\(\int x^5 \log ^3(c (a+b x^2)^p) \, dx\) [91]

Optimal result

Integrand size = 18, antiderivative size = 334 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=-\frac {3 a^2 p^3 x^2}{b^2}+\frac {3 a p^3 \left (a+b x^2\right )^2}{8 b^3}-\frac {p^3 \left (a+b x^2\right )^3}{27 b^3}+\frac {3 a^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}-\frac {3 a p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^3}+\frac {p^2 \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}-\frac {3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^3}-\frac {p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3} \]

[Out]

-3*a^2*p^3*x^2/b^2+3/8*a*p^3*(b*x^2+a)^2/b^3-1/27*p^3*(b*x^2+a)^3/b^3+3*a^2*p^2*(b*x^2+a)*ln(c*(b*x^2+a)^p)/b^
3-3/4*a*p^2*(b*x^2+a)^2*ln(c*(b*x^2+a)^p)/b^3+1/9*p^2*(b*x^2+a)^3*ln(c*(b*x^2+a)^p)/b^3-3/2*a^2*p*(b*x^2+a)*ln
(c*(b*x^2+a)^p)^2/b^3+3/4*a*p*(b*x^2+a)^2*ln(c*(b*x^2+a)^p)^2/b^3-1/6*p*(b*x^2+a)^3*ln(c*(b*x^2+a)^p)^2/b^3+1/
2*a^2*(b*x^2+a)*ln(c*(b*x^2+a)^p)^3/b^3-1/2*a*(b*x^2+a)^2*ln(c*(b*x^2+a)^p)^3/b^3+1/6*(b*x^2+a)^3*ln(c*(b*x^2+
a)^p)^3/b^3

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {3 a^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}+\frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {3 a^2 p^3 x^2}{b^2}+\frac {p^2 \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}-\frac {3 a p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^3}-\frac {p^3 \left (a+b x^2\right )^3}{27 b^3}+\frac {3 a p^3 \left (a+b x^2\right )^2}{8 b^3} \]

[In]

Int[x^5*Log[c*(a + b*x^2)^p]^3,x]

[Out]

(-3*a^2*p^3*x^2)/b^2 + (3*a*p^3*(a + b*x^2)^2)/(8*b^3) - (p^3*(a + b*x^2)^3)/(27*b^3) + (3*a^2*p^2*(a + b*x^2)
*Log[c*(a + b*x^2)^p])/b^3 - (3*a*p^2*(a + b*x^2)^2*Log[c*(a + b*x^2)^p])/(4*b^3) + (p^2*(a + b*x^2)^3*Log[c*(
a + b*x^2)^p])/(9*b^3) - (3*a^2*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*b^3) + (3*a*p*(a + b*x^2)^2*Log[c*(a
+ b*x^2)^p]^2)/(4*b^3) - (p*(a + b*x^2)^3*Log[c*(a + b*x^2)^p]^2)/(6*b^3) + (a^2*(a + b*x^2)*Log[c*(a + b*x^2)
^p]^3)/(2*b^3) - (a*(a + b*x^2)^2*Log[c*(a + b*x^2)^p]^3)/(2*b^3) + ((a + b*x^2)^3*Log[c*(a + b*x^2)^p]^3)/(6*
b^3)

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2341

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

Rule 2342

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

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

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

Rule 2448

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

Rule 2504

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*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^2 \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2 \log ^3\left (c (a+b x)^p\right )}{b^2}-\frac {2 a (a+b x) \log ^3\left (c (a+b x)^p\right )}{b^2}+\frac {(a+b x)^2 \log ^3\left (c (a+b x)^p\right )}{b^2}\right ) \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int (a+b x)^2 \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b^2}-\frac {a \text {Subst}\left (\int (a+b x) \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{b^2}+\frac {a^2 \text {Subst}\left (\int \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b^2} \\ & = \frac {\text {Subst}\left (\int x^2 \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3}-\frac {a \text {Subst}\left (\int x \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3} \\ & = \frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac {p \text {Subst}\left (\int x^2 \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3}+\frac {(3 a p) \text {Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3}-\frac {\left (3 a^2 p\right ) \text {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3} \\ & = -\frac {3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^3}-\frac {p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {p^2 \text {Subst}\left (\int x^2 \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{3 b^3}-\frac {\left (3 a p^2\right ) \text {Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3}+\frac {\left (3 a^2 p^2\right ) \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{b^3} \\ & = -\frac {3 a^2 p^3 x^2}{b^2}+\frac {3 a p^3 \left (a+b x^2\right )^2}{8 b^3}-\frac {p^3 \left (a+b x^2\right )^3}{27 b^3}+\frac {3 a^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}-\frac {3 a p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^3}+\frac {p^2 \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}-\frac {3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^3}-\frac {p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.53 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {b p^3 x^2 \left (-510 a^2+57 a b x^2-8 b^2 x^4\right )+114 a^3 p^3 \log \left (a+b x^2\right )+6 p^2 \left (66 a^3+66 a^2 b x^2-15 a b^2 x^4+4 b^3 x^6\right ) \log \left (c \left (a+b x^2\right )^p\right )-18 p \left (11 a^3+6 a^2 b x^2-3 a b^2 x^4+2 b^3 x^6\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+36 \left (a^3+b^3 x^6\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{216 b^3} \]

