Integrand size = 16, antiderivative size = 52 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2}{27} b^2 n^2 x^3-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \]
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{3} \left (\frac {2}{9} b n x^3 \left (-3 a+b n-3 b \log \left (c x^n\right )\right )+x^3 \left (a+b \log \left (c x^n\right )\right )^2\right ) \]
Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2742, 2741}
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.
\(\displaystyle \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\) |
\(\Big \downarrow \) 2742 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{3} b n \int x^2 \left (a+b \log \left (c x^n\right )\right )dx\) |
\(\Big \downarrow \) 2741 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{3} b n \left (\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3\right )\) |
3.1.51.3.1 Defintions of rubi rules used
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]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[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]
Time = 0.14 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.40
method | result | size |
parallelrisch | \(\frac {x^{3} b^{2} \ln \left (c \,x^{n}\right )^{2}}{3}-\frac {2 \ln \left (c \,x^{n}\right ) x^{3} b^{2} n}{9}+\frac {2 b^{2} n^{2} x^{3}}{27}+\frac {2 x^{3} a b \ln \left (c \,x^{n}\right )}{3}-\frac {2 a b n \,x^{3}}{9}+\frac {x^{3} a^{2}}{3}\) | \(73\) |
risch | \(\frac {x^{3} b^{2} \ln \left (x^{n}\right )^{2}}{3}+\frac {b \,x^{3} \left (-3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+6 b \ln \left (c \right )-2 b n +6 a \right ) \ln \left (x^{n}\right )}{9}+\frac {x^{3} \left (36 a^{2}+12 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-36 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+12 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-36 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-36 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+8 b^{2} n^{2}+72 \ln \left (c \right ) a b +36 \ln \left (c \right )^{2} b^{2}-24 b^{2} \ln \left (c \right ) n -24 a b n -9 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+36 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-12 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-12 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}\right )}{108}\) | \(692\) |
1/3*x^3*b^2*ln(c*x^n)^2-2/9*ln(c*x^n)*x^3*b^2*n+2/27*b^2*n^2*x^3+2/3*x^3*a *b*ln(c*x^n)-2/9*a*b*n*x^3+1/3*x^3*a^2
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (46) = 92\).
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.98 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{3} \, b^{2} n^{2} x^{3} \log \left (x\right )^{2} + \frac {1}{3} \, b^{2} x^{3} \log \left (c\right )^{2} - \frac {2}{9} \, {\left (b^{2} n - 3 \, a b\right )} x^{3} \log \left (c\right ) + \frac {1}{27} \, {\left (2 \, b^{2} n^{2} - 6 \, a b n + 9 \, a^{2}\right )} x^{3} + \frac {2}{9} \, {\left (3 \, b^{2} n x^{3} \log \left (c\right ) - {\left (b^{2} n^{2} - 3 \, a b n\right )} x^{3}\right )} \log \left (x\right ) \]
1/3*b^2*n^2*x^3*log(x)^2 + 1/3*b^2*x^3*log(c)^2 - 2/9*(b^2*n - 3*a*b)*x^3* log(c) + 1/27*(2*b^2*n^2 - 6*a*b*n + 9*a^2)*x^3 + 2/9*(3*b^2*n*x^3*log(c) - (b^2*n^2 - 3*a*b*n)*x^3)*log(x)
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.63 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {a^{2} x^{3}}{3} - \frac {2 a b n x^{3}}{9} + \frac {2 a b x^{3} \log {\left (c x^{n} \right )}}{3} + \frac {2 b^{2} n^{2} x^{3}}{27} - \frac {2 b^{2} n x^{3} \log {\left (c x^{n} \right )}}{9} + \frac {b^{2} x^{3} \log {\left (c x^{n} \right )}^{2}}{3} \]
a**2*x**3/3 - 2*a*b*n*x**3/9 + 2*a*b*x**3*log(c*x**n)/3 + 2*b**2*n**2*x**3 /27 - 2*b**2*n*x**3*log(c*x**n)/9 + b**2*x**3*log(c*x**n)**2/3
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.37 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{3} \, b^{2} x^{3} \log \left (c x^{n}\right )^{2} - \frac {2}{9} \, a b n x^{3} + \frac {2}{3} \, a b x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a^{2} x^{3} + \frac {2}{27} \, {\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} b^{2} \]
1/3*b^2*x^3*log(c*x^n)^2 - 2/9*a*b*n*x^3 + 2/3*a*b*x^3*log(c*x^n) + 1/3*a^ 2*x^3 + 2/27*(n^2*x^3 - 3*n*x^3*log(c*x^n))*b^2
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (46) = 92\).
Time = 0.32 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.13 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{3} \, b^{2} n^{2} x^{3} \log \left (x\right )^{2} - \frac {2}{9} \, b^{2} n^{2} x^{3} \log \left (x\right ) + \frac {2}{3} \, b^{2} n x^{3} \log \left (c\right ) \log \left (x\right ) + \frac {2}{27} \, b^{2} n^{2} x^{3} - \frac {2}{9} \, b^{2} n x^{3} \log \left (c\right ) + \frac {1}{3} \, b^{2} x^{3} \log \left (c\right )^{2} + \frac {2}{3} \, a b n x^{3} \log \left (x\right ) - \frac {2}{9} \, a b n x^{3} + \frac {2}{3} \, a b x^{3} \log \left (c\right ) + \frac {1}{3} \, a^{2} x^{3} \]
1/3*b^2*n^2*x^3*log(x)^2 - 2/9*b^2*n^2*x^3*log(x) + 2/3*b^2*n*x^3*log(c)*l og(x) + 2/27*b^2*n^2*x^3 - 2/9*b^2*n*x^3*log(c) + 1/3*b^2*x^3*log(c)^2 + 2 /3*a*b*n*x^3*log(x) - 2/9*a*b*n*x^3 + 2/3*a*b*x^3*log(c) + 1/3*a^2*x^3
Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.19 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=x^3\,\left (\frac {a^2}{3}-\frac {2\,a\,b\,n}{9}+\frac {2\,b^2\,n^2}{27}\right )+\frac {x^3\,\ln \left (c\,x^n\right )\,\left (2\,a\,b-\frac {2\,b^2\,n}{3}\right )}{3}+\frac {b^2\,x^3\,{\ln \left (c\,x^n\right )}^2}{3} \]