Integrand size = 18, antiderivative size = 85 \[ \int (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-b d^3 n x-\frac {3}{4} b d^2 e n x^2-\frac {1}{3} b d e^2 n x^3-\frac {1}{16} b e^3 n x^4-\frac {b d^4 n \log (x)}{4 e}+\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e} \] Output:
-b*d^3*n*x-3/4*b*d^2*e*n*x^2-1/3*b*d*e^2*n*x^3-1/16*b*e^3*n*x^4-1/4*b*d^4* n*ln(x)/e+1/4*(e*x+d)^4*(a+b*ln(c*x^n))/e
Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.29 \[ \int (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{48} x \left (12 a \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-b n \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )+12 b \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \log \left (c x^n\right )\right ) \] Input:
Integrate[(d + e*x)^3*(a + b*Log[c*x^n]),x]
Output:
(x*(12*a*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - b*n*(48*d^3 + 36*d^ 2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3) + 12*b*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*Log[c*x^n]))/48
Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2750, 27, 49, 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.
\(\displaystyle \int (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2750 |
\(\displaystyle \frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-b n \int \frac {(d+e x)^4}{4 e x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {b n \int \frac {(d+e x)^4}{x}dx}{4 e}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {b n \int \left (\frac {d^4}{x}+4 e d^3+6 e^2 x d^2+4 e^3 x^2 d+e^4 x^3\right )dx}{4 e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {b n \left (d^4 \log (x)+4 d^3 e x+3 d^2 e^2 x^2+\frac {4}{3} d e^3 x^3+\frac {e^4 x^4}{4}\right )}{4 e}\) |
Input:
Int[(d + e*x)^3*(a + b*Log[c*x^n]),x]
Output:
-1/4*(b*n*(4*d^3*e*x + 3*d^2*e^2*x^2 + (4*d*e^3*x^3)/3 + (e^4*x^4)/4 + d^4 *Log[x]))/e + ((d + e*x)^4*(a + b*Log[c*x^n]))/(4*e)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(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]
Time = 0.60 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.58
method | result | size |
parallelrisch | \(\frac {b \ln \left (c \,x^{n}\right ) e^{3} x^{4}}{4}-\frac {b \,e^{3} n \,x^{4}}{16}+\frac {a \,e^{3} x^{4}}{4}+b \ln \left (c \,x^{n}\right ) d \,e^{2} x^{3}-\frac {b d \,e^{2} n \,x^{3}}{3}+a \,e^{2} x^{3} d +\frac {3 b \ln \left (c \,x^{n}\right ) d^{2} e \,x^{2}}{2}-\frac {3 b \,d^{2} e n \,x^{2}}{4}+\frac {3 a \,d^{2} e \,x^{2}}{2}+x b \ln \left (c \,x^{n}\right ) d^{3}-b \,d^{3} n x +x a \,d^{3}\) | \(134\) |
risch | \(-\frac {i e^{3} \pi b \,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 e^{2} \pi b d \,x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+x a \,d^{3}+\frac {a \,e^{3} x^{4}}{4}+a \,e^{2} x^{3} d +\frac {3 a \,d^{2} e \,x^{2}}{2}+\frac {3 \ln \left (c \right ) b \,d^{2} e \,x^{2}}{2}+\frac {i e^{2} \pi b d \,x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\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}+\frac {3 i e \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 \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) x}{2}+\frac {\ln \left (c \right ) b \,e^{3} x^{4}}{4}+x \ln \left (c \right ) b \,d^{3}+\frac {\left (e x +d \right )^{4} b \ln \left (x^{n}\right )}{4 e}+\ln \left (c \right ) b d \,e^{2} x^{3}+\frac {i e^{3} \pi b \,x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i e^{3} \pi b \,x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{8}-\frac {i e^{2} \pi b d \,x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}-\frac {3 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{4}-\frac {i e^{3} \pi b \,x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{8}-\frac {i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x}{2}+\frac {i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x}{2}+\frac {i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right ) x}{2}-\frac {i e^{2} \pi b d \,x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\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 ) \operatorname {csgn}\left (i c \right )}{4}-\frac {3 b \,d^{2} e n \,x^{2}}{4}-\frac {b d \,e^{2} n \,x^{3}}{3}-\frac {b \,d^{4} n \ln \left (x \right )}{4 e}-\frac {b \,e^{3} n \,x^{4}}{16}-b \,d^{3} n x\) | \(571\) |
Input:
int((e*x+d)^3*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)
Output:
1/4*b*ln(c*x^n)*e^3*x^4-1/16*b*e^3*n*x^4+1/4*a*e^3*x^4+b*ln(c*x^n)*d*e^2*x ^3-1/3*b*d*e^2*n*x^3+a*e^2*x^3*d+3/2*b*ln(c*x^n)*d^2*e*x^2-3/4*b*d^2*e*n*x ^2+3/2*a*d^2*e*x^2+x*b*ln(c*x^n)*d^3-b*d^3*n*x+x*a*d^3
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (75) = 150\).
Time = 0.07 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.87 \[ \int (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, {\left (b e^{3} n - 4 \, a e^{3}\right )} x^{4} - \frac {1}{3} \, {\left (b d e^{2} n - 3 \, a d e^{2}\right )} x^{3} - \frac {3}{4} \, {\left (b d^{2} e n - 2 \, a d^{2} e\right )} x^{2} - {\left (b d^{3} n - a d^{3}\right )} x + \frac {1}{4} \, {\left (b e^{3} x^{4} + 4 \, b d e^{2} x^{3} + 6 \, b d^{2} e x^{2} + 4 \, b d^{3} x\right )} \log \left (c\right ) + \frac {1}{4} \, {\left (b e^{3} n x^{4} + 4 \, b d e^{2} n x^{3} + 6 \, b d^{2} e n x^{2} + 4 \, b d^{3} n x\right )} \log \left (x\right ) \] Input:
integrate((e*x+d)^3*(a+b*log(c*x^n)),x, algorithm="fricas")
Output:
-1/16*(b*e^3*n - 4*a*e^3)*x^4 - 1/3*(b*d*e^2*n - 3*a*d*e^2)*x^3 - 3/4*(b*d ^2*e*n - 2*a*d^2*e)*x^2 - (b*d^3*n - a*d^3)*x + 1/4*(b*e^3*x^4 + 4*b*d*e^2 *x^3 + 6*b*d^2*e*x^2 + 4*b*d^3*x)*log(c) + 1/4*(b*e^3*n*x^4 + 4*b*d*e^2*n* x^3 + 6*b*d^2*e*n*x^2 + 4*b*d^3*n*x)*log(x)
Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.84 \[ \int (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} - b d^{3} n x + b d^{3} x \log {\left (c x^{n} \right )} - \frac {3 b d^{2} e n x^{2}}{4} + \frac {3 b d^{2} e x^{2} \log {\left (c x^{n} \right )}}{2} - \frac {b d e^{2} n x^{3}}{3} + b d e^{2} x^{3} \log {\left (c x^{n} \right )} - \frac {b e^{3} n x^{4}}{16} + \frac {b e^{3} x^{4} \log {\left (c x^{n} \right )}}{4} \] Input:
integrate((e*x+d)**3*(a+b*ln(c*x**n)),x)
Output:
a*d**3*x + 3*a*d**2*e*x**2/2 + a*d*e**2*x**3 + a*e**3*x**4/4 - b*d**3*n*x + b*d**3*x*log(c*x**n) - 3*b*d**2*e*n*x**2/4 + 3*b*d**2*e*x**2*log(c*x**n) /2 - b*d*e**2*n*x**3/3 + b*d*e**2*x**3*log(c*x**n) - b*e**3*n*x**4/16 + b* e**3*x**4*log(c*x**n)/4
Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.56 \[ \int (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, b e^{3} n x^{4} + \frac {1}{4} \, b e^{3} x^{4} \log \left (c x^{n}\right ) - \frac {1}{3} \, b d e^{2} n x^{3} + \frac {1}{4} \, a e^{3} x^{4} + b d e^{2} x^{3} \log \left (c x^{n}\right ) - \frac {3}{4} \, b d^{2} e n x^{2} + a d e^{2} x^{3} + \frac {3}{2} \, b d^{2} e x^{2} \log \left (c x^{n}\right ) - b d^{3} n x + \frac {3}{2} \, a d^{2} e x^{2} + b d^{3} x \log \left (c x^{n}\right ) + a d^{3} x \] Input:
integrate((e*x+d)^3*(a+b*log(c*x^n)),x, algorithm="maxima")
Output:
-1/16*b*e^3*n*x^4 + 1/4*b*e^3*x^4*log(c*x^n) - 1/3*b*d*e^2*n*x^3 + 1/4*a*e ^3*x^4 + b*d*e^2*x^3*log(c*x^n) - 3/4*b*d^2*e*n*x^2 + a*d*e^2*x^3 + 3/2*b* d^2*e*x^2*log(c*x^n) - b*d^3*n*x + 3/2*a*d^2*e*x^2 + b*d^3*x*log(c*x^n) + a*d^3*x
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (75) = 150\).
Time = 0.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.92 \[ \int (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} \, b e^{3} n x^{4} \log \left (x\right ) - \frac {1}{16} \, b e^{3} n x^{4} + \frac {1}{4} \, b e^{3} x^{4} \log \left (c\right ) + b d e^{2} n x^{3} \log \left (x\right ) - \frac {1}{3} \, b d e^{2} n x^{3} + \frac {1}{4} \, a e^{3} x^{4} + b d e^{2} x^{3} \log \left (c\right ) + \frac {3}{2} \, b d^{2} e n x^{2} \log \left (x\right ) - \frac {3}{4} \, b d^{2} e n x^{2} + a d e^{2} x^{3} + \frac {3}{2} \, b d^{2} e x^{2} \log \left (c\right ) + b d^{3} n x \log \left (x\right ) - b d^{3} n x + \frac {3}{2} \, a d^{2} e x^{2} + b d^{3} x \log \left (c\right ) + a d^{3} x \] Input:
integrate((e*x+d)^3*(a+b*log(c*x^n)),x, algorithm="giac")
Output:
1/4*b*e^3*n*x^4*log(x) - 1/16*b*e^3*n*x^4 + 1/4*b*e^3*x^4*log(c) + b*d*e^2 *n*x^3*log(x) - 1/3*b*d*e^2*n*x^3 + 1/4*a*e^3*x^4 + b*d*e^2*x^3*log(c) + 3 /2*b*d^2*e*n*x^2*log(x) - 3/4*b*d^2*e*n*x^2 + a*d*e^2*x^3 + 3/2*b*d^2*e*x^ 2*log(c) + b*d^3*n*x*log(x) - b*d^3*n*x + 3/2*a*d^2*e*x^2 + b*d^3*x*log(c) + a*d^3*x
Time = 27.71 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.22 \[ \int (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (b\,d^3\,x+\frac {3\,b\,d^2\,e\,x^2}{2}+b\,d\,e^2\,x^3+\frac {b\,e^3\,x^4}{4}\right )+\frac {e^3\,x^4\,\left (4\,a-b\,n\right )}{16}+d^3\,x\,\left (a-b\,n\right )+\frac {3\,d^2\,e\,x^2\,\left (2\,a-b\,n\right )}{4}+\frac {d\,e^2\,x^3\,\left (3\,a-b\,n\right )}{3} \] Input:
int((a + b*log(c*x^n))*(d + e*x)^3,x)
Output:
log(c*x^n)*((b*e^3*x^4)/4 + b*d^3*x + (3*b*d^2*e*x^2)/2 + b*d*e^2*x^3) + ( e^3*x^4*(4*a - b*n))/16 + d^3*x*(a - b*n) + (3*d^2*e*x^2*(2*a - b*n))/4 + (d*e^2*x^3*(3*a - b*n))/3
Time = 0.52 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.54 \[ \int (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (48 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3}+72 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e x +48 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{2} x^{2}+12 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{3} x^{3}+48 a \,d^{3}+72 a \,d^{2} e x +48 a d \,e^{2} x^{2}+12 a \,e^{3} x^{3}-48 b \,d^{3} n -36 b \,d^{2} e n x -16 b d \,e^{2} n \,x^{2}-3 b \,e^{3} n \,x^{3}\right )}{48} \] Input:
int((e*x+d)^3*(a+b*log(c*x^n)),x)
Output:
(x*(48*log(x**n*c)*b*d**3 + 72*log(x**n*c)*b*d**2*e*x + 48*log(x**n*c)*b*d *e**2*x**2 + 12*log(x**n*c)*b*e**3*x**3 + 48*a*d**3 + 72*a*d**2*e*x + 48*a *d*e**2*x**2 + 12*a*e**3*x**3 - 48*b*d**3*n - 36*b*d**2*e*n*x - 16*b*d*e** 2*n*x**2 - 3*b*e**3*n*x**3))/48