3.93 \(\int x \log ^3(c (a+b x^2)^p) \, dx\)
Optimal. Leaf size=93 \[ \frac{3 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}-\frac{3 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}+\frac{\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b}-3 p^3 x^2 \]
[Out]
-3*p^3*x^2 + (3*p^2*(a + b*x^2)*Log[c*(a + b*x^2)^p])/b - (3*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*b) + ((a
+ b*x^2)*Log[c*(a + b*x^2)^p]^3)/(2*b)
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Rubi [A] time = 0.0656985, antiderivative size = 93, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used =
{2454, 2389, 2296, 2295} \[ \frac{3 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}-\frac{3 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}+\frac{\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b}-3 p^3 x^2 \]
Antiderivative was successfully verified.
[In]
Int[x*Log[c*(a + b*x^2)^p]^3,x]
[Out]
-3*p^3*x^2 + (3*p^2*(a + b*x^2)*Log[c*(a + b*x^2)^p])/b - (3*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*b) + ((a
+ b*x^2)*Log[c*(a + b*x^2)^p]^3)/(2*b)
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 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]
Rubi steps
\begin{align*} \int x \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac{\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b}-\frac{(3 p) \operatorname{Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b}\\ &=-\frac{3 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}+\frac{\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b}+\frac{\left (3 p^2\right ) \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{b}\\ &=-3 p^3 x^2+\frac{3 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}-\frac{3 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}+\frac{\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0121931, size = 87, normalized size = 0.94 \[ \frac{6 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-3 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )-6 b p^3 x^2}{2 b} \]
Antiderivative was successfully verified.
[In]
Integrate[x*Log[c*(a + b*x^2)^p]^3,x]
[Out]
(-6*b*p^3*x^2 + 6*p^2*(a + b*x^2)*Log[c*(a + b*x^2)^p] - 3*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2 + (a + b*x^2)*
Log[c*(a + b*x^2)^p]^3)/(2*b)
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Maple [C] time = 0.843, size = 3925, normalized size = 42.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*ln(c*(b*x^2+a)^p)^3,x)
[Out]
-3/2/b*Pi^2*ln(b*x^2+a)*a*p*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)+3/4/b*Pi^2*ln(b*x^2+a)*a*p*c
sgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)^2+3/4/b*Pi^2*ln(b*x^2+a)*a*p*csgn(I*(b*x^2+a)^p)^2*csgn(I
*c*(b*x^2+a)^p)^3*csgn(I*c)+1/2/b*a*p^3*ln(b*x^2+a)^3+3/2/b*a*p^3*ln(b*x^2+a)^2-3/8*ln(c)*Pi^2*x^2*csgn(I*c*(b
*x^2+a)^p)^6+3/8*Pi^2*p*x^2*csgn(I*c*(b*x^2+a)^p)^6+1/16*I*Pi^3*x^2*csgn(I*c*(b*x^2+a)^p)^9-3*p^3*x^2+3/4*(I*P
i*b*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-I*Pi*b*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(
I*c)-I*Pi*b*x^2*csgn(I*c*(b*x^2+a)^p)^3+I*Pi*b*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+2*ln(c)*b*x^2-2*x^2*p*b+2
*a*p*ln(b*x^2+a))/b*ln((b*x^2+a)^p)^2+3*a*p^3/b*ln(b*x^2+a)+3/8*(8*x^2*b*p^2+4*ln(c)^2*b*x^2-4*a*p^2*ln(b*x^2+
a)^2-8*ln(c)*b*p*x^2+8*ln(c)*ln(b*x^2+a)*a*p-8*ln(b*x^2+a)*a*p^2-Pi^2*b*x^2*csgn(I*c*(b*x^2+a)^p)^6+2*Pi^2*b*x
^2*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)-Pi^2*b*x^2*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)^2+4*I*Pi*ln(b*x^2+a)*a*p*csg
