3.92 \(\int x^3 \log ^3(c (a+b x^2)^p) \, dx\)

Optimal. Leaf size=211 \[ \frac{3 p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{8 b^2}-\frac{3 a p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}-\frac{3 p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{8 b^2}+\frac{3 a p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac{\left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac{a \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^2}-\frac{3 p^3 \left (a+b x^2\right )^2}{16 b^2}+\frac{3 a p^3 x^2}{b} \]

[Out]

(3*a*p^3*x^2)/b - (3*p^3*(a + b*x^2)^2)/(16*b^2) - (3*a*p^2*(a + b*x^2)*Log[c*(a + b*x^2)^p])/b^2 + (3*p^2*(a
+ b*x^2)^2*Log[c*(a + b*x^2)^p])/(8*b^2) + (3*a*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*b^2) - (3*p*(a + b*x^
2)^2*Log[c*(a + b*x^2)^p]^2)/(8*b^2) - (a*(a + b*x^2)*Log[c*(a + b*x^2)^p]^3)/(2*b^2) + ((a + b*x^2)^2*Log[c*(
a + b*x^2)^p]^3)/(4*b^2)

________________________________________________________________________________________

Rubi [A]  time = 0.205337, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{3 p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{8 b^2}-\frac{3 a p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}-\frac{3 p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{8 b^2}+\frac{3 a p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac{\left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac{a \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^2}-\frac{3 p^3 \left (a+b x^2\right )^2}{16 b^2}+\frac{3 a p^3 x^2}{b} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Log[c*(a + b*x^2)^p]^3,x]

[Out]

(3*a*p^3*x^2)/b - (3*p^3*(a + b*x^2)^2)/(16*b^2) - (3*a*p^2*(a + b*x^2)*Log[c*(a + b*x^2)^p])/b^2 + (3*p^2*(a
+ b*x^2)^2*Log[c*(a + b*x^2)^p])/(8*b^2) + (3*a*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*b^2) - (3*p*(a + b*x^
2)^2*Log[c*(a + b*x^2)^p]^2)/(8*b^2) - (a*(a + b*x^2)*Log[c*(a + b*x^2)^p]^3)/(2*b^2) + ((a + b*x^2)^2*Log[c*(
a + b*x^2)^p]^3)/(4*b^2)

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])

Rule 2401

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 2389

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 2296

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 2295

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

Rule 2390

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 2305

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 2304

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
]

Rubi steps

\begin{align*} \int x^3 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a \log ^3\left (c (a+b x)^p\right )}{b}+\frac{(a+b x) \log ^3\left (c (a+b x)^p\right )}{b}\right ) \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int (a+b x) \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b}-\frac{a \operatorname{Subst}\left (\int \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int x \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2}-\frac{a \operatorname{Subst}\left (\int \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2}\\ &=-\frac{a \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac{\left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac{(3 p) \operatorname{Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{4 b^2}+\frac{(3 a p) \operatorname{Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2}\\ &=\frac{3 a p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}-\frac{3 p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{8 b^2}-\frac{a \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac{\left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{4 b^2}+\frac{\left (3 p^2\right ) \operatorname{Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{4 b^2}-\frac{\left (3 a p^2\right ) \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{b^2}\\ &=\frac{3 a p^3 x^2}{b}-\frac{3 p^3 \left (a+b x^2\right )^2}{16 b^2}-\frac{3 a p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac{3 p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{8 b^2}+\frac{3 a p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}-\frac{3 p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{8 b^2}-\frac{a \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac{\left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{4 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0862747, size = 237, normalized size = 1.12 \[ -\frac{9 a^2 p^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac{a^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{4 b^2}+\frac{9 a^2 p \log ^2\left (c \left (a+b x^2\right )^p\right )}{8 b^2}-\frac{3 a^2 p^3 \log \left (a+b x^2\right )}{8 b^2}+\frac{3}{8} p^2 x^4 \log \left (c \left (a+b x^2\right )^p\right )-\frac{9 a p^2 x^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b}+\frac{1}{4} x^4 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{3}{8} p x^4 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{3 a p x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b}+\frac{21 a p^3 x^2}{8 b}-\frac{3}{16} p^3 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*a*p^3*x^2)/(8*b) - (3*p^3*x^4)/16 - (3*a^2*p^3*Log[a + b*x^2])/(8*b^2) - (9*a^2*p^2*Log[c*(a + b*x^2)^p])/
(4*b^2) - (9*a*p^2*x^2*Log[c*(a + b*x^2)^p])/(4*b) + (3*p^2*x^4*Log[c*(a + b*x^2)^p])/8 + (9*a^2*p*Log[c*(a +
b*x^2)^p]^2)/(8*b^2) + (3*a*p*x^2*Log[c*(a + b*x^2)^p]^2)/(4*b) - (3*p*x^4*Log[c*(a + b*x^2)^p]^2)/8 - (a^2*Lo
g[c*(a + b*x^2)^p]^3)/(4*b^2) + (x^4*Log[c*(a + b*x^2)^p]^3)/4

