Integrand size = 25, antiderivative size = 245 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {6 b^2 d^2 e n^2 x^r}{r^3}+\frac {3 b^2 d e^2 n^2 x^{2 r}}{4 r^3}+\frac {2 b^2 e^3 n^2 x^{3 r}}{27 r^3}-\frac {6 b d^2 e n x^r \left (a+b \log \left (c x^n\right )\right )}{r^2}-\frac {3 b d e^2 n x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r^2}-\frac {2 b e^3 n x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{9 r^2}+\frac {3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}+\frac {3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )^2}{2 r}+\frac {e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )^2}{3 r}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \] Output:
6*b^2*d^2*e*n^2*x^r/r^3+3/4*b^2*d*e^2*n^2*x^(2*r)/r^3+2/27*b^2*e^3*n^2*x^( 3*r)/r^3-6*b*d^2*e*n*x^r*(a+b*ln(c*x^n))/r^2-3/2*b*d*e^2*n*x^(2*r)*(a+b*ln (c*x^n))/r^2-2/9*b*e^3*n*x^(3*r)*(a+b*ln(c*x^n))/r^2+3*d^2*e*x^r*(a+b*ln(c *x^n))^2/r+3/2*d*e^2*x^(2*r)*(a+b*ln(c*x^n))^2/r+1/3*e^3*x^(3*r)*(a+b*ln(c *x^n))^2/r+1/3*d^3*(a+b*ln(c*x^n))^3/b/n
Time = 0.46 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {e n x^r \left (18 a^2 r^2 \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )-6 a b n r \left (108 d^2+27 d e x^r+4 e^2 x^{2 r}\right )+b^2 n^2 \left (648 d^2+81 d e x^r+8 e^2 x^{2 r}\right )\right )+108 a^2 d^3 n r^3 \log (x)-6 b e n r x^r \left (-6 a r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )+b n \left (108 d^2+27 d e x^r+4 e^2 x^{2 r}\right )\right ) \log \left (c x^n\right )+18 b r^2 \left (6 a d^3 r+b e n x^r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )\right ) \log ^2\left (c x^n\right )+36 b^2 d^3 r^3 \log ^3\left (c x^n\right )}{108 n r^3} \] Input:
Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n])^2)/x,x]
Output:
(e*n*x^r*(18*a^2*r^2*(18*d^2 + 9*d*e*x^r + 2*e^2*x^(2*r)) - 6*a*b*n*r*(108 *d^2 + 27*d*e*x^r + 4*e^2*x^(2*r)) + b^2*n^2*(648*d^2 + 81*d*e*x^r + 8*e^2 *x^(2*r))) + 108*a^2*d^3*n*r^3*Log[x] - 6*b*e*n*r*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)))*Log [c*x^n] + 18*b*r^2*(6*a*d^3*r + b*e*n*x^r*(18*d^2 + 9*d*e*x^r + 2*e^2*x^(2 *r)))*Log[c*x^n]^2 + 36*b^2*d^3*r^3*Log[c*x^n]^3)/(108*n*r^3)
Time = 0.58 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2795, 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.
