Integrand size = 19, antiderivative size = 74 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} b d^2 n x^2-\frac {2}{9} b d e n x^3-\frac {1}{16} b e^2 n x^4+\frac {1}{12} \left (6 d^2 x^2+8 d e x^3+3 e^2 x^4\right ) \left (a+b \log \left (c x^n\right )\right ) \] Output:
-1/4*b*d^2*n*x^2-2/9*b*d*e*n*x^3-1/16*b*e^2*n*x^4+1/12*(3*e^2*x^4+8*d*e*x^ 3+6*d^2*x^2)*(a+b*ln(c*x^n))
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.09 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{144} x^2 \left (12 a \left (6 d^2+8 d e x+3 e^2 x^2\right )-b n \left (36 d^2+32 d e x+9 e^2 x^2\right )+12 b \left (6 d^2+8 d e x+3 e^2 x^2\right ) \log \left (c x^n\right )\right ) \] Input:
Integrate[x*(d + e*x)^2*(a + b*Log[c*x^n]),x]
Output:
(x^2*(12*a*(6*d^2 + 8*d*e*x + 3*e^2*x^2) - b*n*(36*d^2 + 32*d*e*x + 9*e^2* x^2) + 12*b*(6*d^2 + 8*d*e*x + 3*e^2*x^2)*Log[c*x^n]))/144
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2771, 27, 1140, 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.
\(\displaystyle \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2771 |
\(\displaystyle \frac {1}{12} \left (6 d^2 x^2+8 d e x^3+3 e^2 x^4\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{12} x \left (6 d^2+8 e x d+3 e^2 x^2\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{12} \left (6 d^2 x^2+8 d e x^3+3 e^2 x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{12} b n \int x \left (6 d^2+8 e x d+3 e^2 x^2\right )dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \frac {1}{12} \left (6 d^2 x^2+8 d e x^3+3 e^2 x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{12} b n \int \left (3 e^2 x^3+8 d e x^2+6 d^2 x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{12} \left (6 d^2 x^2+8 d e x^3+3 e^2 x^4\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{12} b n \left (3 d^2 x^2+\frac {8}{3} d e x^3+\frac {3 e^2 x^4}{4}\right )\) |
Input:
Int[x*(d + e*x)^2*(a + b*Log[c*x^n]),x]
Output:
-1/12*(b*n*(3*d^2*x^2 + (8*d*e*x^3)/3 + (3*e^2*x^4)/4)) + ((6*d^2*x^2 + 8* d*e*x^3 + 3*e^2*x^4)*(a + b*Log[c*x^n]))/12
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
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[u*(a + b*Log[c*x^n]), x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IGtQ[m, 0]
Time = 1.51 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.36
method | result | size |
parallelrisch | \(\frac {x^{4} b \ln \left (c \,x^{n}\right ) e^{2}}{4}-\frac {b \,e^{2} n \,x^{4}}{16}+\frac {x^{4} a \,e^{2}}{4}+\frac {2 x^{3} b \ln \left (c \,x^{n}\right ) e d}{3}-\frac {2 b d e n \,x^{3}}{9}+\frac {2 x^{3} a e d}{3}+\frac {x^{2} b \ln \left (c \,x^{n}\right ) d^{2}}{2}-\frac {b \,d^{2} n \,x^{2}}{4}+\frac {x^{2} a \,d^{2}}{2}\) | \(101\) |
