Integrand size = 18, antiderivative size = 334 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=-\frac {3 a^2 p^3 x^2}{b^2}+\frac {3 a p^3 \left (a+b x^2\right )^2}{8 b^3}-\frac {p^3 \left (a+b x^2\right )^3}{27 b^3}+\frac {3 a^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}-\frac {3 a p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^3}+\frac {p^2 \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}-\frac {3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^3}-\frac {p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{6 b^3}+\frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 b^3} \] Output:
-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
Time = 0.22 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.53 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {b p^3 x^2 \left (-510 a^2+57 a b x^2-8 b^2 x^4\right )+114 a^3 p^3 \log \left (a+b x^2\right )+6 p^2 \left (66 a^3+66 a^2 b x^2-15 a b^2 x^4+4 b^3 x^6\right ) \log \left (c \left (a+b x^2\right )^p\right )-18 p \left (11 a^3+6 a^2 b x^2-3 a b^2 x^4+2 b^3 x^6\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+36 \left (a^3+b^3 x^6\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{216 b^3} \] Input:
Integrate[x^5*Log[c*(a + b*x^2)^p]^3,x]
Output:
(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 - 15*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)
Time = 0.98 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2904, 2848, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle \frac {1}{2} \int x^4 \log ^3\left (c \left (b x^2+a\right )^p\right )dx^2\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\left (b x^2+a\right )^2 \log ^3\left (c \left (b x^2+a\right )^p\right )}{b^2}-\frac {2 a \left (b x^2+a\right ) \log ^3\left (c \left (b x^2+a\right )^p\right )}{b^2}+\frac {a^2 \log ^3\left (c \left (b x^2+a\right )^p\right )}{b^2}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {6 a^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^3}+\frac {a^2 \left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{b^3}-\frac {3 a^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{b^3}-\frac {6 a^2 p^3 x^2}{b^2}+\frac {2 p^2 \left (a+b x^2\right )^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b^3}-\frac {3 a p^2 \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 b^3}+\frac {\left (a+b x^2\right )^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{3 b^3}-\frac {a \left (a+b x^2\right )^2 \log ^3\left (c \left (a+b x^2\right )^p\right )}{b^3}-\frac {p \left (a+b x^2\right )^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{3 b^3}+\frac {3 a p \left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^3}-\frac {2 p^3 \left (a+b x^2\right )^3}{27 b^3}+\frac {3 a p^3 \left (a+b x^2\right )^2}{4 b^3}\right )\) |
Input:
Int[x^5*Log[c*(a + b*x^2)^p]^3,x]
Output:
((-6*a^2*p^3*x^2)/b^2 + (3*a*p^3*(a + b*x^2)^2)/(4*b^3) - (2*p^3*(a + b*x^ 2)^3)/(27*b^3) + (6*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])/(2*b^3) + (2*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)/b ^3 + (3*a*p*(a + b*x^2)^2*Log[c*(a + b*x^2)^p]^2)/(2*b^3) - (p*(a + b*x^2) ^3*Log[c*(a + b*x^2)^p]^2)/(3*b^3) + (a^2*(a + b*x^2)*Log[c*(a + b*x^2)^p] ^3)/b^3 - (a*(a + b*x^2)^2*Log[c*(a + b*x^2)^p]^3)/b^3 + ((a + b*x^2)^3*Lo g[c*(a + b*x^2)^p]^3)/(3*b^3))/2
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[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])
Time = 1.96 (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 | \(\text {Expression too large to display}\) | \(5905\) |
Input:
int(x^5*ln(c*(b*x^2+a)^p)^3,x,method=_RETURNVERBOSE)
Output:
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+9 06*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
Time = 0.10 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.07 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=-\frac {8 \, b^{3} p^{3} x^{6} - 36 \, b^{3} x^{6} \log \left (c\right )^{3} - 57 \, a b^{2} p^{3} x^{4} + 510 \, a^{2} b p^{3} x^{2} - 36 \, {\left (b^{3} p^{3} x^{6} + a^{3} p^{3}\right )} \log \left (b x^{2} + a\right )^{3} + 18 \, {\left (2 \, b^{3} p^{3} x^{6} - 3 \, a b^{2} p^{3} x^{4} + 6 \, a^{2} b p^{3} x^{2} + 11 \, a^{3} p^{3} - 6 \, {\left (b^{3} p^{2} x^{6} + a^{3} p^{2}\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right )^{2} + 18 \, {\left (2 \, b^{3} p x^{6} - 3 \, a b^{2} p x^{4} + 6 \, a^{2} b p x^{2}\right )} \log \left (c\right )^{2} - 6 \, {\left (4 \, b^{3} p^{3} x^{6} - 15 \, a b^{2} p^{3} x^{4} + 66 \, a^{2} b p^{3} x^{2} + 85 \, a^{3} p^{3} + 18 \, {\left (b^{3} p x^{6} + a^{3} p\right )} \log \left (c\right )^{2} - 6 \, {\left (2 \, b^{3} p^{2} x^{6} - 3 \, a b^{2} p^{2} x^{4} + 6 \, a^{2} b p^{2} x^{2} + 11 \, a^{3} p^{2}\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right ) - 6 \, {\left (4 \, b^{3} p^{2} x^{6} - 15 \, a b^{2} p^{2} x^{4} + 66 \, a^{2} b p^{2} x^{2}\right )} \log \left (c\right )}{216 \, b^{3}} \] Input:
integrate(x^5*log(c*(b*x^2+a)^p)^3,x, algorithm="fricas")
Output:
-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