| \(\displaystyle \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \int \left (\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{x}+3 d^2 e x^{r-1} \left (a+b \log \left (c x^n\right )\right )^2+3 d e^2 x^{2 r-1} \left (a+b \log \left (c x^n\right )\right )^2+e^3 x^{3 r-1} \left (a+b \log \left (c x^n\right )\right )^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^3 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {6 b d^2 e n x^r \left (a+b \log \left (c x^n\right )\right )}{r^2}+\frac {3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}-\frac {3 b d e^2 n x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r^2}+\frac {3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )^2}{2 r}-\frac {2 b e^3 n x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{9 r^2}+\frac {e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )^2}{3 r}+\frac {6 b^2 d^2 e n^2 x^r}{r^3}+\frac {3 b^2 d e^2 n^2 x^{2 r}}{4 r^3}+\frac {2 b^2 e^3 n^2 x^{3 r}}{27 r^3}\) |
Input:
Int[((d + e*x^r)^3*(a + b*Log[c*x^n])^2)/x,x]
Output:
(6*b^2*d^2*e*n^2*x^r)/r^3 + (3*b^2*d*e^2*n^2*x^(2*r))/(4*r^3) + (2*b^2*e^3 *n^2*x^(3*r))/(27*r^3) - (6*b*d^2*e*n*x^r*(a + b*Log[c*x^n]))/r^2 - (3*b*d *e^2*n*x^(2*r)*(a + b*Log[c*x^n]))/(2*r^2) - (2*b*e^3*n*x^(3*r)*(a + b*Log [c*x^n]))/(9*r^2) + (3*d^2*e*x^r*(a + b*Log[c*x^n])^2)/r + (3*d*e^2*x^(2*r )*(a + b*Log[c*x^n])^2)/(2*r) + (e^3*x^(3*r)*(a + b*Log[c*x^n])^2)/(3*r) + (d^3*(a + b*Log[c*x^n])^3)/(3*b*n)
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
Time = 10.65 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.71
| method | result | size |
| parallelrisch | \(\frac {36 b^{2} e^{3} \ln \left (c \,x^{n}\right )^{2} x^{3 r} r^{2} n +72 x^{3 r} \ln \left (c \,x^{n}\right ) a b \,e^{3} n \,r^{2}-24 x^{3 r} \ln \left (c \,x^{n}\right ) b^{2} e^{3} n^{2} r +162 b^{2} d \,e^{2} \ln \left (c \,x^{n}\right )^{2} x^{2 r} r^{2} n +36 x^{3 r} a^{2} e^{3} n \,r^{2}-24 x^{3 r} a b \,e^{3} n^{2} r +8 x^{3 r} b^{2} e^{3} n^{3}+324 x^{2 r} \ln \left (c \,x^{n}\right ) a b d \,e^{2} n \,r^{2}-162 x^{2 r} \ln \left (c \,x^{n}\right ) b^{2} d \,e^{2} n^{2} r +324 b^{2} d^{2} e \ln \left (c \,x^{n}\right )^{2} x^{r} r^{2} n +36 d^{3} b^{2} \ln \left (c \,x^{n}\right )^{3} r^{3}+108 \ln \left (x \right ) a^{2} d^{3} n \,r^{3}+162 x^{2 r} a^{2} d \,e^{2} n \,r^{2}-162 x^{2 r} a b d \,e^{2} n^{2} r +81 x^{2 r} b^{2} d \,e^{2} n^{3}+648 x^{r} \ln \left (c \,x^{n}\right ) a b \,d^{2} e n \,r^{2}-648 x^{r} \ln \left (c \,x^{n}\right ) b^{2} d^{2} e \,n^{2} r +108 d^{3} a b \ln \left (c \,x^{n}\right )^{2} r^{3}+324 x^{r} a^{2} d^{2} e n \,r^{2}-648 x^{r} a b \,d^{2} e \,n^{2} r +648 x^{r} b^{2} d^{2} e \,n^{3}}{108 r^{3} n}\) | \(418\) |
| risch | \(\text {Expression too large to display}\) | \(3984\) |
Input:
int((d+e*x^r)^3*(a+b*ln(c*x^n))^2/x,x,method=_RETURNVERBOSE)
Output:
1/108*(36*b^2*e^3*ln(c*x^n)^2*(x^r)^3*r^2*n+72*(x^r)^3*ln(c*x^n)*a*b*e^3*n *r^2-24*(x^r)^3*ln(c*x^n)*b^2*e^3*n^2*r+162*b^2*d*e^2*ln(c*x^n)^2*(x^r)^2* r^2*n+36*(x^r)^3*a^2*e^3*n*r^2-24*(x^r)^3*a*b*e^3*n^2*r+8*(x^r)^3*b^2*e^3* n^3+324*(x^r)^2*ln(c*x^n)*a*b*d*e^2*n*r^2-162*(x^r)^2*ln(c*x^n)*b^2*d*e^2* n^2*r+324*b^2*d^2*e*ln(c*x^n)^2*x^r*r^2*n+36*d^3*b^2*ln(c*x^n)^3*r^3+108*l n(x)*a^2*d^3*n*r^3+162*(x^r)^2*a^2*d*e^2*n*r^2-162*(x^r)^2*a*b*d*e^2*n^2*r +81*(x^r)^2*b^2*d*e^2*n^3+648*x^r*ln(c*x^n)*a*b*d^2*e*n*r^2-648*x^r*ln(c*x ^n)*b^2*d^2*e*n^2*r+108*d^3*a*b*ln(c*x^n)^2*r^3+324*x^r*a^2*d^2*e*n*r^2-64 8*x^r*a*b*d^2*e*n^2*r+648*x^r*b^2*d^2*e*n^3)/r^3/n
Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (231) = 462\).
