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

Optimal. Leaf size=334 \[ \frac{3 a^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}-\frac{3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac{a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac{3 a^2 p^3 x^2}{b^2}+\frac{p^2 \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}-\frac{3 a p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^3}-\frac{p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac{3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^3}+\frac{\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac{a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac{p^3 \left (a+b x^2\right )^3}{27 b^3}+\frac{3 a p^3 \left (a+b x^2\right )^2}{8 b^3} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.357969, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 15, 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} \[ \frac{3 a^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}-\frac{3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac{a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac{3 a^2 p^3 x^2}{b^2}+\frac{p^2 \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}-\frac{3 a p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^3}-\frac{p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac{3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^3}+\frac{\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac{a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac{p^3 \left (a+b x^2\right )^3}{27 b^3}+\frac{3 a p^3 \left (a+b x^2\right )^2}{8 b^3} \]

Antiderivative was successfully verified.

[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 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^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{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{\operatorname{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 \operatorname{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 \operatorname{Subst}\left (\int \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b^2}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^3}-\frac{a \operatorname{Subst}\left (\int x \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{b^3}+\frac{a^2 \operatorname{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 \operatorname{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) \operatorname{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 ) \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.194772, size = 309, normalized size = 0.93 \[ \frac{11 a^2 p^2 x^2 \log \left (c \left (a+b x^2\right )^p\right )}{6 b^2}+\frac{11 a^3 p^2 \log \left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac{a^2 p x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac{a^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3}-\frac{11 a^3 p \log ^2\left (c \left (a+b x^2\right )^p\right )}{12 b^3}-\frac{85 a^2 p^3 x^2}{36 b^2}+\frac{19 a^3 p^3 \log \left (a+b x^2\right )}{36 b^3}+\frac{1}{9} p^2 x^6 \log \left (c \left (a+b x^2\right )^p\right )-\frac{5 a p^2 x^4 \log \left (c \left (a+b x^2\right )^p\right )}{12 b}+\frac{1}{6} x^6 \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{1}{6} p x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{a p x^4 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b}+\frac{19 a p^3 x^4}{72 b}-\frac{1}{27} p^3 x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-85*a^2*p^3*x^2)/(36*b^2) + (19*a*p^3*x^4)/(72*b) - (p^3*x^6)/27 + (19*a^3*p^3*Log[a + b*x^2])/(36*b^3) + (11
*a^3*p^2*Log[c*(a + b*x^2)^p])/(6*b^3) + (11*a^2*p^2*x^2*Log[c*(a + b*x^2)^p])/(6*b^2) - (5*a*p^2*x^4*Log[c*(a
+ b*x^2)^p])/(12*b) + (p^2*x^6*Log[c*(a + b*x^2)^p])/9 - (11*a^3*p*Log[c*(a + b*x^2)^p]^2)/(12*b^3) - (a^2*p*
x^2*Log[c*(a + b*x^2)^p]^2)/(2*b^2) + (a*p*x^4*Log[c*(a + b*x^2)^p]^2)/(4*b) - (p*x^6*Log[c*(a + b*x^2)^p]^2)/
6 + (a^3*Log[c*(a + b*x^2)^p]^3)/(6*b^3) + (x^6*Log[c*(a + b*x^2)^p]^3)/6