[In]

Integrate[x^5*Log[c*(a + b*x^2)^p]^3,x]

[Out]

(b*p^3*x^2*(-510*a^2 + 57*a*b*x^2 - 8*b^2*x^4) + 114*a^3*p^3*Log[a + b*x^2] + 6*p^2*(66*a^3 + 66*a^2*b*x^2 - 1
5*a*b^2*x^4 + 4*b^3*x^6)*Log[c*(a + b*x^2)^p] - 18*p*(11*a^3 + 6*a^2*b*x^2 - 3*a*b^2*x^4 + 2*b^3*x^6)*Log[c*(a
+ b*x^2)^p]^2 + 36*(a^3 + b^3*x^6)*Log[c*(a + b*x^2)^p]^3)/(216*b^3)

Maple [A] (verified)

Time = 1.59 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.87

method	result	size
parallelrisch	\(\frac {36 x^{6} {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3} b^{3}-36 x^{6} {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{2} b^{3} p +24 x^{6} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) b^{3} p^{2}-8 b^{3} p^{3} x^{6}+54 x^{4} {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{2} a \,b^{2} p -90 x^{4} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) a \,b^{2} p^{2}+57 a \,b^{2} p^{3} x^{4}-108 x^{2} {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{2} a^{2} b p +396 x^{2} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) a^{2} b \,p^{2}-510 a^{2} b \,p^{3} x^{2}+906 \ln \left (b \,x^{2}+a \right ) a^{3} p^{3}+36 {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3} a^{3}-198 {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{2} a^{3} p -396 \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) a^{3} p^{2}+510 a^{3} p^{3}}{216 b^{3}}\)	\(289\)
risch	\(\text {Expression too large to display}\)	\(5905\)

[In]

int(x^5*ln(c*(b*x^2+a)^p)^3,x,method=_RETURNVERBOSE)

[Out]

1/216*(36*x^6*ln(c*(b*x^2+a)^p)^3*b^3-36*x^6*ln(c*(b*x^2+a)^p)^2*b^3*p+24*x^6*ln(c*(b*x^2+a)^p)*b^3*p^2-8*b^3*
p^3*x^6+54*x^4*ln(c*(b*x^2+a)^p)^2*a*b^2*p-90*x^4*ln(c*(b*x^2+a)^p)*a*b^2*p^2+57*a*b^2*p^3*x^4-108*x^2*ln(c*(b
*x^2+a)^p)^2*a^2*b*p+396*x^2*ln(c*(b*x^2+a)^p)*a^2*b*p^2-510*a^2*b*p^3*x^2+906*ln(b*x^2+a)*a^3*p^3+36*ln(c*(b*
x^2+a)^p)^3*a^3-198*ln(c*(b*x^2+a)^p)^2*a^3*p-396*ln(c*(b*x^2+a)^p)*a^3*p^2+510*a^3*p^3)/b^3