Time = 0.09 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.13 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {36 \, b^{2} d^{3} n^{2} r^{3} \log \left (x\right )^{3} + 108 \, {\left (b^{2} d^{3} n r^{3} \log \left (c\right ) + a b d^{3} n r^{3}\right )} \log \left (x\right )^{2} + 4 \, {\left (9 \, b^{2} e^{3} n^{2} r^{2} \log \left (x\right )^{2} + 9 \, b^{2} e^{3} r^{2} \log \left (c\right )^{2} + 2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n r + 9 \, a^{2} e^{3} r^{2} - 6 \, {\left (b^{2} e^{3} n r - 3 \, a b e^{3} r^{2}\right )} \log \left (c\right ) + 6 \, {\left (3 \, b^{2} e^{3} n r^{2} \log \left (c\right ) - b^{2} e^{3} n^{2} r + 3 \, a b e^{3} n r^{2}\right )} \log \left (x\right )\right )} x^{3 \, r} + 81 \, {\left (2 \, b^{2} d e^{2} n^{2} r^{2} \log \left (x\right )^{2} + 2 \, b^{2} d e^{2} r^{2} \log \left (c\right )^{2} + b^{2} d e^{2} n^{2} - 2 \, a b d e^{2} n r + 2 \, a^{2} d e^{2} r^{2} - 2 \, {\left (b^{2} d e^{2} n r - 2 \, a b d e^{2} r^{2}\right )} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} d e^{2} n r^{2} \log \left (c\right ) - b^{2} d e^{2} n^{2} r + 2 \, a b d e^{2} n r^{2}\right )} \log \left (x\right )\right )} x^{2 \, r} + 324 \, {\left (b^{2} d^{2} e n^{2} r^{2} \log \left (x\right )^{2} + b^{2} d^{2} e r^{2} \log \left (c\right )^{2} + 2 \, b^{2} d^{2} e n^{2} - 2 \, a b d^{2} e n r + a^{2} d^{2} e r^{2} - 2 \, {\left (b^{2} d^{2} e n r - a b d^{2} e r^{2}\right )} \log \left (c\right ) + 2 \, {\left (b^{2} d^{2} e n r^{2} \log \left (c\right ) - b^{2} d^{2} e n^{2} r + a b d^{2} e n r^{2}\right )} \log \left (x\right )\right )} x^{r} + 108 \, {\left (b^{2} d^{3} r^{3} \log \left (c\right )^{2} + 2 \, a b d^{3} r^{3} \log \left (c\right ) + a^{2} d^{3} r^{3}\right )} \log \left (x\right )}{108 \, r^{3}} \] Input:
integrate((d+e*x^r)^3*(a+b*log(c*x^n))^2/x,x, algorithm="fricas")
Output:
1/108*(36*b^2*d^3*n^2*r^3*log(x)^3 + 108*(b^2*d^3*n*r^3*log(c) + a*b*d^3*n *r^3)*log(x)^2 + 4*(9*b^2*e^3*n^2*r^2*log(x)^2 + 9*b^2*e^3*r^2*log(c)^2 + 2*b^2*e^3*n^2 - 6*a*b*e^3*n*r + 9*a^2*e^3*r^2 - 6*(b^2*e^3*n*r - 3*a*b*e^3 *r^2)*log(c) + 6*(3*b^2*e^3*n*r^2*log(c) - b^2*e^3*n^2*r + 3*a*b*e^3*n*r^2 )*log(x))*x^(3*r) + 81*(2*b^2*d*e^2*n^2*r^2*log(x)^2 + 2*b^2*d*e^2*r^2*log (c)^2 + b^2*d*e^2*n^2 - 2*a*b*d*e^2*n*r + 2*a^2*d*e^2*r^2 - 2*(b^2*d*e^2*n *r - 2*a*b*d*e^2*r^2)*log(c) + 2*(2*b^2*d*e^2*n*r^2*log(c) - b^2*d*e^2*n^2 *r + 2*a*b*d*e^2*n*r^2)*log(x))*x^(2*r) + 324*(b^2*d^2*e*n^2*r^2*log(x)^2 + b^2*d^2*e*r^2*log(c)^2 + 2*b^2*d^2*e*n^2 - 2*a*b*d^2*e*n*r + a^2*d^2*e*r ^2 - 2*(b^2*d^2*e*n*r - a*b*d^2*e*r^2)*log(c) + 2*(b^2*d^2*e*n*r^2*log(c) - b^2*d^2*e*n^2*r + a*b*d^2*e*n*r^2)*log(x))*x^r + 108*(b^2*d^3*r^3*log(c) ^2 + 2*a*b*d^3*r^3*log(c) + a^2*d^3*r^3)*log(x))/r^3
Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (246) = 492\).
