Integrand size = 23, antiderivative size = 179 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {b d^3 n}{x}-\frac {3 b d^2 e n x^{-1+r}}{(1-r)^2}-\frac {3 b d e^2 n x^{-1+2 r}}{(1-2 r)^2}-\frac {b e^3 n x^{-1+3 r}}{(1-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 d^2 e x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac {3 d e^2 x^{-1+2 r} \left (a+b \log \left (c x^n\right )\right )}{1-2 r}-\frac {e^3 x^{-1+3 r} \left (a+b \log \left (c x^n\right )\right )}{1-3 r} \]
-b*d^3*n/x-3*b*d^2*e*n*x^(-1+r)/(1-r)^2-3*b*d*e^2*n*x^(-1+2*r)/(1-2*r)^2-b *e^3*n*x^(-1+3*r)/(1-3*r)^2-d^3*(a+b*ln(c*x^n))/x-3*d^2*e*x^(-1+r)*(a+b*ln (c*x^n))/(1-r)-3*d*e^2*x^(-1+2*r)*(a+b*ln(c*x^n))/(1-2*r)-e^3*x^(-1+3*r)*( a+b*ln(c*x^n))/(1-3*r)
Time = 0.25 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {b n \left (-d^3-\frac {3 d^2 e x^r}{(-1+r)^2}-\frac {3 d e^2 x^{2 r}}{(1-2 r)^2}-\frac {e^3 x^{3 r}}{(1-3 r)^2}\right )+a \left (-d^3+\frac {3 d^2 e x^r}{-1+r}+\frac {3 d e^2 x^{2 r}}{-1+2 r}+\frac {e^3 x^{3 r}}{-1+3 r}\right )+b \left (-d^3+\frac {3 d^2 e x^r}{-1+r}+\frac {3 d e^2 x^{2 r}}{-1+2 r}+\frac {e^3 x^{3 r}}{-1+3 r}\right ) \log \left (c x^n\right )}{x} \]
(b*n*(-d^3 - (3*d^2*e*x^r)/(-1 + r)^2 - (3*d*e^2*x^(2*r))/(1 - 2*r)^2 - (e ^3*x^(3*r))/(1 - 3*r)^2) + a*(-d^3 + (3*d^2*e*x^r)/(-1 + r) + (3*d*e^2*x^( 2*r))/(-1 + 2*r) + (e^3*x^(3*r))/(-1 + 3*r)) + b*(-d^3 + (3*d^2*e*x^r)/(-1 + r) + (3*d*e^2*x^(2*r))/(-1 + 2*r) + (e^3*x^(3*r))/(-1 + 3*r))*Log[c*x^n ])/x
Time = 0.61 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2772, 25, 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.
| \(\displaystyle \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle -b n \int -\frac {\frac {3 d^2 e x^r}{1-r}+\frac {3 d e^2 x^{2 r}}{1-2 r}+\frac {e^3 x^{3 r}}{1-3 r}+d^3}{x^2}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 d^2 e x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac {3 d e^2 x^{2 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-2 r}-\frac {e^3 x^{3 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-3 r}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b n \int \frac {\frac {3 d^2 e x^r}{1-r}+\frac {3 d e^2 x^{2 r}}{1-2 r}+\frac {e^3 x^{3 r}}{1-3 r}+d^3}{x^2}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 d^2 e x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac {3 d e^2 x^{2 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-2 r}-\frac {e^3 x^{3 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-3 r}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle b n \int \left (-\frac {3 d^2 e x^{r-2}}{r-1}+\frac {3 d e^2 x^{2 (r-1)}}{1-2 r}-\frac {e^3 x^{3 r-2}}{3 r-1}+\frac {d^3}{x^2}\right )dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 d^2 e x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac {3 d e^2 x^{2 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-2 r}-\frac {e^3 x^{3 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-3 r}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 d^2 e x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac {3 d e^2 x^{2 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-2 r}-\frac {e^3 x^{3 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-3 r}+b n \left (-\frac {d^3}{x}-\frac {3 d^2 e x^{r-1}}{(1-r)^2}-\frac {3 d e^2 x^{2 r-1}}{(1-2 r)^2}-\frac {e^3 x^{3 r-1}}{(1-3 r)^2}\right )\) |
b*n*(-(d^3/x) - (3*d^2*e*x^(-1 + r))/(1 - r)^2 - (3*d*e^2*x^(-1 + 2*r))/(1 - 2*r)^2 - (e^3*x^(-1 + 3*r))/(1 - 3*r)^2) - (d^3*(a + b*Log[c*x^n]))/x - (3*d^2*e*x^(-1 + r)*(a + b*Log[c*x^n]))/(1 - r) - (3*d*e^2*x^(-1 + 2*r)*( a + b*Log[c*x^n]))/(1 - 2*r) - (e^3*x^(-1 + 3*r)*(a + b*Log[c*x^n]))/(1 - 3*r)
3.5.1.3.1 Defintions of rubi rules used
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]]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(1034\) vs. \(2(179)=358\).