Time = 5.36 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.87 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\begin {cases} \frac {85 a^{3} p^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{36 b^{3}} - \frac {11 a^{3} p \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{12 b^{3}} + \frac {a^{3} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{6 b^{3}} - \frac {85 a^{2} p^{3} x^{2}}{36 b^{2}} + \frac {11 a^{2} p^{2} x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{6 b^{2}} - \frac {a^{2} p x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{2 b^{2}} + \frac {19 a p^{3} x^{4}}{72 b} - \frac {5 a p^{2} x^{4} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{12 b} + \frac {a p x^{4} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{4 b} - \frac {p^{3} x^{6}}{27} + \frac {p^{2} x^{6} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{9} - \frac {p x^{6} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{6} + \frac {x^{6} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{6} & \text {for}\: b \neq 0 \\\frac {x^{6} \log {\left (a^{p} c \right )}^{3}}{6} & \text {otherwise} \end {cases} \] Input:
integrate(x**5*ln(c*(b*x**2+a)**p)**3,x)
Output:
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))
Time = 0.04 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.72 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {1}{6} \, x^{6} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} + \frac {1}{12} \, b p {\left (\frac {6 \, a^{3} \log \left (b x^{2} + a\right )}{b^{4}} - \frac {2 \, b^{2} x^{6} - 3 \, a b x^{4} + 6 \, a^{2} x^{2}}{b^{3}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} - \frac {1}{216} \, b p {\left (\frac {{\left (8 \, b^{3} x^{6} - 57 \, a b^{2} x^{4} - 36 \, a^{3} \log \left (b x^{2} + a\right )^{3} + 510 \, a^{2} b x^{2} - 198 \, a^{3} \log \left (b x^{2} + a\right )^{2} - 510 \, a^{3} \log \left (b x^{2} + a\right )\right )} p^{2}}{b^{4}} - \frac {6 \, {\left (4 \, b^{3} x^{6} - 15 \, a b^{2} x^{4} + 66 \, a^{2} b x^{2} - 18 \, a^{3} \log \left (b x^{2} + a\right )^{2} - 66 \, a^{3} \log \left (b x^{2} + a\right )\right )} p \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{b^{4}}\right )} \] Input:
integrate(x^5*log(c*(b*x^2+a)^p)^3,x, algorithm="maxima")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 662 vs. \(2 (314) = 628\).
Time = 0.13 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.98 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx =\text {Too large to display} \] Input:
integrate(x^5*log(c*(b*x^2+a)^p)^3,x, algorithm="giac")
Output:
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)*log(b*x^2 + a) + a)*a^2*p*log(c)^2 + (b*x^2 + a)*a^2*log(c)^3)/b^3
Time = 14.80 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.56 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx={\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3\,\left (\frac {x^6}{6}+\frac {a^3}{6\,b^3}\right )-{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2\,\left (\frac {p\,x^6}{6}+\frac {11\,a^3\,p}{12\,b^3}+\frac {a^2\,p\,x^2}{2\,b^2}-\frac {a\,p\,x^4}{4\,b}\right )-\frac {p^3\,x^6}{27}+\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )\,\left (\frac {b\,p^2\,x^6}{3}-\frac {5\,a\,p^2\,x^4}{4}+\frac {11\,a^2\,p^2\,x^2}{2\,b}\right )}{3\,b}+\frac {19\,a\,p^3\,x^4}{72\,b}+\frac {85\,a^3\,p^3\,\ln \left (b\,x^2+a\right )}{36\,b^3}-\frac {85\,a^2\,p^3\,x^2}{36\,b^2} \] Input:
int(x^5*log(c*(a + b*x^2)^p)^3,x)
Output:
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 + (11* 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)
Time = 0.17 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.79 \[ \int x^5 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {36 {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}^{3} a^{3}+36 {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}^{3} b^{3} x^{6}-198 {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}^{2} a^{3} p -108 {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}^{2} a^{2} b p \,x^{2}+54 {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}^{2} a \,b^{2} p \,x^{4}-36 {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}^{2} b^{3} p \,x^{6}+510 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) a^{3} p^{2}+396 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) a^{2} b \,p^{2} x^{2}-90 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) a \,b^{2} p^{2} x^{4}+24 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b^{3} p^{2} x^{6}-510 a^{2} b \,p^{3} x^{2}+57 a \,b^{2} p^{3} x^{4}-8 b^{3} p^{3} x^{6}}{216 b^{3}} \] Input:
int(x^5*log(c*(b*x^2+a)^p)^3,x)
Output:
(36*log((a + b*x**2)**p*c)**3*a**3 + 36*log((a + b*x**2)**p*c)**3*b**3*x** 6 - 198*log((a + b*x**2)**p*c)**2*a**3*p - 108*log((a + b*x**2)**p*c)**2*a **2*b*p*x**2 + 54*log((a + b*x**2)**p*c)**2*a*b**2*p*x**4 - 36*log((a + b* x**2)**p*c)**2*b**3*p*x**6 + 510*log((a + b*x**2)**p*c)*a**3*p**2 + 396*lo g((a + b*x**2)**p*c)*a**2*b*p**2*x**2 - 90*log((a + b*x**2)**p*c)*a*b**2*p **2*x**4 + 24*log((a + b*x**2)**p*c)*b**3*p**2*x**6 - 510*a**2*b*p**3*x**2 + 57*a*b**2*p**3*x**4 - 8*b**3*p**3*x**6)/(216*b**3)