Time = 7.78 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.40 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\begin {cases} \left (a + b \log {\left (c \right )}\right )^{2} \left (d + e\right )^{3} \log {\left (x \right )} & \text {for}\: n = 0 \wedge r = 0 \\\left (a + b \log {\left (c \right )}\right )^{2} \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} \frac {a^{2} \log {\left (c x^{n} \right )} + a b \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} \log {\left (c x^{n} \right )}^{3}}{3}}{n} & \text {for}\: n \neq 0 \\\left (a^{2} + 2 a b \log {\left (c \right )} + b^{2} \log {\left (c \right )}^{2}\right ) \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) & \text {for}\: r = 0 \\\frac {a^{2} d^{3} \log {\left (c x^{n} \right )}}{n} + \frac {3 a^{2} d^{2} e x^{r}}{r} + \frac {3 a^{2} d e^{2} x^{2 r}}{2 r} + \frac {a^{2} e^{3} x^{3 r}}{3 r} + \frac {a b d^{3} \log {\left (c x^{n} \right )}^{2}}{n} - \frac {6 a b d^{2} e n x^{r}}{r^{2}} + \frac {6 a b d^{2} e x^{r} \log {\left (c x^{n} \right )}}{r} - \frac {3 a b d e^{2} n x^{2 r}}{2 r^{2}} + \frac {3 a b d e^{2} x^{2 r} \log {\left (c x^{n} \right )}}{r} - \frac {2 a b e^{3} n x^{3 r}}{9 r^{2}} + \frac {2 a b e^{3} x^{3 r} \log {\left (c x^{n} \right )}}{3 r} + \frac {b^{2} d^{3} \log {\left (c x^{n} \right )}^{3}}{3 n} + \frac {6 b^{2} d^{2} e n^{2} x^{r}}{r^{3}} - \frac {6 b^{2} d^{2} e n x^{r} \log {\left (c x^{n} \right )}}{r^{2}} + \frac {3 b^{2} d^{2} e x^{r} \log {\left (c x^{n} \right )}^{2}}{r} + \frac {3 b^{2} d e^{2} n^{2} x^{2 r}}{4 r^{3}} - \frac {3 b^{2} d e^{2} n x^{2 r} \log {\left (c x^{n} \right )}}{2 r^{2}} + \frac {3 b^{2} d e^{2} x^{2 r} \log {\left (c x^{n} \right )}^{2}}{2 r} + \frac {2 b^{2} e^{3} n^{2} x^{3 r}}{27 r^{3}} - \frac {2 b^{2} e^{3} n x^{3 r} \log {\left (c x^{n} \right )}}{9 r^{2}} + \frac {b^{2} e^{3} x^{3 r} \log {\left (c x^{n} \right )}^{2}}{3 r} & \text {otherwise} \end {cases} \] Input:
integrate((d+e*x**r)**3*(a+b*ln(c*x**n))**2/x,x)
Output:
Piecewise(((a + b*log(c))**2*(d + e)**3*log(x), Eq(n, 0) & Eq(r, 0)), ((a + b*log(c))**2*(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**2*log(c*x**n) + a*b*log(c*x**n)**2 + b**2*log(c*x**n)**3/3)/n, Ne(n, 0)), ((a**2 + 2*a*b *log(c) + b**2*log(c)**2)*log(x), True)), Eq(r, 0)), (a**2*d**3*log(c*x**n )/n + 3*a**2*d**2*e*x**r/r + 3*a**2*d*e**2*x**(2*r)/(2*r) + a**2*e**3*x**( 3*r)/(3*r) + a*b*d**3*log(c*x**n)**2/n - 6*a*b*d**2*e*n*x**r/r**2 + 6*a*b* d**2*e*x**r*log(c*x**n)/r - 3*a*b*d*e**2*n*x**(2*r)/(2*r**2) + 3*a*b*d*e** 