Time = 3.77 (sec) , antiderivative size = 1035, normalized size of antiderivative = 5.78
| method | result | size |
| parallelrisch | \(\text {Expression too large to display}\) | \(1035\) |
| risch | \(\text {Expression too large to display}\) | \(4031\) |
-(b*ln(c*x^n)*d^3+3*b*d*e^2*ln(c*x^n)*(x^r)^2+e^3*(x^r)^3*a-144*b*d^3*n*r^ 3+58*b*d^3*n*r^2-12*b*d^3*n*r+114*a*d*e^2*r^2*(x^r)^2-204*a*d*e^2*r^3*(x^r )^2+3*d*e^2*(x^r)^2*a+3*d^2*e*x^r*a+a*d^3-51*a*e^3*r^3*(x^r)^3+31*a*e^3*r^ 2*(x^r)^3-9*a*e^3*r*(x^r)^3-12*a*e^3*r^5*(x^r)^3+40*a*e^3*r^4*(x^r)^3-12*( x^r)^3*ln(c*x^n)*b*e^3*r^5+40*(x^r)^3*ln(c*x^n)*b*e^3*r^4-51*(x^r)^3*ln(c* x^n)*b*e^3*r^3+31*(x^r)^3*ln(c*x^n)*b*e^3*r^2-9*(x^r)^3*ln(c*x^n)*b*e^3*r+ 3*b*d^2*e*ln(c*x^n)*x^r+36*b*d^3*n*r^6-132*b*d^3*n*r^5+193*b*d^3*n*r^4-108 *x^r*ln(c*x^n)*b*d^2*e*r^5+288*x^r*ln(c*x^n)*b*d^2*e*r^4-291*x^r*ln(c*x^n) *b*d^2*e*r^3+141*x^r*ln(c*x^n)*b*d^2*e*r^2-33*x^r*ln(c*x^n)*b*d^2*e*r-54*( x^r)^2*ln(c*x^n)*b*d*e^2*r^5+171*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-204*(x^r)^2 *ln(c*x^n)*b*d*e^2*r^3+114*(x^r)^2*ln(c*x^n)*b*d*e^2*r^2-30*(x^r)^2*ln(c*x ^n)*b*d*e^2*r+b*d^3*n+36*ln(c*x^n)*b*d^3*r^6-132*ln(c*x^n)*b*d^3*r^5+193*l n(c*x^n)*b*d^3*r^4-144*ln(c*x^n)*b*d^3*r^3+58*ln(c*x^n)*b*d^3*r^2-12*ln(c* x^n)*b*d^3*r+e^3*b*ln(c*x^n)*(x^r)^3-144*a*d^3*r^3+58*a*d^3*r^2-12*a*d^3*r +36*a*d^3*r^6-132*a*d^3*r^5+193*a*d^3*r^4-291*a*d^2*e*r^3*x^r+3*b*d*e^2*n* (x^r)^2+3*b*d^2*e*n*x^r+b*e^3*n*(x^r)^3-12*b*e^3*n*r^3*(x^r)^3+13*b*e^3*n* r^2*(x^r)^3-6*b*e^3*n*r*(x^r)^3+141*a*d^2*e*r^2*x^r+288*a*d^2*e*r^4*x^r-33 *a*d^2*e*r*x^r+4*b*e^3*n*r^4*(x^r)^3-30*a*d*e^2*r*(x^r)^2-54*a*d*e^2*r^5*( x^r)^2+171*a*d*e^2*r^4*(x^r)^2-108*a*d^2*e*r^5*x^r+66*b*d*e^2*n*r^2*(x^r)^ 2+111*b*d^2*e*n*r^2*x^r-24*b*d*e^2*n*r*(x^r)^2-30*b*d^2*e*n*r*x^r-180*b...
