Integrand size = 20, antiderivative size = 169 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-b d^3 n x-\frac {3 b d^2 e n x^{1+r}}{(1+r)^2}-\frac {3 b d e^2 n x^{1+2 r}}{(1+2 r)^2}-\frac {b e^3 n x^{1+3 r}}{(1+3 r)^2}+d^3 x \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^{1+r} \left (a+b \log \left (c x^n\right )\right )}{1+r}+\frac {3 d e^2 x^{1+2 r} \left (a+b \log \left (c x^n\right )\right )}{1+2 r}+\frac {e^3 x^{1+3 r} \left (a+b \log \left (c x^n\right )\right )}{1+3 r} \]
-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*x*(a+b*ln(c*x^n))+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 = 0.16 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.94 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=x \left (a d^3-b d^3 n-\frac {3 b d^2 e n x^r}{(1+r)^2}-\frac {3 b d e^2 n x^{2 r}}{(1+2 r)^2}-\frac {b e^3 n x^{3 r}}{(1+3 r)^2}+b d^3 \log \left (c x^n\right )+\frac {3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{1+r}+\frac {3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{1+2 r}+\frac {e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{1+3 r}\right ) \]
x*(a*d^3 - b*d^3*n - (3*b*d^2*e*n*x^r)/(1 + r)^2 - (3*b*d*e^2*n*x^(2*r))/( 1 + 2*r)^2 - (b*e^3*n*x^(3*r))/(1 + 3*r)^2 + b*d^3*Log[c*x^n] + (3*d^2*e*x ^r*(a + b*Log[c*x^n]))/(1 + r) + (3*d*e^2*x^(2*r)*(a + b*Log[c*x^n]))/(1 + 2*r) + (e^3*x^(3*r)*(a + b*Log[c*x^n]))/(1 + 3*r))
Time = 0.35 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2750, 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.
| \(\displaystyle \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2750 |
| \(\displaystyle -b n \int \left (\frac {3 d^2 e x^r}{r+1}+\frac {3 d e^2 x^{2 r}}{2 r+1}+\frac {e^3 x^{3 r}}{3 r+1}+d^3\right )dx+d^3 x \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^{r+1} \left (a+b \log \left (c x^n\right )\right )}{r+1}+\frac {3 d e^2 x^{2 r+1} \left (a+b \log \left (c x^n\right )\right )}{2 r+1}+\frac {e^3 x^{3 r+1} \left (a+b \log \left (c x^n\right )\right )}{3 r+1}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^3 x \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^{r+1} \left (a+b \log \left (c x^n\right )\right )}{r+1}+\frac {3 d e^2 x^{2 r+1} \left (a+b \log \left (c x^n\right )\right )}{2 r+1}+\frac {e^3 x^{3 r+1} \left (a+b \log \left (c x^n\right )\right )}{3 r+1}-b n \left (d^3 x+\frac {3 d^2 e x^{r+1}}{(r+1)^2}+\frac {3 d e^2 x^{2 r+1}}{(2 r+1)^2}+\frac {e^3 x^{3 r+1}}{(3 r+1)^2}\right )\) |
-(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*x*(a + b*Log[c*x^n]) + (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*L og[c*x^n]))/(1 + 2*r) + (e^3*x^(1 + 3*r)*(a + b*Log[c*x^n]))/(1 + 3*r)
3.4.100.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1107\) vs. \(2(169)=338\).