2*x**(2*r)*log(c*x**n)/r - 2*a*b*e**3*n*x**(3*r)/(9*r**2) + 2*a*b*e**3*x** (3*r)*log(c*x**n)/(3*r) + b**2*d**3*log(c*x**n)**3/(3*n) + 6*b**2*d**2*e*n **2*x**r/r**3 - 6*b**2*d**2*e*n*x**r*log(c*x**n)/r**2 + 3*b**2*d**2*e*x**r *log(c*x**n)**2/r + 3*b**2*d*e**2*n**2*x**(2*r)/(4*r**3) - 3*b**2*d*e**2*n *x**(2*r)*log(c*x**n)/(2*r**2) + 3*b**2*d*e**2*x**(2*r)*log(c*x**n)**2/(2* r) + 2*b**2*e**3*n**2*x**(3*r)/(27*r**3) - 2*b**2*e**3*n*x**(3*r)*log(c*x* *n)/(9*r**2) + b**2*e**3*x**(3*r)*log(c*x**n)**2/(3*r), True))
Time = 0.04 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.60 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {b^{2} e^{3} x^{3 \, r} \log \left (c x^{n}\right )^{2}}{3 \, r} + \frac {3 \, b^{2} d e^{2} x^{2 \, r} \log \left (c x^{n}\right )^{2}}{2 \, r} + \frac {3 \, b^{2} d^{2} e x^{r} \log \left (c x^{n}\right )^{2}}{r} + \frac {b^{2} d^{3} \log \left (c x^{n}\right )^{3}}{3 \, n} - \frac {2}{27} \, b^{2} e^{3} {\left (\frac {3 \, n x^{3 \, r} \log \left (c x^{n}\right )}{r^{2}} - \frac {n^{2} x^{3 \, r}}{r^{3}}\right )} - \frac {3}{4} \, b^{2} d e^{2} {\left (\frac {2 \, n x^{2 \, r} \log \left (c x^{n}\right )}{r^{2}} - \frac {n^{2} x^{2 \, r}}{r^{3}}\right )} - 6 \, b^{2} d^{2} e {\left (\frac {n x^{r} \log \left (c x^{n}\right )}{r^{2}} - \frac {n^{2} x^{r}}{r^{3}}\right )} + \frac {2 \, a b e^{3} x^{3 \, r} \log \left (c x^{n}\right )}{3 \, r} + \frac {3 \, a b d e^{2} x^{2 \, r} \log \left (c x^{n}\right )}{r} + \frac {6 \, a b d^{2} e x^{r} \log \left (c x^{n}\right )}{r} + \frac {a b d^{3} \log \left (c x^{n}\right )^{2}}{n} + a^{2} d^{3} \log \left (x\right ) - \frac {2 \, a b e^{3} n x^{3 \, r}}{9 \, r^{2}} + \frac {a^{2} e^{3} x^{3 \, r}}{3 \, r} - \frac {3 \, a b d e^{2} n x^{2 \, r}}{2 \, r^{2}} + \frac {3 \, a^{2} d e^{2} x^{2 \, r}}{2 \, r} - \frac {6 \, a b d^{2} e n x^{r}}{r^{2}} + \frac {3 \, a^{2} d^{2} e x^{r}}{r} \] Input:
integrate((d+e*x^r)^3*(a+b*log(c*x^n))^2/x,x, algorithm="maxima")
Output:
1/3*b^2*e^3*x^(3*r)*log(c*x^n)^2/r + 3/2*b^2*d*e^2*x^(2*r)*log(c*x^n)^2/r + 3*b^2*d^2*e*x^r*log(c*x^n)^2/r + 1/3*b^2*d^3*log(c*x^n)^3/n - 2/27*b^2*e ^3*(3*n*x^(3*r)*log(c*x^n)/r^2 - n^2*x^(3*r)/r^3) - 3/4*b^2*d*e^2*(2*n*x^( 2*r)*log(c*x^n)/r^2 - n^2*x^(2*r)/r^3) - 6*b^2*d^2*e*(n*x^r*log(c*x^n)/r^2 - n^2*x^r/r^3) + 2/3*a*b*e^3*x^(3*r)*log(c*x^n)/r + 3*a*b*d*e^2*x^(2*r)*l og(c*x^n)/r + 6*a*b*d^2*e*x^r*log(c*x^n)/r + a*b*d^3*log(c*x^n)^2/n + a^2* d^3*log(x) - 2/9*a*b*e^3*n*x^(3*r)/r^2 + 1/3*a^2*e^3*x^(3*r)/r - 3/2*a*b*d *e^2*n*x^(2*r)/r^2 + 3/2*a^2*d*e^2*x^(2*r)/r - 6*a*b*d^2*e*n*x^r/r^2 + 3*a ^2*d^2*e*x^r/r
