
*log((b*x^2 + a)^p*c) + 1/108*(4*b^3*x^6 - 15*a*b^2*x^4 + 66*a^2*b*x^2 - 18*a^3*log(b*x^2 + a)^2 - 66*a^3*log(
b*x^2 + a))*p^2/b^3

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mupad [B]  time = 0.31, size = 126, normalized size = 0.59 \[ \frac {p^2\,x^6}{27}+{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2\,\left (\frac {x^6}{6}+\frac {a^3}{6\,b^3}\right )-\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )\,\left (\frac {p\,x^6}{9}+\frac {a^2\,p\,x^2}{3\,b^2}-\frac {a\,p\,x^4}{6\,b}\right )-\frac {5\,a\,p^2\,x^4}{36\,b}-\frac {11\,a^3\,p^2\,\ln \left (b\,x^2+a\right )}{18\,b^3}+\frac {11\,a^2\,p^2\,x^2}{18\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*log(c*(a + b*x^2)^p)^2,x)

[Out]

(p^2*x^6)/27 + log(c*(a + b*x^2)^p)^2*(x^6/6 + a^3/(6*b^3)) - log(c*(a + b*x^2)^p)*((p*x^6)/9 + (a^2*p*x^2)/(3
*b^2) - (a*p*x^4)/(6*b)) - (5*a*p^2*x^4)/(36*b) - (11*a^3*p^2*log(a + b*x^2))/(18*b^3) + (11*a^2*p^2*x^2)/(18*
b^2)

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sympy [A]  time = 22.72, size = 267, normalized size = 1.24 \[ \begin {cases} \frac {a^{3} p^{2} \log {\left (a + b x^{2} \right )}^{2}}{6 b^{3}} - \frac {11 a^{3} p^{2} \log {\left (a + b x^{2} \right )}}{18 b^{3}} + \frac {a^{3} p \log {\relax (c )} \log {\left (a + b x^{2} \right )}}{3 b^{3}} - \frac {a^{2} p^{2} x^{2} \log {\left (a + b x^{2} \right )}}{3 b^{2}} + \frac {11 a^{2} p^{2} x^{2}}{18 b^{2}} - \frac {a^{2} p x^{2} \log {\relax (c )}}{3 b^{2}} + \frac {a p^{2} x^{4} \log {\left (a + b x^{2} \right )}}{6 b} - \frac {5 a p^{2} x^{4}}{36 b} + \frac {a p x^{4} \log {\relax (c )}}{6 b} + \frac {p^{2} x^{6} \log {\left (a + b x^{2} \right )}^{2}}{6} - \frac {p^{2} x^{6} \log {\left (a + b x^{2} \right )}}{9} + \frac {p^{2} x^{6}}{27} + \frac {p x^{6} \log {\relax (c )} \log {\left (a + b x^{2} \right )}}{3} - \frac {p x^{6} \log {\relax (c )}}{9} + \frac {x^{6} \log {\relax (c )}^{2}}{6} & \text {for}\: b \neq 0 \\\frac {x^{6} \log {\left (a^{p} c \right )}^{2}}{6} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*ln(c*(b*x**2+a)**p)**2,x)

[Out]

Piecewise((a**3*p**2*log(a + b*x**2)**2/(6*b**3) - 11*a**3*p**2*log(a + b*x**2)/(18*b**3) + a**3*p*log(c)*log(
a + b*x**2)/(3*b**3) - a**2*p**2*x**2*log(a + b*x**2)/(3*b**2) + 11*a**2*p**2*x**2/(18*b**2) - a**2*p*x**2*log
(c)/(3*b**2) + a*p**2*x**4*log(a + b*x**2)/(6*b) - 5*a*p**2*x**4/(36*b) + a*p*x**4*log(c)/(6*b) + p**2*x**6*lo
g(a + b*x**2)**2/6 - p**2*x**6*log(a + b*x**2)/9 + p**2*x**6/27 + p*x**6*log(c)*log(a + b*x**2)/3 - p*x**6*log
(c)/9 + x**6*log(c)**2/6, Ne(b, 0)), (x**6*log(a**p*c)**2/6, True))

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