\(\int \frac {(d+e x^r)^3 (a+b \log (c x^n))}{x} \, dx\) [393]

Optimal result

Integrand size = 23, antiderivative size = 152 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {3 b d^2 e n x^r}{r^2}-\frac {3 b d e^2 n x^{2 r}}{4 r^2}-\frac {b e^3 n x^{3 r}}{9 r^2}-\frac {1}{2} b d^3 n \log ^2(x)+\frac {3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+\frac {3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r}+\frac {e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{3 r}+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right ) \] Output:

-3*b*d^2*e*n*x^r/r^2-3/4*b*d*e^2*n*x^(2*r)/r^2-1/9*b*e^3*n*x^(3*r)/r^2-1/2 
*b*d^3*n*ln(x)^2+3*d^2*e*x^r*(a+b*ln(c*x^n))/r+3/2*d*e^2*x^(2*r)*(a+b*ln(c 
*x^n))/r+1/3*e^3*x^(3*r)*(a+b*ln(c*x^n))/r+d^3*ln(x)*(a+b*ln(c*x^n))
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=a d^3 \log (x)+\frac {1}{36} \left (\frac {e x^r \left (6 a r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )-b n \left (108 d^2+27 d e x^r+4 e^2 x^{2 r}\right )\right )}{r^2}+\frac {6 b e x^r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right ) \log \left (c x^n\right )}{r}+\frac {18 b d^3 \log ^2\left (c x^n\right )}{n}\right ) \] Input:

Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x,x]
 

Output:

a*d^3*Log[x] + ((e*x^r*(6*a*r*(18*d^2 + 9*d*e*x^r + 2*e^2*x^(2*r)) - b*n*( 
108*d^2 + 27*d*e*x^r + 4*e^2*x^(2*r))))/r^2 + (6*b*e*x^r*(18*d^2 + 9*d*e*x 
^r + 2*e^2*x^(2*r))*Log[c*x^n])/r + (18*b*d^3*Log[c*x^n]^2)/n)/36
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2772, 27, 2010, 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.

\(\displaystyle \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\)

\(\Big \downarrow \) 2772

\(\displaystyle -b n \int \frac {e \left (9 d e x^r+2 e^2 x^{2 r}+18 d^2\right ) x^r+6 d^3 r \log (x)}{6 r x}dx+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+\frac {3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r}+\frac {e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{3 r}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b n \int \frac {e \left (9 d e x^r+2 e^2 x^{2 r}+18 d^2\right ) x^r+6 d^3 r \log (x)}{x}dx}{6 r}+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+\frac {3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r}+\frac {e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{3 r}\)

\(\Big \downarrow \) 2010

\(\displaystyle -\frac {b n \int \left (18 d^2 e x^{r-1}+9 d e^2 x^{2 r-1}+2 e^3 x^{3 r-1}+\frac {6 d^3 r \log (x)}{x}\right )dx}{6 r}+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+\frac {3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r}+\frac {e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{3 r}\)

\(\Big \downarrow \) 2009

\(\displaystyle d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{r}+\frac {3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r}+\frac {e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{3 r}-\frac {b n \left (3 d^3 r \log ^2(x)+\frac {18 d^2 e x^r}{r}+\frac {9 d e^2 x^{2 r}}{2 r}+\frac {2 e^3 x^{3 r}}{3 r}\right )}{6 r}\)

Input:

Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x,x]
 

Output:

-1/6*(b*n*((18*d^2*e*x^r)/r + (9*d*e^2*x^(2*r))/(2*r) + (2*e^3*x^(3*r))/(3 
*r) + 3*d^3*r*Log[x]^2))/r + (3*d^2*e*x^r*(a + b*Log[c*x^n]))/r + (3*d*e^2 
*x^(2*r)*(a + b*Log[c*x^n]))/(2*r) + (e^3*x^(3*r)*(a + b*Log[c*x^n]))/(3*r 
) + d^3*Log[x]*(a + b*Log[c*x^n])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

rule 2772

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])
 

Maple [A] (warning: unable to verify)