risch | \(\frac {b \,x^{2} \left (3 e^{2} x^{2}+8 e x d +6 d^{2}\right ) \ln \left (x^{n}\right )}{12}-\frac {i \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{4}-\frac {i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{8}+\frac {i \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}+\frac {i \pi b d e \,x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{3}+\frac {\ln \left (c \right ) b \,e^{2} x^{4}}{4}-\frac {b \,e^{2} n \,x^{4}}{16}+\frac {x^{4} a \,e^{2}}{4}+\frac {i \pi b d e \,x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{3}-\frac {i \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{4}-\frac {i \pi b d e \,x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{3}-\frac {i \pi b d e \,x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{3}+\frac {2 \ln \left (c \right ) b d e \,x^{3}}{3}-\frac {2 b d e n \,x^{3}}{9}+\frac {2 x^{3} a e d}{3}+\frac {i \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{4}+\frac {i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{8}-\frac {i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{8}+\frac {\ln \left (c \right ) b \,d^{2} x^{2}}{2}-\frac {b \,d^{2} n \,x^{2}}{4}+\frac {x^{2} a \,d^{2}}{2}\) | \(432\) |
Input:
int(x*(e*x+d)^2*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)
Output:
1/4*x^4*b*ln(c*x^n)*e^2-1/16*b*e^2*n*x^4+1/4*x^4*a*e^2+2/3*x^3*b*ln(c*x^n) *e*d-2/9*b*d*e*n*x^3+2/3*x^3*a*e*d+1/2*x^2*b*ln(c*x^n)*d^2-1/4*b*d^2*n*x^2 +1/2*x^2*a*d^2
Time = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.59 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, {\left (b e^{2} n - 4 \, a e^{2}\right )} x^{4} - \frac {2}{9} \, {\left (b d e n - 3 \, a d e\right )} x^{3} - \frac {1}{4} \, {\left (b d^{2} n - 2 \, a d^{2}\right )} x^{2} + \frac {1}{12} \, {\left (3 \, b e^{2} x^{4} + 8 \, b d e x^{3} + 6 \, b d^{2} x^{2}\right )} \log \left (c\right ) + \frac {1}{12} \, {\left (3 \, b e^{2} n x^{4} + 8 \, b d e n x^{3} + 6 \, b d^{2} n x^{2}\right )} \log \left (x\right ) \] Input:
integrate(x*(e*x+d)^2*(a+b*log(c*x^n)),x, algorithm="fricas")
Output:
-1/16*(b*e^2*n - 4*a*e^2)*x^4 - 2/9*(b*d*e*n - 3*a*d*e)*x^3 - 1/4*(b*d^2*n - 2*a*d^2)*x^2 + 1/12*(3*b*e^2*x^4 + 8*b*d*e*x^3 + 6*b*d^2*x^2)*log(c) + 1/12*(3*b*e^2*n*x^4 + 8*b*d*e*n*x^3 + 6*b*d^2*n*x^2)*log(x)
Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.64 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d^{2} x^{2}}{2} + \frac {2 a d e x^{3}}{3} + \frac {a e^{2} x^{4}}{4} - \frac {b d^{2} n x^{2}}{4} + \frac {b d^{2} x^{2} \log {\left (c x^{n} \right )}}{2} - \frac {2 b d e n x^{3}}{9} + \frac {2 b d e x^{3} \log {\left (c x^{n} \right )}}{3} - \frac {b e^{2} n x^{4}}{16} + \frac {b e^{2} x^{4} \log {\left (c x^{n} \right )}}{4} \] Input:
integrate(x*(e*x+d)**2*(a+b*ln(c*x**n)),x)
Output:
a*d**2*x**2/2 + 2*a*d*e*x**3/3 + a*e**2*x**4/4 - b*d**2*n*x**2/4 + b*d**2* x**2*log(c*x**n)/2 - 2*b*d*e*n*x**3/9 + 2*b*d*e*x**3*log(c*x**n)/3 - b*e** 2*n*x**4/16 + b*e**2*x**4*log(c*x**n)/4