\[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{x} \,d x } \] Input:
integrate((d+e*x^r)^3*(a+b*log(c*x^n))^2/x,x, algorithm="giac")
Output:
integrate((e*x^r + d)^3*(b*log(c*x^n) + a)^2/x, x)
Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \] Input:
int(((d + e*x^r)^3*(a + b*log(c*x^n))^2)/x,x)
Output:
int(((d + e*x^r)^3*(a + b*log(c*x^n))^2)/x, x)
Time = 0.16 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.70 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {36 x^{3 r} \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e^{3} n \,r^{2}+72 x^{3 r} \mathrm {log}\left (x^{n} c \right ) a b \,e^{3} n \,r^{2}-24 x^{3 r} \mathrm {log}\left (x^{n} c \right ) b^{2} e^{3} n^{2} r +36 x^{3 r} a^{2} e^{3} n \,r^{2}-24 x^{3 r} a b \,e^{3} n^{2} r +8 x^{3 r} b^{2} e^{3} n^{3}+162 x^{2 r} \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d \,e^{2} n \,r^{2}+324 x^{2 r} \mathrm {log}\left (x^{n} c \right ) a b d \,e^{2} n \,r^{2}-162 x^{2 r} \mathrm {log}\left (x^{n} c \right ) b^{2} d \,e^{2} n^{2} r +162 x^{2 r} a^{2} d \,e^{2} n \,r^{2}-162 x^{2 r} a b d \,e^{2} n^{2} r +81 x^{2 r} b^{2} d \,e^{2} n^{3}+324 x^{r} \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d^{2} e n \,r^{2}+648 x^{r} \mathrm {log}\left (x^{n} c \right ) a b \,d^{2} e n \,r^{2}-648 x^{r} \mathrm {log}\left (x^{n} c \right ) b^{2} d^{2} e \,n^{2} r +324 x^{r} a^{2} d^{2} e n \,r^{2}-648 x^{r} a b \,d^{2} e \,n^{2} r +648 x^{r} b^{2} d^{2} e \,n^{3}+36 \mathrm {log}\left (x^{n} c \right )^{3} b^{2} d^{3} r^{3}+108 \mathrm {log}\left (x^{n} c \right )^{2} a b \,d^{3} r^{3}+108 \,\mathrm {log}\left (x \right ) a^{2} d^{3} n \,r^{3}}{108 n \,r^{3}} \] Input:
int((d+e*x^r)^3*(a+b*log(c*x^n))^2/x,x)
Output:
(36*x**(3*r)*log(x**n*c)**2*b**2*e**3*n*r**2 + 72*x**(3*r)*log(x**n*c)*a*b *e**3*n*r**2 - 24*x**(3*r)*log(x**n*c)*b**2*e**3*n**2*r + 36*x**(3*r)*a**2 *e**3*n*r**2 - 24*x**(3*r)*a*b*e**3*n**2*r + 8*x**(3*r)*b**2*e**3*n**3 + 1 62*x**(2*r)*log(x**n*c)**2*b**2*d*e**2*n*r**2 + 324*x**(2*r)*log(x**n*c)*a *b*d*e**2*n*r**2 - 162*x**(2*r)*log(x**n*c)*b**2*d*e**2*n**2*r + 162*x**(2 *r)*a**2*d*e**2*n*r**2 - 162*x**(2*r)*a*b*d*e**2*n**2*r + 81*x**(2*r)*b**2 *d*e**2*n**3 + 324*x**r*log(x**n*c)**2*b**2*d**2*e*n*r**2 + 648*x**r*log(x **n*c)*a*b*d**2*e*n*r**2 - 648*x**r*log(x**n*c)*b**2*d**2*e*n**2*r + 324*x **r*a**2*d**2*e*n*r**2 - 648*x**r*a*b*d**2*e*n**2*r + 648*x**r*b**2*d**2*e *n**3 + 36*log(x**n*c)**3*b**2*d**3*r**3 + 108*log(x**n*c)**2*a*b*d**3*r** 3 + 108*log(x)*a**2*d**3*n*r**3)/(108*n*r**3)