________________________________________________________________________________________

Maple [C]  time = 0.87, size = 4942, normalized size = 23.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*ln(c*(b*x^2+a)^p)^3,x)

[Out]

-3/16*x^4*p^3+3/8*(I*Pi*b^2*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-I*Pi*b^2*x^4*csgn(I*(b*x^2+a)^p)*c
sgn(I*c*(b*x^2+a)^p)*csgn(I*c)-I*Pi*b^2*x^4*csgn(I*c*(b*x^2+a)^p)^3+I*Pi*b^2*x^4*csgn(I*c*(b*x^2+a)^p)^2*csgn(
I*c)+2*ln(c)*b^2*x^4-b^2*p*x^4+2*a*b*p*x^2-2*a^2*p*ln(b*x^2+a))/b^2*ln((b*x^2+a)^p)^2+3/8*ln(c)*Pi^2*x^4*csgn(
I*c*(b*x^2+a)^p)^5*csgn(I*c)-3/16*ln(c)*Pi^2*x^4*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)^2+3/32*Pi^2*p*x^4*csgn(I*(b
*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^4-3/16*Pi^2*p*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^5-3/16*Pi^2*p*x
^4*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)+3/32*Pi^2*p*x^4*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)^2-1/32*I*Pi^3*x^4*csgn(
I*(b*x^2+a)^p)^3*csgn(I*c*(b*x^2+a)^p)^6+3/32*I*Pi^3*x^4*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^7-3/32*I*
Pi^3*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^8-3/32*I*Pi^3*x^4*csgn(I*c*(b*x^2+a)^p)^8*csgn(I*c)+3/32*I*
Pi^3*x^4*csgn(I*c*(b*x^2+a)^p)^7*csgn(I*c)^2-1/32*I*Pi^3*x^4*csgn(I*c*(b*x^2+a)^p)^6*csgn(I*c)^3-3/8*I*ln(c)^2
*Pi*x^4*csgn(I*c*(b*x^2+a)^p)^3-3/16*I*Pi*p^2*x^4*csgn(I*c*(b*x^2+a)^p)^3+3/8*ln(c)*p^2*x^4-3/8*ln(c)^2*p*x^4-
1/4/b^2*a^2*p^3*ln(b*x^2+a)^3-9/8/b^2*a^2*p^3*ln(b*x^2+a)^2-3/16*ln(c)*Pi^2*x^4*csgn(I*c*(b*x^2+a)^p)^6+3/32*P
i^2*p*x^4*csgn(I*c*(b*x^2+a)^p)^6+1/32*I*Pi^3*x^4*csgn(I*c*(b*x^2+a)^p)^9+21/8*a*p^3*x^2/b-21/8*a^2*p^3/b^2*ln
(b*x^2+a)-3/16*ln(c)*Pi^2*x^4*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^4+3/4/b*ln(c)^2*a*p*x^2+3/4/b^2*ln(c
)*a^2*p^2*ln(b*x^2+a)^2-9/4/b*ln(c)*a*p^2*x^2-3/4/b^2*ln(c)^2*ln(b*x^2+a)*a^2*p+9/4/b^2*ln(c)*ln(b*x^2+a)*a^2*
p^2+3/8*ln(c)*Pi^2*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^5+1/4*x^4*ln((b*x^2+a)^p)^3+3/16*(-12*x^2*b*a
*p^2-4*I*Pi*ln(b*x^2+a)*a^2*p*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-4*I*Pi*a*b*p*x^2*csgn(I*c*(b*x^2+a)^p)^3+4*I*l
n(c)*Pi*b^2*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+12*ln(b*x^2+a)*a^2*p^2+4*ln(c)^2*b^2*x^4+4*a^2*p^2
*ln(b*x^2+a)^2-4*ln(c)*b^2*p*x^4-8*ln(c)*ln(b*x^2+a)*a^2*p+2*x^4*b^2*p^2+8*ln(c)*a*b*p*x^2-Pi^2*b^2*x^4*csgn(I
*c*(b*x^2+a)^p)^6-4*I*Pi*ln(b*x^2+a)*a^2*p*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+4*I*Pi*ln(b*x^2+a)*a^2*
p*csgn(I*c*(b*x^2+a)^p)^3-4*I*Pi*a*b*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-Pi^2*b^2*x^4*cs
gn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^4+2*Pi^2*b^2*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^5+2*Pi^2*
b^2*x^4*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)-Pi^2*b^2*x^4*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)^2+2*Pi^2*b^2*x^4*csgn
(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)-Pi^2*b^2*x^4*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^2
*csgn(I*c)^2-4*Pi^2*b^2*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)+2*Pi^2*b^2*x^4*csgn(I*(b*x^2
+a)^p)*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)^2-4*I*ln(c)*Pi*b^2*x^4*csgn(I*c*(b*x^2+a)^p)^3+2*I*Pi*b^2*p*x^4*csgn(
I*c*(b*x^2+a)^p)^3+2*I*Pi*b^2*p*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+4*I*Pi*ln(b*x^2+a)*a^2
*p*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-2*I*Pi*b^2*p*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)
^p)^2-2*I*Pi*b^2*p*x^4*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+4*I*Pi*a*b*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+
a)^p)^2+4*I*Pi*a*b*p*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-4*I*ln(c)*Pi*b^2*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(
b*x^2+a)^p)*csgn(I*c)+4*I*ln(c)*Pi*b^2*x^4*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c))/b^2*ln((b*x^2+a)^p)+9/8*I/b*Pi*a
*p^2*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-3/4*I/b^2*ln(c)*Pi*ln(b*x^2+a)*a^2*p*csgn(I*(b*x^
2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-3/4*I/b^2*ln(c)*Pi*ln(b*x^2+a)*a^2*p*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-9/8*I/b
^2*Pi*ln(b*x^2+a)*a^2*p^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+3/4*I/b*ln(c)*Pi*a*p*x^2*csgn(I*
(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+3/4*I/b*ln(c)*Pi*a*p*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-3/8*I/b^2*Pi*a
^2*p^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)*ln(b*x^2+a)^2-3/4*I/b*ln(c)*Pi*a*p*x^2*csgn(I*c*(b*
x^2+a)^p)^3+3/8*I/b^2*Pi*a^2*p^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2*ln(b*x^2+a)^2-9/8*I/b*Pi*a*p^2*x^
2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+3/8*I/b^2*Pi*a^2*p^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)*ln(b*x^2+
a)^2-9/8*I/b*Pi*a*p^2*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+3/4*I/b^2*ln(c)*Pi*ln(b*x^2+a)*a^2*p*csgn(I*c*(b*x
^2+a)^p)^3+9/8*I/b^2*Pi*ln(b*x^2+a)*a^2*p^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+9/8*I/b^2*Pi*ln(b*x^2+
a)*a^2*p^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+3/8/b*Pi^2*a*p*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^3*
csgn(I*c)-3/16/b*Pi^2*a*p*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)^2+3/8*I*ln(c)*Pi*p*x^4*c
sgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-3/4/b*Pi^2*a*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p
)^4*csgn(I*c)+3/8/b*Pi^2*a*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)^2-3/8/b^2*Pi^2*ln(b*x^2
+a)*a^2*p*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)+3/16/b^2*Pi^2*ln(b*x^2+a)*a^2*p*csgn(I*(b*x^
2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)^2+3/4/b^2*Pi^2*ln(b*x^2+a)*a^2*p*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x
^2+a)^p)^4*csgn(I*c)-3/8/b^2*Pi^2*ln(b*x^2+a)*a^2*p*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)^2+3/
8*ln(c)*Pi^2*x^4*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)-3/16*ln(c)*Pi^2*x^4*csgn(I*(b*x^2+a)^
p)^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)^2-3/16/b*Pi^2*a*p*x^2*csgn(I*c*(b*x^2+a)^p)^6+3/16/b^2*Pi^2*ln(b*x^2+a)
*a^2*p*csgn(I*c*(b*x^2+a)^p)^6-3/4*ln(c)*Pi^2*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)+3/8*ln
(c)*Pi^2*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)^2-3/16*Pi^2*p*x^4*csgn(I*(b*x^2+a)^p)^2*csg
n(I*c*(b*x^2+a)^p)^3*csgn(I*c)+3/32*Pi^2*p*x^4*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)^2+3/8*P
i^2*p*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)-3/16*Pi^2*p*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(
b*x^2+a)^p)^3*csgn(I*c)^2+3/32*I*Pi^3*x^4*csgn(I*(b*x^2+a)^p)^3*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)-3/32*I*Pi^3*
x^4*csgn(I*(b*x^2+a)^p)^3*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)^2+1/32*I*Pi^3*x^4*csgn(I*(b*x^2+a)^p)^3*csgn(I*c*(
b*x^2+a)^p)^3*csgn(I*c)^3-9/32*I*Pi^3*x^4*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^6*csgn(I*c)+9/32*I*Pi^3*
x^4*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)^2-3/32*I*Pi^3*x^4*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(
b*x^2+a)^p)^4*csgn(I*c)^3+9/32*I*Pi^3*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^7*csgn(I*c)-9/32*I*Pi^3*x^
4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^6*csgn(I*c)^2+3/32*I*Pi^3*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+
a)^p)^5*csgn(I*c)^3+3/8*I*ln(c)^2*Pi*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+3/8*I*ln(c)^2*Pi*x^4*csgn
(I*c*(b*x^2+a)^p)^2*csgn(I*c)+3/8*I*ln(c)*Pi*p*x^4*csgn(I*c*(b*x^2+a)^p)^3+3/16*I*Pi*p^2*x^4*csgn(I*(b*x^2+a)^
p)*csgn(I*c*(b*x^2+a)^p)^2+3/16*I*Pi*p^2*x^4*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+1/4*ln(c)^3*x^4-3/4*I/b*ln(c)*P
i*a*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+3/4*I/b^2*ln(c)*Pi*ln(b*x^2+a)*a^2*p*csgn(I*(b*x
^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+3/16/b^2*Pi^2*ln(b*x^2+a)*a^2*p*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)^2-3
/16*I*Pi*p^2*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-3/8*I*ln(c)^2*Pi*x^4*csgn(I*(b*x^2+a)^p)*
csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-3/8*I*ln(c)*Pi*p*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-3/8*I*ln(c)*P
i*p*x^4*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-3/8*I/b^2*Pi*a^2*p^2*csgn(I*c*(b*x^2+a)^p)^3*ln(b*x^2+a)^2+9/8*I/b*P
i*a*p^2*x^2*csgn(I*c*(b*x^2+a)^p)^3-9/8*I/b^2*Pi*ln(b*x^2+a)*a^2*p^2*csgn(I*c*(b*x^2+a)^p)^3-3/16/b*Pi^2*a*p*x
^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^4+3/8/b*Pi^2*a*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^
5+3/8/b*Pi^2*a*p*x^2*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)-3/16/b*Pi^2*a*p*x^2*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)^2
+3/16/b^2*Pi^2*ln(b*x^2+a)*a^2*p*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^4-3/8/b^2*Pi^2*ln(b*x^2+a)*a^2*p*
csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^5-3/8/b^2*Pi^2*ln(b*x^2+a)*a^2*p*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)

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Maxima [A]  time = 1.06158, size = 274, normalized size = 1.3 \begin{align*} \frac{1}{4} \, x^{4} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} - \frac{3}{8} \, b p{\left (\frac{2 \, a^{2} \log \left (b x^{2} + a\right )}{b^{3}} + \frac{b x^{4} - 2 \, a x^{2}}{b^{2}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} - \frac{1}{16} \, b p{\left (\frac{{\left (3 \, b^{2} x^{4} + 4 \, a^{2} \log \left (b x^{2} + a\right )^{3} - 42 \, a b x^{2} + 18 \, a^{2} \log \left (b x^{2} + a\right )^{2} + 42 \, a^{2} \log \left (b x^{2} + a\right )\right )} p^{2}}{b^{3}} - \frac{6 \,{\left (b^{2} x^{4} - 6 \, a b x^{2} + 2 \, a^{2} \log \left (b x^{2} + a\right )^{2} + 6 \, a^{2} \log \left (b x^{2} + a\right )\right )} p \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{b^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*log((b*x^2 + a)^p*c)^3 - 3/8*b*p*(2*a^2*log(b*x^2 + a)/b^3 + (b*x^4 - 2*a*x^2)/b^2)*log((b*x^2 + a)^p*
c)^2 - 1/16*b*p*((3*b^2*x^4 + 4*a^2*log(b*x^2 + a)^3 - 42*a*b*x^2 + 18*a^2*log(b*x^2 + a)^2 + 42*a^2*log(b*x^2
+ a))*p^2/b^3 - 6*(b^2*x^4 - 6*a*b*x^2 + 2*a^2*log(b*x^2 + a)^2 + 6*a^2*log(b*x^2 + a))*p*log((b*x^2 + a)^p*c
)/b^3)

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Fricas [A]  time = 2.18504, size = 587, normalized size = 2.78 \begin{align*} -\frac{3 \, b^{2} p^{3} x^{4} - 4 \, b^{2} x^{4} \log \left (c\right )^{3} - 42 \, a b p^{3} x^{2} - 4 \,{\left (b^{2} p^{3} x^{4} - a^{2} p^{3}\right )} \log \left (b x^{2} + a\right )^{3} + 6 \,{\left (b^{2} p^{3} x^{4} - 2 \, a b p^{3} x^{2} - 3 \, a^{2} p^{3} - 2 \,{\left (b^{2} p^{2} x^{4} - a^{2} p^{2}\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right )^{2} + 6 \,{\left (b^{2} p x^{4} - 2 \, a b p x^{2}\right )} \log \left (c\right )^{2} - 6 \,{\left (b^{2} p^{3} x^{4} - 6 \, a b p^{3} x^{2} - 7 \, a^{2} p^{3} + 2 \,{\left (b^{2} p x^{4} - a^{2} p\right )} \log \left (c\right )^{2} - 2 \,{\left (b^{2} p^{2} x^{4} - 2 \, a b p^{2} x^{2} - 3 \, a^{2} p^{2}\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right ) - 6 \,{\left (b^{2} p^{2} x^{4} - 6 \, a b p^{2} x^{2}\right )} \log \left (c\right )}{16 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(3*b^2*p^3*x^4 - 4*b^2*x^4*log(c)^3 - 42*a*b*p^3*x^2 - 4*(b^2*p^3*x^4 - a^2*p^3)*log(b*x^2 + a)^3 + 6*(b
^2*p^3*x^4 - 2*a*b*p^3*x^2 - 3*a^2*p^3 - 2*(b^2*p^2*x^4 - a^2*p^2)*log(c))*log(b*x^2 + a)^2 + 6*(b^2*p*x^4 - 2
*a*b*p*x^2)*log(c)^2 - 6*(b^2*p^3*x^4 - 6*a*b*p^3*x^2 - 7*a^2*p^3 + 2*(b^2*p*x^4 - a^2*p)*log(c)^2 - 2*(b^2*p^
2*x^4 - 2*a*b*p^2*x^2 - 3*a^2*p^2)*log(c))*log(b*x^2 + a) - 6*(b^2*p^2*x^4 - 6*a*b*p^2*x^2)*log(c))/b^2

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Sympy [A]  time = 24.0075, size = 450, normalized size = 2.13 \begin{align*} \begin{cases} - \frac{a^{2} p^{3} \log{\left (a + b x^{2} \right )}^{3}}{4 b^{2}} + \frac{9 a^{2} p^{3} \log{\left (a + b x^{2} \right )}^{2}}{8 b^{2}} - \frac{21 a^{2} p^{3} \log{\left (a + b x^{2} \right )}}{8 b^{2}} - \frac{3 a^{2} p^{2} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}^{2}}{4 b^{2}} + \frac{9 a^{2} p^{2} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{4 b^{2}} - \frac{3 a^{2} p \log{\left (c \right )}^{2} \log{\left (a + b x^{2} \right )}}{4 b^{2}} + \frac{3 a p^{3} x^{2} \log{\left (a + b x^{2} \right )}^{2}}{4 b} - \frac{9 a p^{3} x^{2} \log{\left (a + b x^{2} \right )}}{4 b} + \frac{21 a p^{3} x^{2}}{8 b} + \frac{3 a p^{2} x^{2} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{2 b} - \frac{9 a p^{2} x^{2} \log{\left (c \right )}}{4 b} + \frac{3 a p x^{2} \log{\left (c \right )}^{2}}{4 b} + \frac{p^{3} x^{4} \log{\left (a + b x^{2} \right )}^{3}}{4} - \frac{3 p^{3} x^{4} \log{\left (a + b x^{2} \right )}^{2}}{8} + \frac{3 p^{3} x^{4} \log{\left (a + b x^{2} \right )}}{8} - \frac{3 p^{3} x^{4}}{16} + \frac{3 p^{2} x^{4} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}^{2}}{4} - \frac{3 p^{2} x^{4} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{4} + \frac{3 p^{2} x^{4} \log{\left (c \right )}}{8} + \frac{3 p x^{4} \log{\left (c \right )}^{2} \log{\left (a + b x^{2} \right )}}{4} - \frac{3 p x^{4} \log{\left (c \right )}^{2}}{8} + \frac{x^{4} \log{\left (c \right )}^{3}}{4} & \text{for}\: b \neq 0 \\\frac{x^{4} \log{\left (a^{p} c \right )}^{3}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*p**3*log(a + b*x**2)**3/(4*b**2) + 9*a**2*p**3*log(a + b*x**2)**2/(8*b**2) - 21*a**2*p**3*log
(a + b*x**2)/(8*b**2) - 3*a**2*p**2*log(c)*log(a + b*x**2)**2/(4*b**2) + 9*a**2*p**2*log(c)*log(a + b*x**2)/(4
*b**2) - 3*a**2*p*log(c)**2*log(a + b*x**2)/(4*b**2) + 3*a*p**3*x**2*log(a + b*x**2)**2/(4*b) - 9*a*p**3*x**2*
log(a + b*x**2)/(4*b) + 21*a*p**3*x**2/(8*b) + 3*a*p**2*x**2*log(c)*log(a + b*x**2)/(2*b) - 9*a*p**2*x**2*log(
c)/(4*b) + 3*a*p*x**2*log(c)**2/(4*b) + p**3*x**4*log(a + b*x**2)**3/4 - 3*p**3*x**4*log(a + b*x**2)**2/8 + 3*
p**3*x**4*log(a + b*x**2)/8 - 3*p**3*x**4/16 + 3*p**2*x**4*log(c)*log(a + b*x**2)**2/4 - 3*p**2*x**4*log(c)*lo
g(a + b*x**2)/4 + 3*p**2*x**4*log(c)/8 + 3*p*x**4*log(c)**2*log(a + b*x**2)/4 - 3*p*x**4*log(c)**2/8 + x**4*lo
g(c)**3/4, Ne(b, 0)), (x**4*log(a**p*c)**3/4, True))

