Integrand size = 21, antiderivative size = 74 \[ \int x^3 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} b d^2 n x^4-\frac {2}{25} b d e n x^5-\frac {1}{36} b e^2 n x^6+\frac {1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right ) \] Output:
-1/16*b*d^2*n*x^4-2/25*b*d*e*n*x^5-1/36*b*e^2*n*x^6+1/60*(10*e^2*x^6+24*d* e*x^5+15*d^2*x^4)*(a+b*ln(c*x^n))
Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.09 \[ \int x^3 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^4 \left (60 a \left (15 d^2+24 d e x+10 e^2 x^2\right )-b n \left (225 d^2+288 d e x+100 e^2 x^2\right )+60 b \left (15 d^2+24 d e x+10 e^2 x^2\right ) \log \left (c x^n\right )\right )}{3600} \] Input:
Integrate[x^3*(d + e*x)^2*(a + b*Log[c*x^n]),x]
Output:
(x^4*(60*a*(15*d^2 + 24*d*e*x + 10*e^2*x^2) - b*n*(225*d^2 + 288*d*e*x + 1 00*e^2*x^2) + 60*b*(15*d^2 + 24*d*e*x + 10*e^2*x^2)*Log[c*x^n]))/3600
Time = 0.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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.
\(\displaystyle \int x^3 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2771 |
\(\displaystyle \frac {1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{60} x^3 \left (15 d^2+24 e x d+10 e^2 x^2\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{60} b n \int x^3 \left (15 d^2+24 e x d+10 e^2 x^2\right )dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \frac {1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{60} b n \int \left (10 e^2 x^5+24 d e x^4+15 d^2 x^3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{60} \left (15 d^2 x^4+24 d e x^5+10 e^2 x^6\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{60} b n \left (\frac {15 d^2 x^4}{4}+\frac {24}{5} d e x^5+\frac {5 e^2 x^6}{3}\right )\) |
Input:
Int[x^3*(d + e*x)^2*(a + b*Log[c*x^n]),x]
Output:
-1/60*(b*n*((15*d^2*x^4)/4 + (24*d*e*x^5)/5 + (5*e^2*x^6)/3)) + ((15*d^2*x ^4 + 24*d*e*x^5 + 10*e^2*x^6)*(a + b*Log[c*x^n]))/60
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 = 9.98 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.36
method | result | size |
parallelrisch | \(\frac {x^{6} \ln \left (c \,x^{n}\right ) b \,e^{2}}{6}-\frac {b \,e^{2} n \,x^{6}}{36}+\frac {a \,e^{2} x^{6}}{6}+\frac {2 x^{5} \ln \left (c \,x^{n}\right ) b d e}{5}-\frac {2 b d e n \,x^{5}}{25}+\frac {2 a d e \,x^{5}}{5}+\frac {x^{4} \ln \left (c \,x^{n}\right ) b \,d^{2}}{4}-\frac {b \,d^{2} n \,x^{4}}{16}+\frac {a \,d^{2} x^{4}}{4}\) | \(101\) |
risch | \(\frac {b \,x^{4} \left (10 e^{2} x^{2}+24 e x d +15 d^{2}\right ) \ln \left (x^{n}\right )}{60}+\frac {i \pi b \,e^{2} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{12}-\frac {i \pi b \,e^{2} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{12}+\frac {i \pi b d e \,x^{5} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{5}+\frac {i \pi b \,d^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {\ln \left (c \right ) b \,e^{2} x^{6}}{6}-\frac {b \,e^{2} n \,x^{6}}{36}+\frac {a \,e^{2} x^{6}}{6}+\frac {i \pi b d e \,x^{5} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{5}-\frac {i \pi b d e \,x^{5} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{5}+\frac {i \pi b \,d^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{8}+\frac {i \pi b \,e^{2} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{12}+\frac {2 \ln \left (c \right ) b d e \,x^{5}}{5}-\frac {2 b d e n \,x^{5}}{25}+\frac {2 a d e \,x^{5}}{5}-\frac {i \pi b \,d^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{8}-\frac {i \pi b \,d^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{8}-\frac {i \pi b \,e^{2} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{12}-\frac {i \pi b d e \,x^{5} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{5}+\frac {\ln \left (c \right ) b \,d^{2} x^{4}}{4}-\frac {b \,d^{2} n \,x^{4}}{16}+\frac {a \,d^{2} x^{4}}{4}\) | \(432\) |
Input:
int(x^3*(e*x+d)^2*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)
Output:
1/6*x^6*ln(c*x^n)*b*e^2-1/36*b*e^2*n*x^6+1/6*a*e^2*x^6+2/5*x^5*ln(c*x^n)*b *d*e-2/25*b*d*e*n*x^5+2/5*a*d*e*x^5+1/4*x^4*ln(c*x^n)*b*d^2-1/16*b*d^2*n*x ^4+1/4*a*d^2*x^4
Time = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.59 \[ \int x^3 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{36} \, {\left (b e^{2} n - 6 \, a e^{2}\right )} x^{6} - \frac {2}{25} \, {\left (b d e n - 5 \, a d e\right )} x^{5} - \frac {1}{16} \, {\left (b d^{2} n - 4 \, a d^{2}\right )} x^{4} + \frac {1}{60} \, {\left (10 \, b e^{2} x^{6} + 24 \, b d e x^{5} + 15 \, b d^{2} x^{4}\right )} \log \left (c\right ) + \frac {1}{60} \, {\left (10 \, b e^{2} n x^{6} + 24 \, b d e n x^{5} + 15 \, b d^{2} n x^{4}\right )} \log \left (x\right ) \] Input:
integrate(x^3*(e*x+d)^2*(a+b*log(c*x^n)),x, algorithm="fricas")
Output:
-1/36*(b*e^2*n - 6*a*e^2)*x^6 - 2/25*(b*d*e*n - 5*a*d*e)*x^5 - 1/16*(b*d^2 *n - 4*a*d^2)*x^4 + 1/60*(10*b*e^2*x^6 + 24*b*d*e*x^5 + 15*b*d^2*x^4)*log( c) + 1/60*(10*b*e^2*n*x^6 + 24*b*d*e*n*x^5 + 15*b*d^2*n*x^4)*log(x)
Time = 0.51 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.64 \[ \int x^3 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d^{2} x^{4}}{4} + \frac {2 a d e x^{5}}{5} + \frac {a e^{2} x^{6}}{6} - \frac {b d^{2} n x^{4}}{16} + \frac {b d^{2} x^{4} \log {\left (c x^{n} \right )}}{4} - \frac {2 b d e n x^{5}}{25} + \frac {2 b d e x^{5} \log {\left (c x^{n} \right )}}{5} - \frac {b e^{2} n x^{6}}{36} + \frac {b e^{2} x^{6} \log {\left (c x^{n} \right )}}{6} \] Input:
integrate(x**3*(e*x+d)**2*(a+b*ln(c*x**n)),x)
Output:
a*d**2*x**4/4 + 2*a*d*e*x**5/5 + a*e**2*x**6/6 - b*d**2*n*x**4/16 + b*d**2 *x**4*log(c*x**n)/4 - 2*b*d*e*n*x**5/25 + 2*b*d*e*x**5*log(c*x**n)/5 - b*e **2*n*x**6/36 + b*e**2*x**6*log(c*x**n)/6
Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.35 \[ \int x^3 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{36} \, b e^{2} n x^{6} + \frac {1}{6} \, b e^{2} x^{6} \log \left (c x^{n}\right ) - \frac {2}{25} \, b d e n x^{5} + \frac {1}{6} \, a e^{2} x^{6} + \frac {2}{5} \, b d e x^{5} \log \left (c x^{n}\right ) - \frac {1}{16} \, b d^{2} n x^{4} + \frac {2}{5} \, a d e x^{5} + \frac {1}{4} \, b d^{2} x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d^{2} x^{4} \] Input:
integrate(x^3*(e*x+d)^2*(a+b*log(c*x^n)),x, algorithm="maxima")
Output:
-1/36*b*e^2*n*x^6 + 1/6*b*e^2*x^6*log(c*x^n) - 2/25*b*d*e*n*x^5 + 1/6*a*e^ 2*x^6 + 2/5*b*d*e*x^5*log(c*x^n) - 1/16*b*d^2*n*x^4 + 2/5*a*d*e*x^5 + 1/4* b*d^2*x^4*log(c*x^n) + 1/4*a*d^2*x^4
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.66 \[ \int x^3 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{6} \, b e^{2} n x^{6} \log \left (x\right ) - \frac {1}{36} \, b e^{2} n x^{6} + \frac {1}{6} \, b e^{2} x^{6} \log \left (c\right ) + \frac {2}{5} \, b d e n x^{5} \log \left (x\right ) - \frac {2}{25} \, b d e n x^{5} + \frac {1}{6} \, a e^{2} x^{6} + \frac {2}{5} \, b d e x^{5} \log \left (c\right ) + \frac {1}{4} \, b d^{2} n x^{4} \log \left (x\right ) - \frac {1}{16} \, b d^{2} n x^{4} + \frac {2}{5} \, a d e x^{5} + \frac {1}{4} \, b d^{2} x^{4} \log \left (c\right ) + \frac {1}{4} \, a d^{2} x^{4} \] Input:
integrate(x^3*(e*x+d)^2*(a+b*log(c*x^n)),x, algorithm="giac")
Output:
1/6*b*e^2*n*x^6*log(x) - 1/36*b*e^2*n*x^6 + 1/6*b*e^2*x^6*log(c) + 2/5*b*d *e*n*x^5*log(x) - 2/25*b*d*e*n*x^5 + 1/6*a*e^2*x^6 + 2/5*b*d*e*x^5*log(c) + 1/4*b*d^2*n*x^4*log(x) - 1/16*b*d^2*n*x^4 + 2/5*a*d*e*x^5 + 1/4*b*d^2*x^ 4*log(c) + 1/4*a*d^2*x^4
Time = 28.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.11 \[ \int x^3 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^4}{4}+\frac {2\,b\,d\,e\,x^5}{5}+\frac {b\,e^2\,x^6}{6}\right )+\frac {d^2\,x^4\,\left (4\,a-b\,n\right )}{16}+\frac {e^2\,x^6\,\left (6\,a-b\,n\right )}{36}+\frac {2\,d\,e\,x^5\,\left (5\,a-b\,n\right )}{25} \] Input:
int(x^3*(a + b*log(c*x^n))*(d + e*x)^2,x)
Output:
log(c*x^n)*((b*d^2*x^4)/4 + (b*e^2*x^6)/6 + (2*b*d*e*x^5)/5) + (d^2*x^4*(4 *a - b*n))/16 + (e^2*x^6*(6*a - b*n))/36 + (2*d*e*x^5*(5*a - b*n))/25
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.22 \[ \int x^3 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^{4} \left (900 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2}+1440 \,\mathrm {log}\left (x^{n} c \right ) b d e x +600 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{2} x^{2}+900 a \,d^{2}+1440 a d e x +600 a \,e^{2} x^{2}-225 b \,d^{2} n -288 b d e n x -100 b \,e^{2} n \,x^{2}\right )}{3600} \] Input:
int(x^3*(e*x+d)^2*(a+b*log(c*x^n)),x)
Output:
(x**4*(900*log(x**n*c)*b*d**2 + 1440*log(x**n*c)*b*d*e*x + 600*log(x**n*c) *b*e**2*x**2 + 900*a*d**2 + 1440*a*d*e*x + 600*a*e**2*x**2 - 225*b*d**2*n - 288*b*d*e*n*x - 100*b*e**2*n*x**2))/3600