Integrand size = 23, antiderivative size = 130 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {3}{4} b d^2 e n x^2-\frac {3}{16} b d e^2 n x^4-\frac {1}{36} b e^3 n x^6-\frac {1}{2} b d^3 n \log ^2(x)+\frac {3}{2} d^2 e x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} d e^2 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{6} e^3 x^6 \left (a+b \log \left (c x^n\right )\right )+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right ) \]
-3/4*b*d^2*e*n*x^2-3/16*b*d*e^2*n*x^4-1/36*b*e^3*n*x^6-1/2*b*d^3*n*ln(x)^2 +3/2*d^2*e*x^2*(a+b*ln(c*x^n))+3/4*d*e^2*x^4*(a+b*ln(c*x^n))+1/6*e^3*x^6*( a+b*ln(c*x^n))+d^3*ln(x)*(a+b*ln(c*x^n))
Time = 0.05 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.89 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {1}{144} \left (-108 b d^2 e n x^2-27 b d e^2 n x^4-4 b e^3 n x^6+216 d^2 e x^2 \left (a+b \log \left (c x^n\right )\right )+108 d e^2 x^4 \left (a+b \log \left (c x^n\right )\right )+24 e^3 x^6 \left (a+b \log \left (c x^n\right )\right )+\frac {72 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{b n}\right ) \]
(-108*b*d^2*e*n*x^2 - 27*b*d*e^2*n*x^4 - 4*b*e^3*n*x^6 + 216*d^2*e*x^2*(a + b*Log[c*x^n]) + 108*d*e^2*x^4*(a + b*Log[c*x^n]) + 24*e^3*x^6*(a + b*Log [c*x^n]) + (72*d^3*(a + b*Log[c*x^n])^2)/(b*n))/144
Time = 0.32 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2772, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle -b n \int \left (\frac {\log (x) d^3}{x}+\frac {1}{12} e x \left (2 e^2 x^4+9 d e x^2+18 d^2\right )\right )dx+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} d^2 e x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} d e^2 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{6} e^3 x^6 \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} d^2 e x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} d e^2 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{6} e^3 x^6 \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {1}{2} d^3 \log ^2(x)+\frac {3}{4} d^2 e x^2+\frac {3}{16} d e^2 x^4+\frac {e^3 x^6}{36}\right )\) |
-(b*n*((3*d^2*e*x^2)/4 + (3*d*e^2*x^4)/16 + (e^3*x^6)/36 + (d^3*Log[x]^2)/ 2)) + (3*d^2*e*x^2*(a + b*Log[c*x^n]))/2 + (3*d*e^2*x^4*(a + b*Log[c*x^n]) )/4 + (e^3*x^6*(a + b*Log[c*x^n]))/6 + d^3*Log[x]*(a + b*Log[c*x^n])
3.2.99.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
Time = 0.72 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.15
| method | result | size |
| parallelrisch | \(\frac {24 x^{6} \ln \left (c \,x^{n}\right ) b \,e^{3} n -4 x^{6} b \,e^{3} n^{2}+24 x^{6} a \,e^{3} n +108 x^{4} \ln \left (c \,x^{n}\right ) b d \,e^{2} n -27 x^{4} b d \,e^{2} n^{2}+108 x^{4} a d \,e^{2} n +216 x^{2} \ln \left (c \,x^{n}\right ) b \,d^{2} e n -108 x^{2} b \,d^{2} e \,n^{2}+216 x^{2} a \,d^{2} e n +144 \ln \left (x \right ) a \,d^{3} n +72 b \,d^{3} \ln \left (c \,x^{n}\right )^{2}}{144 n}\) | \(150\) |