Time = 3.41 (sec) , antiderivative size = 1108, normalized size of antiderivative = 6.56
| method | result | size |
| parallelrisch | \(\text {Expression too large to display}\) | \(1108\) |
| risch | \(\text {Expression too large to display}\) | \(4023\) |
-(-36*x*a*d^3*r^6-132*x*a*d^3*r^5-193*x*a*d^3*r^4-144*x*a*d^3*r^3-58*x*a*d ^3*r^2-12*x*a*d^3*r-40*x*(x^r)^3*a*e^3*r^4-51*x*(x^r)^3*a*e^3*r^3-31*x*(x^ r)^3*a*e^3*r^2+36*x*b*d^3*n*r^6+132*x*b*d^3*n*r^5+193*x*b*d^3*n*r^4-3*x*d^ 2*e*x^r*b*ln(c*x^n)-3*x*d*e^2*(x^r)^2*b*ln(c*x^n)-x*e^3*(x^r)^3*a-x*b*ln(c *x^n)*d^3-a*d^3*x-3*x*d*e^2*(x^r)^2*a-x*e^3*(x^r)^3*b*ln(c*x^n)-3*x*d^2*e* x^r*a+4*x*(x^r)^3*b*e^3*n*r^4+x*(x^r)^3*b*e^3*n+144*x*b*d^3*n*r^3+58*x*b*d ^3*n*r^2+12*x*b*d^3*n*r-9*x*(x^r)^3*a*e^3*r-33*x*x^r*r*a*d^2*e+30*x*x^r*b* d^2*e*n*r+27*x*(x^r)^2*b*d*e^2*n*r^4+72*x*(x^r)^2*b*d*e^2*n*r^3+108*x*x^r* b*d^2*e*n*r^4+66*x*(x^r)^2*b*d*e^2*n*r^2+180*x*x^r*b*d^2*e*n*r^3+24*x*(x^r )^2*b*d*e^2*n*r+111*x*x^r*b*d^2*e*n*r^2+b*d^3*n*x-114*x*(x^r)^2*ln(c*x^n)* b*d*e^2*r^2-30*x*(x^r)^2*ln(c*x^n)*b*d*e^2*r-12*x*ln(c*x^n)*b*d^3*r-36*x*l n(c*x^n)*b*d^3*r^6-132*x*ln(c*x^n)*b*d^3*r^5-193*x*ln(c*x^n)*b*d^3*r^4-144 *x*ln(c*x^n)*b*d^3*r^3-58*x*ln(c*x^n)*b*d^3*r^2-291*x*x^r*ln(c*x^n)*b*d^2* e*r^3-141*x*x^r*ln(c*x^n)*b*d^2*e*r^2-33*x*x^r*ln(c*x^n)*b*d^2*e*r-54*x*(x ^r)^2*ln(c*x^n)*b*d*e^2*r^5-171*x*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-204*x*(x^r )^2*ln(c*x^n)*b*d*e^2*r^3-12*x*(x^r)^3*a*e^3*r^5-30*x*(x^r)^2*a*d*e^2*r-9* e^3*b*ln(c*x^n)*(x^r)^3*x*r-108*x*x^r*ln(c*x^n)*b*d^2*e*r^5-288*x*x^r*ln(c *x^n)*b*d^2*e*r^4+12*x*(x^r)^3*b*e^3*n*r^3+13*x*(x^r)^3*b*e^3*n*r^2+6*x*(x ^r)^3*b*e^3*n*r+3*x*(x^r)^2*b*d*e^2*n-141*x*x^r*a*d^2*e*r^2-54*x*(x^r)^2*a *d*e^2*r^5-171*x*(x^r)^2*a*d*e^2*r^4-204*x*(x^r)^2*a*d*e^2*r^3-114*x*(x...
Leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (169) = 338\).
Time = 0.31 (sec) , antiderivative size = 983, normalized size of antiderivative = 5.82 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\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 )} x \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 )} x \log \left (x\right ) - {\left (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\right )} x + {\left ({\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 )} x \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 )} x \log \left (x\right ) + {\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\right )} x\right )} x^{3 \, r} + 3 \, {\left ({\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 )} x \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 )} x \log \left (x\right ) + {\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\right )} x\right )} x^{2 \, r} + 3 \, {\left ({\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 )} x \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 )} x \log \left (x\right ) + {\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\right )} x\right )} x^{r}}{36 \, r^{6} + 132 \, r^{5} + 193 \, r^{4} + 144 \, r^{3} + 58 \, r^{2} + 12 \, r + 1} \]
((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)*x*log(c) + (36*b*d^3*n*r^6 + 132*b*d^3*n*r^5 + 1 93*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 )*x*log(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)*x + ((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)*x*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)*x*log(x) + (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)*x)*x^(3*r) + 3*((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)*x*log(c) + (1 8*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)*x*log(x) + (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)*x)*x^ (2*r) + 3*((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)*x*log(c) + (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)*x* log(x) + (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...