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Maple [C]  time = 0.941, size = 5905, normalized size = 17.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [A]  time = 1.11961, size = 323, normalized size = 0.97 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.80952, size = 780, normalized size = 2.34 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 50.4087, size = 561, normalized size = 1.68 \begin{align*} \begin{cases} \frac{a^{3} p^{3} \log{\left (a + b x^{2} \right )}^{3}}{6 b^{3}} - \frac{11 a^{3} p^{3} \log{\left (a + b x^{2} \right )}^{2}}{12 b^{3}} + \frac{85 a^{3} p^{3} \log{\left (a + b x^{2} \right )}}{36 b^{3}} + \frac{a^{3} p^{2} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}^{2}}{2 b^{3}} - \frac{11 a^{3} p^{2} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{6 b^{3}} + \frac{a^{3} p \log{\left (c \right )}^{2} \log{\left (a + b x^{2} \right )}}{2 b^{3}} - \frac{a^{2} p^{3} x^{2} \log{\left (a + b x^{2} \right )}^{2}}{2 b^{2}} + \frac{11 a^{2} p^{3} x^{2} \log{\left (a + b x^{2} \right )}}{6 b^{2}} - \frac{85 a^{2} p^{3} x^{2}}{36 b^{2}} - \frac{a^{2} p^{2} x^{2} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{b^{2}} + \frac{11 a^{2} p^{2} x^{2} \log{\left (c \right )}}{6 b^{2}} - \frac{a^{2} p x^{2} \log{\left (c \right )}^{2}}{2 b^{2}} + \frac{a p^{3} x^{4} \log{\left (a + b x^{2} \right )}^{2}}{4 b} - \frac{5 a p^{3} x^{4} \log{\left (a + b x^{2} \right )}}{12 b} + \frac{19 a p^{3} x^{4}}{72 b} + \frac{a p^{2} x^{4} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{2 b} - \frac{5 a p^{2} x^{4} \log{\left (c \right )}}{12 b} + \frac{a p x^{4} \log{\left (c \right )}^{2}}{4 b} + \frac{p^{3} x^{6} \log{\left (a + b x^{2} \right )}^{3}}{6} - \frac{p^{3} x^{6} \log{\left (a + b x^{2} \right )}^{2}}{6} + \frac{p^{3} x^{6} \log{\left (a + b x^{2} \right )}}{9} - \frac{p^{3} x^{6}}{27} + \frac{p^{2} x^{6} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}^{2}}{2} - \frac{p^{2} x^{6} \log{\left (c \right )} \log{\left (a + b x^{2} \right )}}{3} + \frac{p^{2} x^{6} \log{\left (c \right )}}{9} + \frac{p x^{6} \log{\left (c \right )}^{2} \log{\left (a + b x^{2} \right )}}{2} - \frac{p x^{6} \log{\left (c \right )}^{2}}{6} + \frac{x^{6} \log{\left (c \right )}^{3}}{6} & \text{for}\: b \neq 0 \\\frac{x^{6} \log{\left (a^{p} c \right )}^{3}}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*p**3*log(a + b*x**2)**3/(6*b**3) - 11*a**3*p**3*log(a + b*x**2)**2/(12*b**3) + 85*a**3*p**3*lo
g(a + b*x**2)/(36*b**3) + a**3*p**2*log(c)*log(a + b*x**2)**2/(2*b**3) - 11*a**3*p**2*log(c)*log(a + b*x**2)/(
6*b**3) + a**3*p*log(c)**2*log(a + b*x**2)/(2*b**3) - a**2*p**3*x**2*log(a + b*x**2)**2/(2*b**2) + 11*a**2*p**
3*x**2*log(a + b*x**2)/(6*b**2) - 85*a**2*p**3*x**2/(36*b**2) - a**2*p**2*x**2*log(c)*log(a + b*x**2)/b**2 + 1
1*a**2*p**2*x**2*log(c)/(6*b**2) - a**2*p*x**2*log(c)**2/(2*b**2) + a*p**3*x**4*log(a + b*x**2)**2/(4*b) - 5*a
*p**3*x**4*log(a + b*x**2)/(12*b) + 19*a*p**3*x**4/(72*b) + a*p**2*x**4*log(c)*log(a + b*x**2)/(2*b) - 5*a*p**
2*x**4*log(c)/(12*b) + a*p*x**4*log(c)**2/(4*b) + p**3*x**6*log(a + b*x**2)**3/6 - p**3*x**6*log(a + b*x**2)**
2/6 + p**3*x**6*log(a + b*x**2)/9 - p**3*x**6/27 + p**2*x**6*log(c)*log(a + b*x**2)**2/2 - p**2*x**6*log(c)*lo
g(a + b*x**2)/3 + p**2*x**6*log(c)/9 + p*x**6*log(c)**2*log(a + b*x**2)/2 - p*x**6*log(c)**2/6 + x**6*log(c)**
3/6, Ne(b, 0)), (x**6*log(a**p*c)**3/6, True))

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Giac [A]  time = 1.25596, size = 803, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*(36*b*x^6*log(c)^3 + (36*(b*x^2 + a)^3*log(b*x^2 + a)^3/b^2 - 108*(b*x^2 + a)^2*a*log(b*x^2 + a)^3/b^2 +
108*(b*x^2 + a)*a^2*log(b*x^2 + a)^3/b^2 - 36*(b*x^2 + a)^3*log(b*x^2 + a)^2/b^2 + 162*(b*x^2 + a)^2*a*log(b*
x^2 + a)^2/b^2 - 324*(b*x^2 + a)*a^2*log(b*x^2 + a)^2/b^2 + 24*(b*x^2 + a)^3*log(b*x^2 + a)/b^2 - 162*(b*x^2 +
a)^2*a*log(b*x^2 + a)/b^2 + 648*(b*x^2 + a)*a^2*log(b*x^2 + a)/b^2 - 8*(b*x^2 + a)^3/b^2 + 81*(b*x^2 + a)^2*a
/b^2 - 648*(b*x^2 + a)*a^2/b^2)*p^3 + 6*(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*log(c) + 18*(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)^2)/b