Leaf count of result is larger than twice the leaf count of optimal. 967 vs. \(2 (174) = 348\).
Time = 0.31 (sec) , antiderivative size = 967, normalized size of antiderivative = 5.40 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {36 \, {\left (b d^{3} n + a d^{3}\right )} r^{6} - 132 \, {\left (b d^{3} n + a d^{3}\right )} r^{5} + b d^{3} n + 193 \, {\left (b d^{3} n + a d^{3}\right )} r^{4} + a d^{3} - 144 \, {\left (b d^{3} n + a d^{3}\right )} r^{3} + 58 \, {\left (b d^{3} n + a d^{3}\right )} r^{2} - 12 \, {\left (b d^{3} n + a d^{3}\right )} r - {\left (12 \, a e^{3} r^{5} - b e^{3} n - 4 \, {\left (b e^{3} n + 10 \, a e^{3}\right )} r^{4} - a e^{3} + 3 \, {\left (4 \, b e^{3} n + 17 \, a e^{3}\right )} r^{3} - {\left (13 \, b e^{3} n + 31 \, a e^{3}\right )} r^{2} + 3 \, {\left (2 \, b e^{3} n + 3 \, a e^{3}\right )} r + {\left (12 \, b e^{3} r^{5} - 40 \, b e^{3} r^{4} + 51 \, b e^{3} r^{3} - 31 \, b e^{3} r^{2} + 9 \, b e^{3} r - b e^{3}\right )} \log \left (c\right ) + {\left (12 \, b e^{3} n r^{5} - 40 \, b e^{3} n r^{4} + 51 \, b e^{3} n r^{3} - 31 \, b e^{3} n r^{2} + 9 \, b e^{3} n r - b e^{3} n\right )} \log \left (x\right )\right )} x^{3 \, r} - 3 \, {\left (18 \, a d e^{2} r^{5} - b d e^{2} n - 3 \, {\left (3 \, b d e^{2} n + 19 \, a d e^{2}\right )} r^{4} - a d e^{2} + 4 \, {\left (6 \, b d e^{2} n + 17 \, a d e^{2}\right )} r^{3} - 2 \, {\left (11 \, b d e^{2} n + 19 \, a d e^{2}\right )} r^{2} + 2 \, {\left (4 \, b d e^{2} n + 5 \, a d e^{2}\right )} r + {\left (18 \, b d e^{2} r^{5} - 57 \, b d e^{2} r^{4} + 68 \, b d e^{2} r^{3} - 38 \, b d e^{2} r^{2} + 10 \, b d e^{2} r - b d e^{2}\right )} \log \left (c\right ) + {\left (18 \, b d e^{2} n r^{5} - 57 \, b d e^{2} n r^{4} + 68 \, b d e^{2} n r^{3} - 38 \, b d e^{2} n r^{2} + 10 \, b d e^{2} n r - b d e^{2} n\right )} \log \left (x\right )\right )} x^{2 \, r} - 3 \, {\left (36 \, a d^{2} e r^{5} - b d^{2} e n - 12 \, {\left (3 \, b d^{2} e n + 8 \, a d^{2} e\right )} r^{4} - a d^{2} e + {\left (60 \, b d^{2} e n + 97 \, a d^{2} e\right )} r^{3} - {\left (37 \, b d^{2} e n + 47 \, a d^{2} e\right )} r^{2} + {\left (10 \, b d^{2} e n + 11 \, a d^{2} e\right )} r + {\left (36 \, b d^{2} e r^{5} - 96 \, b d^{2} e r^{4} + 97 \, b d^{2} e r^{3} - 