| risch | \(\frac {x^{6} a \,e^{3}}{6}+\frac {3 \ln \left (c \right ) b d \,e^{2} x^{4}}{4}-\frac {3 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{8}-\frac {3 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{4}-\frac {i \ln \left (x \right ) \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {3 x^{4} a d \,e^{2}}{4}+\frac {3 a \,d^{2} e \,x^{2}}{2}+\ln \left (x \right ) \ln \left (c \right ) b \,d^{3}+\frac {\ln \left (c \right ) b \,e^{3} x^{6}}{6}+\frac {3 e \ln \left (c \right ) b \,d^{2} x^{2}}{2}+\frac {i \ln \left (x \right ) \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \ln \left (x \right ) \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{12}-\frac {3 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{8}-\frac {i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{12}-\frac {3 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{4}+\ln \left (x \right ) a \,d^{3}+\frac {i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{12}+\frac {3 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}+\frac {3 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}+\left (\frac {x^{6} b \,e^{3}}{6}+\frac {3 b d \,e^{2} x^{4}}{4}+\frac {3 e \,d^{2} b \,x^{2}}{2}+b \,d^{3} \ln \left (x \right )\right ) \ln \left (x^{n}\right )-\frac {i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{12}+\frac {3 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {3 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {3 b d \,e^{2} n \,x^{4}}{16}-\frac {3 b \,d^{2} e n \,x^{2}}{4}-\frac {b \,e^{3} n \,x^{6}}{36}-\frac {b \,d^{3} n \ln \left (x \right )^{2}}{2}-\frac {i \ln \left (x \right ) \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}\) | \(595\) |
1/144*(24*x^6*ln(c*x^n)*b*e^3*n-4*x^6*b*e^3*n^2+24*x^6*a*e^3*n+108*x^4*ln( c*x^n)*b*d*e^2*n-27*x^4*b*d*e^2*n^2+108*x^4*a*d*e^2*n+216*x^2*ln(c*x^n)*b* d^2*e*n-108*x^2*b*d^2*e*n^2+216*x^2*a*d^2*e*n+144*ln(x)*a*d^3*n+72*b*d^3*l n(c*x^n)^2)/n
Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.19 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {1}{36} \, {\left (b e^{3} n - 6 \, a e^{3}\right )} x^{6} + \frac {1}{2} \, b d^{3} n \log \left (x\right )^{2} - \frac {3}{16} \, {\left (b d e^{2} n - 4 \, a d e^{2}\right )} x^{4} - \frac {3}{4} \, {\left (b d^{2} e n - 2 \, a d^{2} e\right )} x^{2} + \frac {1}{12} \, {\left (2 \, b e^{3} x^{6} + 9 \, b d e^{2} x^{4} + 18 \, b d^{2} e x^{2}\right )} \log \left (c\right ) + \frac {1}{12} \, {\left (2 \, b e^{3} n x^{6} + 9 \, b d e^{2} n x^{4} + 18 \, b d^{2} e n x^{2} + 12 \, b d^{3} \log \left (c\right ) + 12 \, a d^{3}\right )} \log \left (x\right ) \]
-1/36*(b*e^3*n - 6*a*e^3)*x^6 + 1/2*b*d^3*n*log(x)^2 - 3/16*(b*d*e^2*n - 4 *a*d*e^2)*x^4 - 3/4*(b*d^2*e*n - 2*a*d^2*e)*x^2 + 1/12*(2*b*e^3*x^6 + 9*b* d*e^2*x^4 + 18*b*d^2*e*x^2)*log(c) + 1/12*(2*b*e^3*n*x^6 + 9*b*d*e^2*n*x^4 + 18*b*d^2*e*n*x^2 + 12*b*d^3*log(c) + 12*a*d^3)*log(x)