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.35 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, b e^{2} n x^{4} + \frac {1}{4} \, b e^{2} x^{4} \log \left (c x^{n}\right ) - \frac {2}{9} \, b d e n x^{3} + \frac {1}{4} \, a e^{2} x^{4} + \frac {2}{3} \, b d e x^{3} \log \left (c x^{n}\right ) - \frac {1}{4} \, b d^{2} n x^{2} + \frac {2}{3} \, a d e x^{3} + \frac {1}{2} \, b d^{2} x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a d^{2} x^{2} \] Input:
integrate(x*(e*x+d)^2*(a+b*log(c*x^n)),x, algorithm="maxima")
Output:
-1/16*b*e^2*n*x^4 + 1/4*b*e^2*x^4*log(c*x^n) - 2/9*b*d*e*n*x^3 + 1/4*a*e^2 *x^4 + 2/3*b*d*e*x^3*log(c*x^n) - 1/4*b*d^2*n*x^2 + 2/3*a*d*e*x^3 + 1/2*b* d^2*x^2*log(c*x^n) + 1/2*a*d^2*x^2
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.66 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} \, b e^{2} n x^{4} \log \left (x\right ) - \frac {1}{16} \, b e^{2} n x^{4} + \frac {1}{4} \, b e^{2} x^{4} \log \left (c\right ) + \frac {2}{3} \, b d e n x^{3} \log \left (x\right ) - \frac {2}{9} \, b d e n x^{3} + \frac {1}{4} \, a e^{2} x^{4} + \frac {2}{3} \, b d e x^{3} \log \left (c\right ) + \frac {1}{2} \, b d^{2} n x^{2} \log \left (x\right ) - \frac {1}{4} \, b d^{2} n x^{2} + \frac {2}{3} \, a d e x^{3} + \frac {1}{2} \, b d^{2} x^{2} \log \left (c\right ) + \frac {1}{2} \, a d^{2} x^{2} \] Input:
integrate(x*(e*x+d)^2*(a+b*log(c*x^n)),x, algorithm="giac")
Output:
1/4*b*e^2*n*x^4*log(x) - 1/16*b*e^2*n*x^4 + 1/4*b*e^2*x^4*log(c) + 2/3*b*d *e*n*x^3*log(x) - 2/9*b*d*e*n*x^3 + 1/4*a*e^2*x^4 + 2/3*b*d*e*x^3*log(c) + 1/2*b*d^2*n*x^2*log(x) - 1/4*b*d^2*n*x^2 + 2/3*a*d*e*x^3 + 1/2*b*d^2*x^2* log(c) + 1/2*a*d^2*x^2
Time = 29.49 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.11 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^2}{2}+\frac {2\,b\,d\,e\,x^3}{3}+\frac {b\,e^2\,x^4}{4}\right )+\frac {d^2\,x^2\,\left (2\,a-b\,n\right )}{4}+\frac {e^2\,x^4\,\left (4\,a-b\,n\right )}{16}+\frac {2\,d\,e\,x^3\,\left (3\,a-b\,n\right )}{9} \] Input:
int(x*(a + b*log(c*x^n))*(d + e*x)^2,x)
Output:
log(c*x^n)*((b*d^2*x^2)/2 + (b*e^2*x^4)/4 + (2*b*d*e*x^3)/3) + (d^2*x^2*(2 *a - b*n))/4 + (e^2*x^4*(4*a - b*n))/16 + (2*d*e*x^3*(3*a - b*n))/9
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.22 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^{2} \left (72 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2}+96 \,\mathrm {log}\left (x^{n} c \right ) b d e x +36 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{2} x^{2}+72 a \,d^{2}+96 a d e x +36 a \,e^{2} x^{2}-36 b \,d^{2} n -32 b d e n x -9 b \,e^{2} n \,x^{2}\right )}{144} \] Input:
int(x*(e*x+d)^2*(a+b*log(c*x^n)),x)
Output:
(x**2*(72*log(x**n*c)*b*d**2 + 96*log(x**n*c)*b*d*e*x + 36*log(x**n*c)*b*e **2*x**2 + 72*a*d**2 + 96*a*d*e*x + 36*a*e**2*x**2 - 36*b*d**2*n - 32*b*d* e*n*x - 9*b*e**2*n*x**2))/144