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.07 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=-\frac {8 \, b^{3} p^{3} x^{6} - 36 \, b^{3} x^{6} \log \left (c\right )^{3} - 57 \, a b^{2} p^{3} x^{4} + 510 \, a^{2} b p^{3} x^{2} - 36 \, {\left (b^{3} p^{3} x^{6} + a^{3} p^{3}\right )} \log \left (b x^{2} + a\right )^{3} + 18 \, {\left (2 \, b^{3} p^{3} x^{6} - 3 \, a b^{2} p^{3} x^{4} + 6 \, a^{2} b p^{3} x^{2} + 11 \, a^{3} p^{3} - 6 \, {\left (b^{3} p^{2} x^{6} + a^{3} p^{2}\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right )^{2} + 18 \, {\left (2 \, b^{3} p x^{6} - 3 \, a b^{2} p x^{4} + 6 \, a^{2} b p x^{2}\right )} \log \left (c\right )^{2} - 6 \, {\left (4 \, b^{3} p^{3} x^{6} - 15 \, a b^{2} p^{3} x^{4} + 66 \, a^{2} b p^{3} x^{2} + 85 \, a^{3} p^{3} + 18 \, {\left (b^{3} p x^{6} + a^{3} p\right )} \log \left (c\right )^{2} - 6 \, {\left (2 \, b^{3} p^{2} x^{6} - 3 \, a b^{2} p^{2} x^{4} + 6 \, a^{2} b p^{2} x^{2} + 11 \, a^{3} p^{2}\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right ) - 6 \, {\left (4 \, b^{3} p^{2} x^{6} - 15 \, a b^{2} p^{2} x^{4} + 66 \, a^{2} b p^{2} x^{2}\right )} \log \left (c\right )}{216 \, b^{3}} \]

[In]

integrate(x^5*log(c*(b*x^2+a)^p)^3,x, algorithm="fricas")

[Out]

-1/216*(8*b^3*p^3*x^6 - 36*b^3*x^6*log(c)^3 - 57*a*b^2*p^3*x^4 + 510*a^2*b*p^3*x^2 - 36*(b^3*p^3*x^6 + a^3*p^3
)*log(b*x^2 + a)^3 + 18*(2*b^3*p^3*x^6 - 3*a*b^2*p^3*x^4 + 6*a^2*b*p^3*x^2 + 11*a^3*p^3 - 6*(b^3*p^2*x^6 + a^3
*p^2)*log(c))*log(b*x^2 + a)^2 + 18*(2*b^3*p*x^6 - 3*a*b^2*p*x^4 + 6*a^2*b*p*x^2)*log(c)^2 - 6*(4*b^3*p^3*x^6
- 15*a*b^2*p^3*x^4 + 66*a^2*b*p^3*x^2 + 85*a^3*p^3 + 18*(b^3*p*x^6 + a^3*p)*log(c)^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)*log(c))*log(b*x^2 + a) - 6*(4*b^3*p^2*x^6 - 15*a*b^2*p^2*x^4 + 66*
a^2*b*p^2*x^2)*log(c))/b^3

Sympy [A] (verification not implemented)

Time = 5.00 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.87 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\begin {cases} \frac {85 a^{3} p^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{36 b^{3}} - \frac {11 a^{3} p \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{12 b^{3}} + \frac {a^{3} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{6 b^{3}} - \frac {85 a^{2} p^{3} x^{2}}{36 b^{2}} + \frac {11 a^{2} p^{2} x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{6 b^{2}} - \frac {a^{2} p x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{2 b^{2}} + \frac {19 a p^{3} x^{4}}{72 b} - \frac {5 a p^{2} x^{4} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{12 b} + \frac {a p x^{4} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{4 b} - \frac {p^{3} x^{6}}{27} + \frac {p^{2} x^{6} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{9} - \frac {p x^{6} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{6} + \frac {x^{6} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{6} & \text {for}\: b \neq 0 \\\frac {x^{6} \log {\left (a^{p} c \right )}^{3}}{6} & \text {otherwise} \end {cases} \]

[In]

integrate(x**5*ln(c*(b*x**2+a)**p)**3,x)

[Out]