n(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+4*I*Pi*ln(b*x^2+a)*a*p*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-4*I*ln(c)*Pi
*b*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-Pi^2*b*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)
^p)^4+2*Pi^2*b*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^5+4*I*ln(c)*Pi*b*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn
(I*c)-4*I*Pi*b*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-4*I*Pi*b*p*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I
*c)-4*I*Pi*ln(b*x^2+a)*a*p*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+4*I*Pi*b*p*x^2*csgn(I*(b*x^2+a)
^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+2*Pi^2*b*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)-Pi^2*
b*x^2*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*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x
^2+a)^p)^4*csgn(I*c)+2*Pi^2*b*x^2*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*x^2*c
sgn(I*c*(b*x^2+a)^p)^3+4*I*Pi*b*p*x^2*csgn(I*c*(b*x^2+a)^p)^3-4*I*Pi*ln(b*x^2+a)*a*p*csgn(I*c*(b*x^2+a)^p)^3+4
*I*ln(c)*Pi*b*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2)/b*ln((b*x^2+a)^p)-3/2*ln(c)^2*p*x^2+3*ln(c)*p^2
*x^2-3/8*ln(c)*Pi^2*x^2*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)^2-3/8*ln(c)*Pi^2*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*
(b*x^2+a)^p)^4-3/4*Pi^2*p*x^2*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)-3/4*Pi^2*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b
*x^2+a)^p)^5+3/8*Pi^2*p*x^2*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)^2+3/8*Pi^2*p*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*
(b*x^2+a)^p)^4-3/16*I*Pi^3*x^2*csgn(I*c*(b*x^2+a)^p)^8*csgn(I*c)-3/16*I*Pi^3*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*
(b*x^2+a)^p)^8+3/16*I*Pi^3*x^2*csgn(I*c*(b*x^2+a)^p)^7*csgn(I*c)^2+3/16*I*Pi^3*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(
I*c*(b*x^2+a)^p)^7-1/16*I*Pi^3*x^2*csgn(I*c*(b*x^2+a)^p)^6*csgn(I*c)^3-1/16*I*Pi^3*x^2*csgn(I*(b*x^2+a)^p)^3*c
sgn(I*c*(b*x^2+a)^p)^6-3/4*I*ln(c)^2*Pi*x^2*csgn(I*c*(b*x^2+a)^p)^3-3/2*I*Pi*p^2*x^2*csgn(I*c*(b*x^2+a)^p)^3-3
/2*I/b*Pi*ln(b*x^2+a)*a*p^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+3/2*I*ln(c)*Pi*p*x^2*csgn(I*(b*x^2+a)^
p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+3/2*I/b*ln(c)*Pi*ln(b*x^2+a)*a*p*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+3/2*I/b*
ln(c)*Pi*ln(b*x^2+a)*a*p*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+3/2*I/b*Pi*ln(b*x^2+a)*a*p^2*csgn(I*(b*x^
2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+3/4*I/b*Pi*a*p^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)*l
n(b*x^2+a)^2+1/2*x^2*ln((b*x^2+a)^p)^3-3/8/b*Pi^2*ln(b*x^2+a)*a*p*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^
2*csgn(I*c)^2-3/4*I/b*Pi*a*p^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)*ln(b*x^2+a)^2-3/4*I/b*Pi*a*p^2*csgn(I*(b*x^2+
a)^p)*csgn(I*c*(b*x^2+a)^p)^2*ln(b*x^2+a)^2-3/2*I/b*ln(c)*Pi*ln(b*x^2+a)*a*p*csgn(I*c*(b*x^2+a)^p)^3-3/2*I/b*P
i*ln(b*x^2+a)*a*p^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+3/4*ln(c)*Pi^2*x^2*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)+3/4
*ln(c)*Pi^2*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^5-3/2/b*ln(c)*a*p^2*ln(b*x^2+a)^2+3/2/b*ln(c)^2*ln(b
*x^2+a)*a*p-3/b*ln(c)*ln(b*x^2+a)*a*p^2+1/2*ln(c)^3*x^2+9/16*I*Pi^3*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+
a)^p)^5*csgn(I*c)^2+3/16*I*Pi^3*x^2*csgn(I*(b*x^2+a)^p)^3*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)-3/16*I*Pi^3*x^2*cs