47 \, b d^{2} e r^{2} + 11 \, b d^{2} e r - b d^{2} e\right )} \log \left (c\right ) + {\left (36 \, b d^{2} e n r^{5} - 96 \, b d^{2} e n r^{4} + 97 \, b d^{2} e n r^{3} - 47 \, b d^{2} e n r^{2} + 11 \, b d^{2} e n r - b d^{2} e n\right )} \log \left (x\right )\right )} x^{r} + {\left (36 \, b d^{3} r^{6} - 132 \, b d^{3} r^{5} + 193 \, b d^{3} r^{4} - 144 \, b d^{3} r^{3} + 58 \, b d^{3} r^{2} - 12 \, b d^{3} r + b d^{3}\right )} \log \left (c\right ) + {\left (36 \, b d^{3} n r^{6} - 132 \, b d^{3} n r^{5} + 193 \, b d^{3} n r^{4} - 144 \, b d^{3} n r^{3} + 58 \, b d^{3} n r^{2} - 12 \, b d^{3} n r + b d^{3} n\right )} \log \left (x\right )}{{\left (36 \, r^{6} - 132 \, r^{5} + 193 \, r^{4} - 144 \, r^{3} + 58 \, r^{2} - 12 \, r + 1\right )} x} \]
-(36*(b*d^3*n + a*d^3)*r^6 - 132*(b*d^3*n + a*d^3)*r^5 + b*d^3*n + 193*(b* d^3*n + a*d^3)*r^4 + a*d^3 - 144*(b*d^3*n + a*d^3)*r^3 + 58*(b*d^3*n + a*d ^3)*r^2 - 12*(b*d^3*n + a*d^3)*r - (12*a*e^3*r^5 - b*e^3*n - 4*(b*e^3*n + 10*a*e^3)*r^4 - a*e^3 + 3*(4*b*e^3*n + 17*a*e^3)*r^3 - (13*b*e^3*n + 31*a* e^3)*r^2 + 3*(2*b*e^3*n + 3*a*e^3)*r + (12*b*e^3*r^5 - 40*b*e^3*r^4 + 51*b *e^3*r^3 - 31*b*e^3*r^2 + 9*b*e^3*r - b*e^3)*log(c) + (12*b*e^3*n*r^5 - 40 *b*e^3*n*r^4 + 51*b*e^3*n*r^3 - 31*b*e^3*n*r^2 + 9*b*e^3*n*r - b*e^3*n)*lo g(x))*x^(3*r) - 3*(18*a*d*e^2*r^5 - b*d*e^2*n - 3*(3*b*d*e^2*n + 19*a*d*e^ 2)*r^4 - a*d*e^2 + 4*(6*b*d*e^2*n + 17*a*d*e^2)*r^3 - 2*(11*b*d*e^2*n + 19 *a*d*e^2)*r^2 + 2*(4*b*d*e^2*n + 5*a*d*e^2)*r + (18*b*d*e^2*r^5 - 57*b*d*e ^2*r^4 + 68*b*d*e^2*r^3 - 38*b*d*e^2*r^2 + 10*b*d*e^2*r - b*d*e^2)*log(c) + (18*b*d*e^2*n*r^5 - 57*b*d*e^2*n*r^4 + 68*b*d*e^2*n*r^3 - 38*b*d*e^2*n*r ^2 + 10*b*d*e^2*n*r - b*d*e^2*n)*log(x))*x^(2*r) - 3*(36*a*d^2*e*r^5 - b*d ^2*e*n - 12*(3*b*d^2*e*n + 8*a*d^2*e)*r^4 - a*d^2*e + (60*b*d^2*e*n + 97*a *d^2*e)*r^3 - (37*b*d^2*e*n + 47*a*d^2*e)*r^2 + (10*b*d^2*e*n + 11*a*d^2*e )*r + (36*b*d^2*e*r^5 - 96*b*d^2*e*r^4 + 97*b*d^2*e*r^3 - 47*b*d^2*e*r^2 + 11*b*d^2*e*r - b*d^2*e)*log(c) + (36*b*d^2*e*n*r^5 - 96*b*d^2*e*n*r^4 + 9 7*b*d^2*e*n*r^3 - 47*b*d^2*e*n*r^2 + 11*b*d^2*e*n*r - b*d^2*e*n)*log(x))*x ^r + (36*b*d^3*r^6 - 132*b*d^3*r^5 + 193*b*d^3*r^4 - 144*b*d^3*r^3 + 58*b* d^3*r^2 - 12*b*d^3*r + b*d^3)*log(c) + (36*b*d^3*n*r^6 - 132*b*d^3*n*r^...