Time = 0.94 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.63 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \frac {a d^{3} \log {\left (c x^{n} \right )}}{n} + \frac {3 a d^{2} e x^{2}}{2} + \frac {3 a d e^{2} x^{4}}{4} + \frac {a e^{3} x^{6}}{6} + \frac {b d^{3} \log {\left (c x^{n} \right )}^{2}}{2 n} - \frac {3 b d^{2} e n x^{2}}{4} + \frac {3 b d^{2} e x^{2} \log {\left (c x^{n} \right )}}{2} - \frac {3 b d e^{2} n x^{4}}{16} + \frac {3 b d e^{2} x^{4} \log {\left (c x^{n} \right )}}{4} - \frac {b e^{3} n x^{6}}{36} + \frac {b e^{3} x^{6} \log {\left (c x^{n} \right )}}{6} & \text {for}\: n \neq 0 \\\left (a + b \log {\left (c \right )}\right ) \left (d^{3} \log {\left (x \right )} + \frac {3 d^{2} e x^{2}}{2} + \frac {3 d e^{2} x^{4}}{4} + \frac {e^{3} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d**3*log(c*x**n)/n + 3*a*d**2*e*x**2/2 + 3*a*d*e**2*x**4/4 + a*e**3*x**6/6 + b*d**3*log(c*x**n)**2/(2*n) - 3*b*d**2*e*n*x**2/4 + 3*b*d* *2*e*x**2*log(c*x**n)/2 - 3*b*d*e**2*n*x**4/16 + 3*b*d*e**2*x**4*log(c*x** n)/4 - b*e**3*n*x**6/36 + b*e**3*x**6*log(c*x**n)/6, Ne(n, 0)), ((a + b*lo g(c))*(d**3*log(x) + 3*d**2*e*x**2/2 + 3*d*e**2*x**4/4 + e**3*x**6/6), Tru e))
Time = 0.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {1}{36} \, b e^{3} n x^{6} + \frac {1}{6} \, b e^{3} x^{6} \log \left (c x^{n}\right ) + \frac {1}{6} \, a e^{3} x^{6} - \frac {3}{16} \, b d e^{2} n x^{4} + \frac {3}{4} \, b d e^{2} x^{4} \log \left (c x^{n}\right ) + \frac {3}{4} \, a d e^{2} x^{4} - \frac {3}{4} \, b d^{2} e n x^{2} + \frac {3}{2} \, b d^{2} e x^{2} \log \left (c x^{n}\right ) + \frac {3}{2} \, a d^{2} e x^{2} + \frac {b d^{3} \log \left (c x^{n}\right )^{2}}{2 \, n} + a d^{3} \log \left (x\right ) \]
-1/36*b*e^3*n*x^6 + 1/6*b*e^3*x^6*log(c*x^n) + 1/6*a*e^3*x^6 - 3/16*b*d*e^ 2*n*x^4 + 3/4*b*d*e^2*x^4*log(c*x^n) + 3/4*a*d*e^2*x^4 - 3/4*b*d^2*e*n*x^2 + 3/2*b*d^2*e*x^2*log(c*x^n) + 3/2*a*d^2*e*x^2 + 1/2*b*d^3*log(c*x^n)^2/n + a*d^3*log(x)
Time = 0.34 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.15 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {1}{36} \, {\left (b e^{3} n - 6 \, b e^{3} \log \left (c\right ) - 6 \, a e^{3}\right )} x^{6} + \frac {1}{2} \, b d^{3} n \log \left (x\right )^{2} - \frac {3}{16} \, {\left (b d e^{2} n - 4 \, b d e^{2} \log \left (c\right ) - 4 \, a d e^{2}\right )} x^{4} - \frac {3}{4} \, {\left (b d^{2} e n - 2 \, b d^{2} e \log \left (c\right ) - 2 \, a d^{2} e\right )} x^{2} + \frac {1}{12} \, {\left (2 \, b e^{3} n x^{6} + 9 \, b d e^{2} n x^{4} + 18 \, b d^{2} e n x^{2}\right )} \log \left (x\right ) + {\left (b d^{3} \log \left (c\right ) + a d^{3}\right )} \log \left (x\right ) \]
-1/36*(b*e^3*n - 6*b*e^3*log(c) - 6*a*e^3)*x^6 + 1/2*b*d^3*n*log(x)^2 - 3/ 16*(b*d*e^2*n - 4*b*d*e^2*log(c) - 4*a*d*e^2)*x^4 - 3/4*(b*d^2*e*n - 2*b*d ^2*e*log(c) - 2*a*d^2*e)*x^2 + 1/12*(2*b*e^3*n*x^6 + 9*b*d*e^2*n*x^4 + 18* b*d^2*e*n*x^2)*log(x) + (b*d^3*log(c) + a*d^3)*log(x)
Time = 0.38 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\ln \left (c\,x^n\right )\,\left (\frac {3\,b\,d^2\,e\,x^2}{2}+\frac {3\,b\,d\,e^2\,x^4}{4}+\frac {b\,e^3\,x^6}{6}\right )+\frac {e^3\,x^6\,\left (6\,a-b\,n\right )}{36}+a\,d^3\,\ln \left (x\right )+\frac {b\,d^3\,{\ln \left (c\,x^n\right )}^2}{2\,n}+\frac {3\,d^2\,e\,x^2\,\left (2\,a-b\,n\right )}{4}+\frac {3\,d\,e^2\,x^4\,\left (4\,a-b\,n\right )}{16} \]