Piecewise((85*a**3*p**2*log(c*(a + b*x**2)**p)/(36*b**3) - 11*a**3*p*log(c*(a + b*x**2)**p)**2/(12*b**3) + a**
3*log(c*(a + b*x**2)**p)**3/(6*b**3) - 85*a**2*p**3*x**2/(36*b**2) + 11*a**2*p**2*x**2*log(c*(a + b*x**2)**p)/
(6*b**2) - a**2*p*x**2*log(c*(a + b*x**2)**p)**2/(2*b**2) + 19*a*p**3*x**4/(72*b) - 5*a*p**2*x**4*log(c*(a + b
*x**2)**p)/(12*b) + a*p*x**4*log(c*(a + b*x**2)**p)**2/(4*b) - p**3*x**6/27 + p**2*x**6*log(c*(a + b*x**2)**p)
/9 - p*x**6*log(c*(a + b*x**2)**p)**2/6 + x**6*log(c*(a + b*x**2)**p)**3/6, Ne(b, 0)), (x**6*log(a**p*c)**3/6,
True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.72 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {1}{6} \, x^{6} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} + \frac {1}{12} \, b p {\left (\frac {6 \, a^{3} \log \left (b x^{2} + a\right )}{b^{4}} - \frac {2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{b^{3}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} - \frac {1}{216} \, b p {\left (\frac {{\left (8 \, b^{3} x^{6} - 57 \, a b^{2} x^{4} - 36 \, a^{3} \log \left (b x^{2} + a\right )^{3} + 510 \, a^{2} b x^{2} - 198 \, a^{3} \log \left (b x^{2} + a\right )^{2} - 510 \, a^{3} \log \left (b x^{2} + a\right )\right )} p^{2}}{b^{4}} - \frac {6 \, {\left (4 \, b^{3} x^{6} - 15 \, a b^{2} x^{4} + 66 \, a^{2} b x^{2} - 18 \, a^{3} \log \left (b x^{2} + a\right )^{2} - 66 \, a^{3} \log \left (b x^{2} + a\right )\right )} p \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{b^{4}}\right )} \]

[In]

integrate(x^5*log(c*(b*x^2+a)^p)^3,x, algorithm="maxima")

[Out]

1/6*x^6*log((b*x^2 + a)^p*c)^3 + 1/12*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)^2 - 1/216*b*p*((8*b^3*x^6 - 57*a*b^2*x^4 - 36*a^3*log(b*x^2 + a)^3 + 510*a^2*b*x^2 - 198
*a^3*log(b*x^2 + a)^2 - 510*a^3*log(b*x^2 + a))*p^2/b^4 - 6*(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*log((b*x^2 + a)^p*c)/b^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 662 vs. \(2 (314) = 628\).

Time = 0.32 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.98 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{3} p^{3} \log \left (b x^{2} + a\right )^{3}}{6 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{2} a p^{3} \log \left (b x^{2} + a\right )^{3}}{2 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{3} p^{3} \log \left (b x^{2} + a\right )^{2}}{6 \, b^{3}} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} a p^{3} \log \left (b x^{2} + a\right )^{2}}{4 \, b^{3}} + \frac {{\left (b x^{2} + a\right )}^{3} p^{2} \log \left (b x^{2} + a\right )^{2} \log \left (c\right )}{2 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{2} a p^{2} \log \left (b x^{2} + a\right )^{2} \log \left (c\right )}{2 \, b^{3}} + \frac {{\left (b x^{2} + a\right )}^{3} p^{3} \log \left (b x^{2} + a\right )}{9 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{2} a p^{3} \log \left (b x^{2} + a\right )}{4 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{3} p^{2} \log \left (b x^{2} + a\right ) \log \left (c\right )}{3 \, b^{3}} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} a p^{2} \log \left (b x^{2} + a\right ) \log \left (c\right )}{2 \, b^{3}} + \frac {{\left (b x^{2} + a\right )}^{3} p \log \left (b x^{2} + a\right ) \log \left (c\right )^{2}}{2 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{2} a p \log \left (b x^{2} + a\right ) \log \left (c\right )^{2}}{2 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{3} p^{3}}{27 \, b^{3}} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} a p^{3}}{8 \, b^{3}} + \frac {{\left (b x^{2} + a\right )}^{3} p^{2} \log \left (c\right )}{9 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{2} a p^{2} \log \left (c\right )}{4 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{3} p \log \left (c\right )^{2}}{6 \, b^{3}} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} a p \log \left (c\right )^{2}}{4 \, b^{3}} + \frac {{\left (b x^{2} + a\right )}^{3} \log \left (c\right )^{3}}{6 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{2} a \log \left (c\right )^{3}}{2 \, b^{3}} + \frac {{\left ({\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right )^{3} - 6 \, b x^{2} - 3 \, {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right )^{2} + 6 \, {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) - 6 \, a\right )} a^{2} p^{3} + 3 \, {\left (2 \, b x^{2} + {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right )^{2} - 2 \, {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + 2 \, a\right )} a^{2} p^{2} \log \left (c\right ) - 3 \, {\left (b x^{2} - {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + a\right )} a^{2} p \log \left (c\right )^{2} + {\left (b x^{2} + a\right )} a^{2} \log \left (c\right )^{3}}{2 \, b^{3}} \]

[In]

integrate(x^5*log(c*(b*x^2+a)^p)^3,x, algorithm="giac")

[Out]

1/6*(b*x^2 + a)^3*p^3*log(b*x^2 + a)^3/b^3 - 1/2*(b*x^2 + a)^2*a*p^3*log(b*x^2 + a)^3/b^3 - 1/6*(b*x^2 + a)^3*
p^3*log(b*x^2 + a)^2/b^3 + 3/4*(b*x^2 + a)^2*a*p^3*log(b*x^2 + a)^2/b^3 + 1/2*(b*x^2 + a)^3*p^2*log(b*x^2 + a)
^2*log(c)/b^3 - 3/2*(b*x^2 + a)^2*a*p^2*log(b*x^2 + a)^2*log(c)/b^3 + 1/9*(b*x^2 + a)^3*p^3*log(b*x^2 + a)/b^3
- 3/4*(b*x^2 + a)^2*a*p^3*log(b*x^2 + a)/b^3 - 1/3*(b*x^2 + a)^3*p^2*log(b*x^2 + a)*log(c)/b^3 + 3/2*(b*x^2 +
a)^2*a*p^2*log(b*x^2 + a)*log(c)/b^3 + 1/2*(b*x^2 + a)^3*p*log(b*x^2 + a)*log(c)^2/b^3 - 3/2*(b*x^2 + a)^2*a*
p*log(b*x^2 + a)*log(c)^2/b^3 - 1/27*(b*x^2 + a)^3*p^3/b^3 + 3/8*(b*x^2 + a)^2*a*p^3/b^3 + 1/9*(b*x^2 + a)^3*p
^2*log(c)/b^3 - 3/4*(b*x^2 + a)^2*a*p^2*log(c)/b^3 - 1/6*(b*x^2 + a)^3*p*log(c)^2/b^3 + 3/4*(b*x^2 + a)^2*a*p*
log(c)^2/b^3 + 1/6*(b*x^2 + a)^3*log(c)^3/b^3 - 1/2*(b*x^2 + a)^2*a*log(c)^3/b^3 + 1/2*(((b*x^2 + a)*log(b*x^2
+ a)^3 - 6*b*x^2 - 3*(b*x^2 + a)*log(b*x^2 + a)^2 + 6*(b*x^2 + a)*log(b*x^2 + a) - 6*a)*a^2*p^3 + 3*(2*b*x^2
+ (b*x^2 + a)*log(b*x^2 + a)^2 - 2*(b*x^2 + a)*log(b*x^2 + a) + 2*a)*a^2*p^2*log(c) - 3*(b*x^2 - (b*x^2 + a)*l
og(b*x^2 + a) + a)*a^2*p*log(c)^2 + (b*x^2 + a)*a^2*log(c)^3)/b^3

Mupad [B] (verification not implemented)

Time = 1.43 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.56 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx={\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3\,\left (\frac {x^6}{6}+\frac {a^3}{6\,b^3}\right )-{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2\,\left (\frac {p\,x^6}{6}+\frac {11\,a^3\,p}{12\,b^3}+\frac {a^2\,p\,x^2}{2\,b^2}-\frac {a\,p\,x^4}{4\,b}\right )-\frac {p^3\,x^6}{27}+\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )\,\left (\frac {b\,p^2\,x^6}{3}-\frac {5\,a\,p^2\,x^4}{4}+\frac {11\,a^2\,p^2\,x^2}{2\,b}\right )}{3\,b}+\frac {19\,a\,p^3\,x^4}{72\,b}+\frac {85\,a^3\,p^3\,\ln \left (b\,x^2+a\right )}{36\,b^3}-\frac {85\,a^2\,p^3\,x^2}{36\,b^2} \]

[In]

int(x^5*log(c*(a + b*x^2)^p)^3,x)

[Out]

log(c*(a + b*x^2)^p)^3*(x^6/6 + a^3/(6*b^3)) - log(c*(a + b*x^2)^p)^2*((p*x^6)/6 + (11*a^3*p)/(12*b^3) + (a^2*
p*x^2)/(2*b^2) - (a*p*x^4)/(4*b)) - (p^3*x^6)/27 + (log(c*(a + b*x^2)^p)*((b*p^2*x^6)/3 - (5*a*p^2*x^4)/4 + (1
1*a^2*p^2*x^2)/(2*b)))/(3*b) + (19*a*p^3*x^4)/(72*b) + (85*a^3*p^3*log(a + b*x^2))/(36*b^3) - (85*a^2*p^3*x^2)
/(36*b^2)