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Giac [A]  time = 1.309, size = 486, normalized size = 2.3 \begin{align*} \frac{\frac{{\left (4 \,{\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right )^{3} - 8 \,{\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right )^{3} - 6 \,{\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right )^{2} + 24 \,{\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right )^{2} + 6 \,{\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right ) - 48 \,{\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right ) - 3 \,{\left (b x^{2} + a\right )}^{2} + 48 \,{\left (b x^{2} + a\right )} a\right )} p^{3}}{b} + \frac{6 \,{\left (2 \,{\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right )^{2} - 4 \,{\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right )^{2} - 2 \,{\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right ) + 8 \,{\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right ) +{\left (b x^{2} + a\right )}^{2} - 8 \,{\left (b x^{2} + a\right )} a\right )} p^{2} \log \left (c\right )}{b} + \frac{6 \,{\left (2 \,{\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right ) - 4 \,{\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right ) -{\left (b x^{2} + a\right )}^{2} + 4 \,{\left (b x^{2} + a\right )} a\right )} p \log \left (c\right )^{2}}{b} + \frac{4 \,{\left ({\left (b x^{2} + a\right )}^{2} - 2 \,{\left (b x^{2} + a\right )} a\right )} \log \left (c\right )^{3}}{b}}{16 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*((4*(b*x^2 + a)^2*log(b*x^2 + a)^3 - 8*(b*x^2 + a)*a*log(b*x^2 + a)^3 - 6*(b*x^2 + a)^2*log(b*x^2 + a)^2
+ 24*(b*x^2 + a)*a*log(b*x^2 + a)^2 + 6*(b*x^2 + a)^2*log(b*x^2 + a) - 48*(b*x^2 + a)*a*log(b*x^2 + a) - 3*(b*
x^2 + a)^2 + 48*(b*x^2 + a)*a)*p^3/b + 6*(2*(b*x^2 + a)^2*log(b*x^2 + a)^2 - 4*(b*x^2 + a)*a*log(b*x^2 + a)^2
- 2*(b*x^2 + a)^2*log(b*x^2 + a) + 8*(b*x^2 + a)*a*log(b*x^2 + a) + (b*x^2 + a)^2 - 8*(b*x^2 + a)*a)*p^2*log(c
)/b + 6*(2*(b*x^2 + a)^2*log(b*x^2 + a) - 4*(b*x^2 + a)*a*log(b*x^2 + a) - (b*x^2 + a)^2 + 4*(b*x^2 + a)*a)*p*
log(c)^2/b + 4*((b*x^2 + a)^2 - 2*(b*x^2 + a)*a)*log(c)^3/b)/b