gn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)^3-3/16*I*Pi^3*x^2*csgn(I*(b*x^2+a)^p)^3*csgn(I*c*(b*x^2+
a)^p)^4*csgn(I*c)^2+1/16*I*Pi^3*x^2*csgn(I*(b*x^2+a)^p)^3*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)^3+3/4*I*ln(c)^2*Pi
*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+3/4*I*ln(c)^2*Pi*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+3/2*I*
ln(c)*Pi*p*x^2*csgn(I*c*(b*x^2+a)^p)^3+3/2*I*Pi*p^2*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+3/2*I*Pi*p^2*x^2*csg
n(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-3/2*I/b*ln(c)*Pi*ln(b*x^2+a)*a*p*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+
a)^p)*csgn(I*c)-3/2*ln(c)*Pi^2*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)+3/4*ln(c)*Pi^2*x^2*cs
gn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)^2+3/4*ln(c)*Pi^2*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2
+a)^p)^3*csgn(I*c)-3/8*ln(c)*Pi^2*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)^2+3/2*Pi^2*p*x^2
*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^4*csgn(I*c)-3/4*Pi^2*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p
)^3*csgn(I*c)^2-3/4*Pi^2*p*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^3*csgn(I*c)+3/8*Pi^2*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/b*Pi^2*ln(b*x^2+a)*a*p*csgn(I*c*(b*x^2+a)^p)^6+9/16*I*Pi
^3*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^7*csgn(I*c)-9/16*I*Pi^3*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x
^2+a)^p)^6*csgn(I*c)^2-9/16*I*Pi^3*x^2*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^6*csgn(I*c)+3/16*I*Pi^3*x^2
*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^5*csgn(I*c)^3+3/4/b*Pi^2*ln(b*x^2+a)*a*p*csgn(I*c*(b*x^2+a)^p)^5*cs
gn(I*c)+3/4/b*Pi^2*ln(b*x^2+a)*a*p*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^5-3/8/b*Pi^2*ln(b*x^2+a)*a*p*csgn
(I*c*(b*x^2+a)^p)^4*csgn(I*c)^2-3/8/b*Pi^2*ln(b*x^2+a)*a*p*csgn(I*(b*x^2+a)^p)^2*csgn(I*c*(b*x^2+a)^p)^4-3/4*I
*ln(c)^2*Pi*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-3/2*I*ln(c)*Pi*p*x^2*csgn(I*c*(b*x^2+a)^p)
^2*csgn(I*c)-3/2*I*ln(c)*Pi*p*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-3/2*I*Pi*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*Pi*a*p^2*csgn(I*c*(b*x^2+a)^p)^3*ln(b*x^2+a)^2+3/2*I/b*Pi*ln(b*x^
2+a)*a*p^2*csgn(I*c*(b*x^2+a)^p)^3
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Maxima [A] time = 1.09372, size = 221, normalized size = 2.38 \begin{align*} -\frac{3}{2} \, b p{\left (\frac{x^{2}}{b} - \frac{a \log \left (b x^{2} + a\right )}{b^{2}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} + \frac{1}{2} \, x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} + \frac{1}{2} \, b p{\left (\frac{{\left (a \log \left (b x^{2} + a\right )^{3} - 6 \, b x^{2} + 3 \, a \log \left (b x^{2} + a\right )^{2} + 6 \, a \log \left (b x^{2} + a\right )\right )} p^{2}}{b^{2}} + \frac{3 \,{\left (2 \, b x^{2} - a \log \left (b x^{2} + a\right )^{2} - 2 \, a \log \left (b x^{2} + a\right )\right )} p \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*log(c*(b*x^2+a)^p)^3,x, algorithm="maxima")
[Out]
-3/2*b*p*(x^2/b - a*log(b*x^2 + a)/b^2)*log((b*x^2 + a)^p*c)^2 + 1/2*x^2*log((b*x^2 + a)^p*c)^3 + 1/2*b*p*((a*
log(b*x^2 + a)^3 - 6*b*x^2 + 3*a*log(b*x^2 + a)^2 + 6*a*log(b*x^2 + a))*p^2/b^2 + 3*(2*b*x^2 - a*log(b*x^2 + a
)^2 - 2*a*log(b*x^2 + a))*p*log((b*x^2 + a)^p*c)/b^2)