Time = 3.89 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.92 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=a d^{3} x + 3 a 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 ) + 3 a 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 ) + a 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 ) - b d^{3} n x + b d^{3} x \log {\left (c x^{n} \right )} - 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 + 1)/(r + 1), Ne(r, -1)), (log(x), True)) + 3*a*d*e**2*Piecewise((x**(2*r + 1)/(2*r + 1), Ne(r, -1/2)), (log( x), True)) + a*e**3*Piecewise((x**(3*r + 1)/(3*r + 1), Ne(r, -1/3)), (log( x), True)) - b*d**3*n*x + b*d**3*x*log(c*x**n) - 3*b*d**2*e*n*Piecewise((P iecewise((x**(r + 1)/(r + 1), Ne(r, -1)), (log(x), True))/(r + 1), (r > -o o) & (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)), (log(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)
Time = 0.19 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.30 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-b d^{3} n x + b d^{3} x \log \left (c x^{n}\right ) + a d^{3} x + \frac {b e^{3} x^{3 \, r + 1} \log \left (c x^{n}\right )}{3 \, r + 1} + \frac {3 \, b d e^{2} x^{2 \, r + 1} \log \left (c x^{n}\right )}{2 \, r + 1} + \frac {3 \, b d^{2} e x^{r + 1} \log \left (c x^{n}\right )}{r + 1} - \frac {b e^{3} n x^{3 \, r + 1}}{{\left (3 \, r + 1\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 1}}{3 \, r + 1} - \frac {3 \, b d e^{2} n x^{2 \, r + 1}}{{\left (2 \, r + 1\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 1}}{2 \, r + 1} - \frac {3 \, b d^{2} e n x^{r + 1}}{{\left (r + 1\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 1}}{r + 1} \]
-b*d^3*n*x + b*d^3*x*log(c*x^n) + a*d^3*x + b*e^3*x^(3*r + 1)*log(c*x^n)/( 3*r + 1) + 3*b*d*e^2*x^(2*r + 1)*log(c*x^n)/(2*r + 1) + 3*b*d^2*e*x^(r + 1 )*log(c*x^n)/(r + 1) - b*e^3*n*x^(3*r + 1)/(3*r + 1)^2 + a*e^3*x^(3*r + 1) /(3*r + 1) - 3*b*d*e^2*n*x^(2*r + 1)/(2*r + 1)^2 + 3*a*d*e^2*x^(2*r + 1)/( 2*r + 1) - 3*b*d^2*e*n*x^(r + 1)/(r + 1)^2 + 3*a*d^2*e*x^(r + 1)/(r + 1)
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (169) = 338\).
Time = 0.31 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.24 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3 \, b e^{3} n r x x^{3 \, r} \log \left (x\right )}{9 \, r^{2} + 6 \, r + 1} + \frac {6 \, b d e^{2} n r x x^{2 \, r} \log \left (x\right )}{4 \, r^{2} + 4 \, r + 1} + \frac {3 \, b d^{2} e n r x x^{r} \log \left (x\right )}{r^{2} + 2 \, r + 1} + b d^{3} n x \log \left (x\right ) + \frac {b e^{3} n x x^{3 \, r} \log \left (x\right )}{9 \, r^{2} + 6 \, r + 1} + \frac {3 \, b d e^{2} n x x^{2 \, r} \log \left (x\right )}{4 \, r^{2} + 4 \, r + 1} + \frac {3 \, b d^{2} e n x x^{r} \log \left (x\right )}{r^{2} + 2 \, r + 1} - b d^{3} n x - \frac {b e^{3} n x x^{3 \, r}}{9 \, r^{2} + 6 \, r + 1} - \frac {3 \, b d e^{2} n x x^{2 \, r}}{4 \, r^{2} + 4 \, r + 1} - \frac {3 \, b d^{2} e n x x^{r}}{r^{2} + 2 \, r + 1} + b d^{3} x \log \left (c\right ) + \frac {b e^{3} x x^{3 \, r} \log \left (c\right )}{3 \, r + 1} + \frac {3 \, b d e^{2} x x^{2 \, r} \log \left (c\right )}{2 \, r + 1} + \frac {3 \, b d^{2} e x x^{r} \log \left (c\right )}{r + 1} + a d^{3} x + \frac {a e^{3} x x^{3 \, r}}{3 \, r + 1} + \frac {3 \, a d e^{2} x x^{2 \, r}}{2 \, r + 1} + \frac {3 \, a d^{2} e x x^{r}}{r + 1} \]
3*b*e^3*n*r*x*x^(3*r)*log(x)/(9*r^2 + 6*r + 1) + 6*b*d*e^2*n*r*x*x^(2*r)*l og(x)/(4*r^2 + 4*r + 1) + 3*b*d^2*e*n*r*x*x^r*log(x)/(r^2 + 2*r + 1) + b*d ^3*n*x*log(x) + b*e^3*n*x*x^(3*r)*log(x)/(9*r^2 + 6*r + 1) + 3*b*d*e^2*n*x *x^(2*r)*log(x)/(4*r^2 + 4*r + 1) + 3*b*d^2*e*n*x*x^r*log(x)/(r^2 + 2*r + 1) - b*d^3*n*x - b*e^3*n*x*x^(3*r)/(9*r^2 + 6*r + 1) - 3*b*d*e^2*n*x*x^(2* r)/(4*r^2 + 4*r + 1) - 3*b*d^2*e*n*x*x^r/(r^2 + 2*r + 1) + b*d^3*x*log(c) + b*e^3*x*x^(3*r)*log(c)/(3*r + 1) + 3*b*d*e^2*x*x^(2*r)*log(c)/(2*r + 1) + 3*b*d^2*e*x*x^r*log(c)/(r + 1) + a*d^3*x + a*e^3*x*x^(3*r)/(3*r + 1) + 3 *a*d*e^2*x*x^(2*r)/(2*r + 1) + 3*a*d^2*e*x*x^r/(r + 1)
Timed out. \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]