Time = 3.52 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.16

method	result	size
parallelrisch	\(\frac {12 x^{3 r} \ln \left (c \,x^{n}\right ) b \,e^{3} n r +12 x^{3 r} a \,e^{3} n r -4 x^{3 r} b \,e^{3} n^{2}+54 x^{2 r} \ln \left (c \,x^{n}\right ) b d \,e^{2} n r +36 \ln \left (x \right ) a \,d^{3} n \,r^{2}+54 x^{2 r} a d \,e^{2} n r -27 x^{2 r} b d \,e^{2} n^{2}+108 x^{r} \ln \left (c \,x^{n}\right ) b \,d^{2} e n r +18 b \,d^{3} \ln \left (c \,x^{n}\right )^{2} r^{2}+108 x^{r} a \,d^{2} e n r -108 x^{r} b \,d^{2} e \,n^{2}}{36 n \,r^{2}}\)	\(177\)
risch	\(\frac {3 a \,d^{2} e \,x^{r}}{r}+\ln \left (x \right ) \ln \left (c \right ) b \,d^{3}-\frac {i \pi b \,e^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{3 r}}{6 r}+\frac {i \ln \left (x \right ) \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\frac {3 \ln \left (c \right ) b \,d^{2} e \,x^{r}}{r}+\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 {b \left (2 e^{3} x^{3 r}+6 d^{3} \ln \left (x \right ) r +9 d \,e^{2} x^{2 r}+18 d^{2} e \,x^{r}\right ) \ln \left (x^{n}\right )}{6 r}+\frac {a \,e^{3} x^{3 r}}{3 r}-\frac {3 b \,d^{2} e n \,x^{r}}{r^{2}}-\frac {3 b d \,e^{2} n \,x^{2 r}}{4 r^{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 ) \operatorname {csgn}\left (i c \right )}{2}+\ln \left (x \right ) a \,d^{3}-\frac {3 i \pi b \,d^{2} e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r}}{2 r}+\frac {\ln \left (c \right ) b \,e^{3} x^{3 r}}{3 r}+\frac {3 a d \,e^{2} x^{2 r}}{2 r}+\frac {i \pi b \,e^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right ) x^{3 r}}{6 r}+\frac {i \pi b \,e^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{3 r}}{6 r}-\frac {3 i \pi b d \,e^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{2 r}}{4 r}+\frac {3 \ln \left (c \right ) b d \,e^{2} x^{2 r}}{2 r}-\frac {i \ln \left (x \right ) \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}-\frac {b \,e^{3} n \,x^{3 r}}{9 r^{2}}-\frac {3 i \pi b d \,e^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) x^{2 r}}{4 r}-\frac {i \pi b \,e^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) x^{3 r}}{6 r}+\frac {3 i \pi b d \,e^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right ) x^{2 r}}{4 r}+\frac {3 i \pi b d \,e^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{2 r}}{4 r}+\frac {3 i \pi b \,d^{2} e \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right ) x^{r}}{2 r}+\frac {3 i \pi b \,d^{2} e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}}{2 r}-\frac {3 i \pi b \,d^{2} e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) x^{r}}{2 r}-\frac {b \,d^{3} n \ln \left (x \right )^{2}}{2}\)	\(693\)

Input:

int((d+e*x^r)^3*(a+b*ln(c*x^n))/x,x,method=_RETURNVERBOSE)
 

Output:

1/36*(12*(x^r)^3*ln(c*x^n)*b*e^3*n*r+12*(x^r)^3*a*e^3*n*r-4*(x^r)^3*b*e^3* 
n^2+54*(x^r)^2*ln(c*x^n)*b*d*e^2*n*r+36*ln(x)*a*d^3*n*r^2+54*(x^r)^2*a*d*e 
^2*n*r-27*(x^r)^2*b*d*e^2*n^2+108*x^r*ln(c*x^n)*b*d^2*e*n*r+18*b*d^3*ln(c* 
x^n)^2*r^2+108*x^r*a*d^2*e*n*r-108*x^r*b*d^2*e*n^2)/n/r^2
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.11 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {18 \, b d^{3} n r^{2} \log \left (x\right )^{2} + 4 \, {\left (3 \, b e^{3} n r \log \left (x\right ) + 3 \, b e^{3} r \log \left (c\right ) - b e^{3} n + 3 \, a e^{3} r\right )} x^{3 \, r} + 27 \, {\left (2 \, b d e^{2} n r \log \left (x\right ) + 2 \, b d e^{2} r \log \left (c\right ) - b d e^{2} n + 2 \, a d e^{2} r\right )} x^{2 \, r} + 108 \, {\left (b d^{2} e n r \log \left (x\right ) + b d^{2} e r \log \left (c\right ) - b d^{2} e n + a d^{2} e r\right )} x^{r} + 36 \, {\left (b d^{3} r^{2} \log \left (c\right ) + a d^{3} r^{2}\right )} \log \left (x\right )}{36 \, r^{2}} \] Input:

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x,x, algorithm="fricas")
 

Output:

1/36*(18*b*d^3*n*r^2*log(x)^2 + 4*(3*b*e^3*n*r*log(x) + 3*b*e^3*r*log(c) - 
 b*e^3*n + 3*a*e^3*r)*x^(3*r) + 27*(2*b*d*e^2*n*r*log(x) + 2*b*d*e^2*r*log 
(c) - b*d*e^2*n + 2*a*d*e^2*r)*x^(2*r) + 108*(b*d^2*e*n*r*log(x) + b*d^2*e 
*r*log(c) - b*d^2*e*n + a*d^2*e*r)*x^r + 36*(b*d^3*r^2*log(c) + a*d^3*r^2) 
*log(x))/r^2
 

Sympy [A] (verification not implemented)

Time = 2.74 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.97 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \left (a + b \log {\left (c \right )}\right ) \left (d + e\right )^{3} \log {\left (x \right )} & \text {for}\: n = 0 \wedge r = 0 \\\left (a + b \log {\left (c \right )}\right ) \left (d^{3} \log {\left (x \right )} + \frac {3 d^{2} e x^{r}}{r} + \frac {3 d e^{2} x^{2 r}}{2 r} + \frac {e^{3} x^{3 r}}{3 r}\right ) & \text {for}\: n = 0 \\\left (d + e\right )^{3} \left (\begin {cases} a \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (- a - b \log {\left (c \right )}\right ) \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}\right ) & \text {for}\: r = 0 \\\frac {a d^{3} \log {\left (c x^{n} \right )}}{n} + \frac {3 a d^{2} e x^{r}}{r} + \frac {3 a d e^{2} x^{2 r}}{2 r} + \frac {a e^{3} x^{3 r}}{3 r} + \frac {b d^{3} \log {\left (c x^{n} \right )}^{2}}{2 n} - \frac {3 b d^{2} e n x^{r}}{r^{2}} + \frac {3 b d^{2} e x^{r} \log {\left (c x^{n} \right )}}{r} - \frac {3 b d e^{2} n x^{2 r}}{4 r^{2}} + \frac {3 b d e^{2} x^{2 r} \log {\left (c x^{n} \right )}}{2 r} - \frac {b e^{3} n x^{3 r}}{9 r^{2}} + \frac {b e^{3} x^{3 r} \log {\left (c x^{n} \right )}}{3 r} & \text {otherwise} \end {cases} \] Input:

integrate((d+e*x**r)**3*(a+b*ln(c*x**n))/x,x)
 

Output:

Piecewise(((a + b*log(c))*(d + e)**3*log(x), Eq(n, 0) & Eq(r, 0)), ((a + b 
*log(c))*(d**3*log(x) + 3*d**2*e*x**r/r + 3*d*e**2*x**(2*r)/(2*r) + e**3*x 
**(3*r)/(3*r)), Eq(n, 0)), ((d + e)**3*Piecewise((a*log(x), Eq(b, 0)), (-( 
-a - b*log(c))*log(x), Eq(n, 0)), ((-a - b*log(c*x**n))**2/(2*b*n), True)) 
, Eq(r, 0)), (a*d**3*log(c*x**n)/n + 3*a*d**2*e*x**r/r + 3*a*d*e**2*x**(2* 
r)/(2*r) + a*e**3*x**(3*r)/(3*r) + b*d**3*log(c*x**n)**2/(2*n) - 3*b*d**2* 
e*n*x**r/r**2 + 3*b*d**2*e*x**r*log(c*x**n)/r - 3*b*d*e**2*n*x**(2*r)/(4*r 
**2) + 3*b*d*e**2*x**(2*r)*log(c*x**n)/(2*r) - b*e**3*n*x**(3*r)/(9*r**2) 
+ b*e**3*x**(3*r)*log(c*x**n)/(3*r), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {b e^{3} x^{3 \, r} \log \left (c x^{n}\right )}{3 \, r} + \frac {3 \, b d e^{2} x^{2 \, r} \log \left (c x^{n}\right )}{2 \, r} + \frac {3 \, b d^{2} e x^{r} \log \left (c x^{n}\right )}{r} + \frac {b d^{3} \log \left (c x^{n}\right )^{2}}{2 \, n} + a d^{3} \log \left (x\right ) - \frac {b e^{3} n x^{3 \, r}}{9 \, r^{2}} + \frac {a e^{3} x^{3 \, r}}{3 \, r} - \frac {3 \, b d e^{2} n x^{2 \, r}}{4 \, r^{2}} + \frac {3 \, a d e^{2} x^{2 \, r}}{2 \, r} - \frac {3 \, b d^{2} e n x^{r}}{r^{2}} + \frac {3 \, a d^{2} e x^{r}}{r} \] Input:

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x,x, algorithm="maxima")
 

Output:

1/3*b*e^3*x^(3*r)*log(c*x^n)/r + 3/2*b*d*e^2*x^(2*r)*log(c*x^n)/r + 3*b*d^ 
2*e*x^r*log(c*x^n)/r + 1/2*b*d^3*log(c*x^n)^2/n + a*d^3*log(x) - 1/9*b*e^3 
*n*x^(3*r)/r^2 + 1/3*a*e^3*x^(3*r)/r - 3/4*b*d*e^2*n*x^(2*r)/r^2 + 3/2*a*d 
*e^2*x^(2*r)/r - 3*b*d^2*e*n*x^r/r^2 + 3*a*d^2*e*x^r/r
 

Giac [F]

\[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \] Input:

integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x,x, algorithm="giac")
 

Output:

integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \] Input:

int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x,x)
 

Output:

int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x, x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.16 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {12 x^{3 r} \mathrm {log}\left (x^{n} c \right ) b \,e^{3} n r +12 x^{3 r} a \,e^{3} n r -4 x^{3 r} b \,e^{3} n^{2}+54 x^{2 r} \mathrm {log}\left (x^{n} c \right ) b d \,e^{2} n r +54 x^{2 r} a d \,e^{2} n r -27 x^{2 r} b d \,e^{2} n^{2}+108 x^{r} \mathrm {log}\left (x^{n} c \right ) b \,d^{2} e n r +108 x^{r} a \,d^{2} e n r -108 x^{r} b \,d^{2} e \,n^{2}+18 \mathrm {log}\left (x^{n} c \right )^{2} b \,d^{3} r^{2}+36 \,\mathrm {log}\left (x \right ) a \,d^{3} n \,r^{2}}{36 n \,r^{2}} \] Input:

int((d+e*x^r)^3*(a+b*log(c*x^n))/x,x)
 

Output:

(12*x**(3*r)*log(x**n*c)*b*e**3*n*r + 12*x**(3*r)*a*e**3*n*r - 4*x**(3*r)* 
b*e**3*n**2 + 54*x**(2*r)*log(x**n*c)*b*d*e**2*n*r + 54*x**(2*r)*a*d*e**2* 
n*r - 27*x**(2*r)*b*d*e**2*n**2 + 108*x**r*log(x**n*c)*b*d**2*e*n*r + 108* 
x**r*a*d**2*e*n*r - 108*x**r*b*d**2*e*n**2 + 18*log(x**n*c)**2*b*d**3*r**2 
 + 36*log(x)*a*d**3*n*r**2)/(36*n*r**2)