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Fricas [A] time = 1.86092, size = 393, normalized size = 4.23 \begin{align*} -\frac{6 \, b p^{3} x^{2} - 6 \, b p^{2} x^{2} \log \left (c\right ) + 3 \, b p x^{2} \log \left (c\right )^{2} - b x^{2} \log \left (c\right )^{3} -{\left (b p^{3} x^{2} + a p^{3}\right )} \log \left (b x^{2} + a\right )^{3} + 3 \,{\left (b p^{3} x^{2} + a p^{3} -{\left (b p^{2} x^{2} + a p^{2}\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right )^{2} - 3 \,{\left (2 \, b p^{3} x^{2} + 2 \, a p^{3} +{\left (b p x^{2} + a p\right )} \log \left (c\right )^{2} - 2 \,{\left (b p^{2} x^{2} + a p^{2}\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*log(c*(b*x^2+a)^p)^3,x, algorithm="fricas")
[Out]
-1/2*(6*b*p^3*x^2 - 6*b*p^2*x^2*log(c) + 3*b*p*x^2*log(c)^2 - b*x^2*log(c)^3 - (b*p^3*x^2 + a*p^3)*log(b*x^2 +
a)^3 + 3*(b*p^3*x^2 + a*p^3 - (b*p^2*x^2 + a*p^2)*log(c))*log(b*x^2 + a)^2 - 3*(2*b*p^3*x^2 + 2*a*p^3 + (b*p*
x^2 + a*p)*log(c)^2 - 2*(b*p^2*x^2 + a*p^2)*log(c))*log(b*x^2 + a))/b
________________________________________________________________________________________
Sympy [A] time = 7.31956, size = 301, normalized size = 3.24 \begin{align*} \begin{cases} \frac{a p^{3} \log{\left (a + b x^{2} \right )}^{3}}{2 b} - \frac{3 a p^{3} \log{\left (a + b x^{2} \right )}^{2}}{2 b} + \frac{3 a p^{3} \log{\left (a + b x^{2} \right )}}{b} + \frac{3 a p^{2} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}^{2}}{2 b} - \frac{3 a p^{2} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{b} + \frac{3 a p \log{\left (c \right )}^{2} \log{\left (a + b x^{2} \right )}}{2 b} + \frac{p^{3} x^{2} \log{\left (a + b x^{2} \right )}^{3}}{2} - \frac{3 p^{3} x^{2} \log{\left (a + b x^{2} \right )}^{2}}{2} + 3 p^{3} x^{2} \log{\left (a + b x^{2} \right )} - 3 p^{3} x^{2} + \frac{3 p^{2} x^{2} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}^{2}}{2} - 3 p^{2} x^{2} \log{\left (c \right )} \log{\left (a + b x^{2} \right )} + 3 p^{2} x^{2} \log{\left (c \right )} + \frac{3 p x^{2} \log{\left (c \right )}^{2} \log{\left (a + b x^{2} \right )}}{2} - \frac{3 p x^{2} \log{\left (c \right )}^{2}}{2} + \frac{x^{2} \log{\left (c \right )}^{3}}{2} & \text{for}\: b \neq 0 \\\frac{x^{2} \log{\left (a^{p} c \right )}^{3}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*ln(c*(b*x**2+a)**p)**3,x)
[Out]
Piecewise((a*p**3*log(a + b*x**2)**3/(2*b) - 3*a*p**3*log(a + b*x**2)**2/(2*b) + 3*a*p**3*log(a + b*x**2)/b +
3*a*p**2*log(c)*log(a + b*x**2)**2/(2*b) - 3*a*p**2*log(c)*log(a + b*x**2)/b + 3*a*p*log(c)**2*log(a + b*x**2)
/(2*b) + p**3*x**2*log(a + b*x**2)**3/2 - 3*p**3*x**2*log(a + b*x**2)**2/2 + 3*p**3*x**2*log(a + b*x**2) - 3*p
**3*x**2 + 3*p**2*x**2*log(c)*log(a + b*x**2)**2/2 - 3*p**2*x**2*log(c)*log(a + b*x**2) + 3*p**2*x**2*log(c) +
3*p*x**2*log(c)**2*log(a + b*x**2)/2 - 3*p*x**2*log(c)**2/2 + x**2*log(c)**3/2, Ne(b, 0)), (x**2*log(a**p*c)*
*3/2, True))
________________________________________________________________________________________
Giac [A] time = 1.28473, size = 228, normalized size = 2.45 \begin{align*} \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 )} 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 )} 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 )} p \log \left (c\right )^{2} +{\left (b x^{2} + a\right )} \log \left (c\right )^{3}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*log(c*(b*x^2+a)^p)^3,x, algorithm="giac")
[Out]
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)*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)*p^2*log(c) - 3*(b*
x^2 - (b*x^2 + a)*log(b*x^2 + a) + a)*p*log(c)^2 + (b*x^2 + a)*log(c)^3)/b