Time = 10.76 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.80 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=- \frac {a d^{3}}{x} + 3 a d^{2} e \left (\begin {cases} \frac {x^{r}}{r x - x} & \text {for}\: r \neq 1 \\\frac {x^{r} \log {\left (x \right )}}{x} & \text {otherwise} \end {cases}\right ) + 3 a d e^{2} \left (\begin {cases} \frac {x^{2 r}}{2 r x - x} & \text {for}\: r \neq \frac {1}{2} \\\frac {x^{2 r} \log {\left (x \right )}}{x} & \text {otherwise} \end {cases}\right ) + a e^{3} \left (\begin {cases} \frac {x^{3 r}}{3 r x - x} & \text {for}\: r \neq \frac {1}{3} \\\frac {x^{3 r} \log {\left (x \right )}}{x} & \text {otherwise} \end {cases}\right ) - \frac {b d^{3} n}{x} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{x} - 3 b d^{2} e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r - 1}}{r - 1} & \text {for}\: r \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{r - 1} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 1 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d^{2} e \left (\begin {cases} \frac {x^{r - 1}}{r - 1} & \text {for}\: r \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - 3 b d e^{2} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{2 r - 1}}{2 r - 1} & \text {for}\: r \neq \frac {1}{2} \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{2 r - 1} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq \frac {1}{2} \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d e^{2} \left (\begin {cases} \frac {x^{2 r - 1}}{2 r - 1} & \text {for}\: r \neq \frac {1}{2} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - b e^{3} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{3 r - 1}}{3 r - 1} & \text {for}\: r \neq \frac {1}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{3 r - 1} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq \frac {1}{3} \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + b e^{3} \left (\begin {cases} \frac {x^{3 r - 1}}{3 r - 1} & \text {for}\: r \neq \frac {1}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
-a*d**3/x + 3*a*d**2*e*Piecewise((x**r/(r*x - x), Ne(r, 1)), (x**r*log(x)/ x, True)) + 3*a*d*e**2*Piecewise((x**(2*r)/(2*r*x - x), Ne(r, 1/2)), (x**( 2*r)*log(x)/x, True)) + a*e**3*Piecewise((x**(3*r)/(3*r*x - x), Ne(r, 1/3) ), (x**(3*r)*log(x)/x, True)) - b*d**3*n/x - b*d**3*log(c*x**n)/x - 3*b*d* *2*e*n*Piecewise((Piecewise((x**(r - 1)/(r - 1), Ne(r, 1)), (log(x), True) )/(r - 1), (r > -oo) & (r < oo) & Ne(r, 1)), (log(x)**2/2, True)) + 3*b*d* *2*e*Piecewise((x**(r - 1)/(r - 1), Ne(r, 1)), (log(x), True))*log(c*x**n) - 3*b*d*e**2*n*Piecewise((Piecewise((x**(2*r - 1)/(2*r - 1), Ne(r, 1/2)), (log(x), True))/(2*r - 1), (r > -oo) & (r < oo) & Ne(r, 1/2)), (log(x)**2 /2, True)) + 3*b*d*e**2*Piecewise((x**(2*r - 1)/(2*r - 1), Ne(r, 1/2)), (l og(x), True))*log(c*x**n) - b*e**3*n*Piecewise((Piecewise((x**(3*r - 1)/(3 *r - 1), Ne(r, 1/3)), (log(x), True))/(3*r - 1), (r > -oo) & (r < oo) & Ne (r, 1/3)), (log(x)**2/2, True)) + b*e**3*Piecewise((x**(3*r - 1)/(3*r - 1) , Ne(r, 1/3)), (log(x), True))*log(c*x**n)
Exception generated. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(r-2>0)', see `assume?` for more details)Is
\[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \]