Integrand size = 23, antiderivative size = 168 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-\frac {2 b^2 d^2 n^2}{27 x^3}-\frac {b^2 d e n^2}{2 x^2}-\frac {2 b^2 e^2 n^2}{x}-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {b d e n \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x} \] Output:
-2/27*b^2*d^2*n^2/x^3-1/2*b^2*d*e*n^2/x^2-2*b^2*e^2*n^2/x-2/9*b*d^2*n*(a+b *ln(c*x^n))/x^3-b*d*e*n*(a+b*ln(c*x^n))/x^2-2*b*e^2*n*(a+b*ln(c*x^n))/x-1/ 3*d^2*(a+b*ln(c*x^n))^2/x^3-d*e*(a+b*ln(c*x^n))^2/x^2-e^2*(a+b*ln(c*x^n))^ 2/x
Time = 0.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-\frac {18 d^2 \left (a+b \log \left (c x^n\right )\right )^2+54 d e x \left (a+b \log \left (c x^n\right )\right )^2+54 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+108 b e^2 n x^2 \left (a+b n+b \log \left (c x^n\right )\right )+27 b d e n x \left (2 a+b n+2 b \log \left (c x^n\right )\right )+4 b d^2 n \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{54 x^3} \] Input:
Integrate[((d + e*x)^2*(a + b*Log[c*x^n])^2)/x^4,x]
Output:
-1/54*(18*d^2*(a + b*Log[c*x^n])^2 + 54*d*e*x*(a + b*Log[c*x^n])^2 + 54*e^ 2*x^2*(a + b*Log[c*x^n])^2 + 108*b*e^2*n*x^2*(a + b*n + b*Log[c*x^n]) + 27 *b*d*e*n*x*(2*a + b*n + 2*b*Log[c*x^n]) + 4*b*d^2*n*(3*a + b*n + 3*b*Log[c *x^n]))/x^3
Time = 0.43 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4}+\frac {2 d e \left (a+b \log \left (c x^n\right )\right )^2}{x^3}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {b d e n \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 b^2 d^2 n^2}{27 x^3}-\frac {b^2 d e n^2}{2 x^2}-\frac {2 b^2 e^2 n^2}{x}\) |
Input:
Int[((d + e*x)^2*(a + b*Log[c*x^n])^2)/x^4,x]
Output:
(-2*b^2*d^2*n^2)/(27*x^3) - (b^2*d*e*n^2)/(2*x^2) - (2*b^2*e^2*n^2)/x - (2 *b*d^2*n*(a + b*Log[c*x^n]))/(9*x^3) - (b*d*e*n*(a + b*Log[c*x^n]))/x^2 - (2*b*e^2*n*(a + b*Log[c*x^n]))/x - (d^2*(a + b*Log[c*x^n])^2)/(3*x^3) - (d *e*(a + b*Log[c*x^n])^2)/x^2 - (e^2*(a + b*Log[c*x^n])^2)/x
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 = 0.61 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.42
| method | result | size |
| parallelrisch | \(-\frac {54 b^{2} \ln \left (c \,x^{n}\right )^{2} e^{2} x^{2}+108 x^{2} \ln \left (c \,x^{n}\right ) b^{2} e^{2} n +108 b^{2} e^{2} n^{2} x^{2}+108 a b \ln \left (c \,x^{n}\right ) e^{2} x^{2}+108 n b \,x^{2} a \,e^{2}+54 b^{2} \ln \left (c \,x^{n}\right )^{2} e x d +54 b^{2} d e n x \ln \left (c \,x^{n}\right )+27 b^{2} d e \,n^{2} x +54 a^{2} e^{2} x^{2}+108 a b \ln \left (c \,x^{n}\right ) e x d +54 a b d e n x +18 b^{2} \ln \left (c \,x^{n}\right )^{2} d^{2}+12 \ln \left (c \,x^{n}\right ) n \,b^{2} d^{2}+4 b^{2} d^{2} n^{2}+54 a^{2} e x d +36 a b \ln \left (c \,x^{n}\right ) d^{2}+12 b \,d^{2} n a +18 a^{2} d^{2}}{54 x^{3}}\) | \(238\) |
| risch | \(\text {Expression too large to display}\) | \(2473\) |
Input:
int((e*x+d)^2*(a+b*ln(c*x^n))^2/x^4,x,method=_RETURNVERBOSE)
Output:
-1/54/x^3*(54*b^2*ln(c*x^n)^2*e^2*x^2+108*x^2*ln(c*x^n)*b^2*e^2*n+108*b^2* e^2*n^2*x^2+108*a*b*ln(c*x^n)*e^2*x^2+108*n*b*x^2*a*e^2+54*b^2*ln(c*x^n)^2 *e*x*d+54*b^2*d*e*n*x*ln(c*x^n)+27*b^2*d*e*n^2*x+54*a^2*e^2*x^2+108*a*b*ln (c*x^n)*e*x*d+54*a*b*d*e*n*x+18*b^2*ln(c*x^n)^2*d^2+12*ln(c*x^n)*n*b^2*d^2 +4*b^2*d^2*n^2+54*a^2*e*x*d+36*a*b*ln(c*x^n)*d^2+12*b*d^2*n*a+18*a^2*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (160) = 320\).
Time = 0.08 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.94 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-\frac {4 \, b^{2} d^{2} n^{2} + 12 \, a b d^{2} n + 18 \, a^{2} d^{2} + 54 \, {\left (2 \, b^{2} e^{2} n^{2} + 2 \, a b e^{2} n + a^{2} e^{2}\right )} x^{2} + 18 \, {\left (3 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (c\right )^{2} + 18 \, {\left (3 \, b^{2} e^{2} n^{2} x^{2} + 3 \, b^{2} d e n^{2} x + b^{2} d^{2} n^{2}\right )} \log \left (x\right )^{2} + 27 \, {\left (b^{2} d e n^{2} + 2 \, a b d e n + 2 \, a^{2} d e\right )} x + 6 \, {\left (2 \, b^{2} d^{2} n + 6 \, a b d^{2} + 18 \, {\left (b^{2} e^{2} n + a b e^{2}\right )} x^{2} + 9 \, {\left (b^{2} d e n + 2 \, a b d e\right )} x\right )} \log \left (c\right ) + 6 \, {\left (2 \, b^{2} d^{2} n^{2} + 6 \, a b d^{2} n + 18 \, {\left (b^{2} e^{2} n^{2} + a b e^{2} n\right )} x^{2} + 9 \, {\left (b^{2} d e n^{2} + 2 \, a b d e n\right )} x + 6 \, {\left (3 \, b^{2} e^{2} n x^{2} + 3 \, b^{2} d e n x + b^{2} d^{2} n\right )} \log \left (c\right )\right )} \log \left (x\right )}{54 \, x^{3}} \] Input:
integrate((e*x+d)^2*(a+b*log(c*x^n))^2/x^4,x, algorithm="fricas")
Output:
-1/54*(4*b^2*d^2*n^2 + 12*a*b*d^2*n + 18*a^2*d^2 + 54*(2*b^2*e^2*n^2 + 2*a *b*e^2*n + a^2*e^2)*x^2 + 18*(3*b^2*e^2*x^2 + 3*b^2*d*e*x + b^2*d^2)*log(c )^2 + 18*(3*b^2*e^2*n^2*x^2 + 3*b^2*d*e*n^2*x + b^2*d^2*n^2)*log(x)^2 + 27 *(b^2*d*e*n^2 + 2*a*b*d*e*n + 2*a^2*d*e)*x + 6*(2*b^2*d^2*n + 6*a*b*d^2 + 18*(b^2*e^2*n + a*b*e^2)*x^2 + 9*(b^2*d*e*n + 2*a*b*d*e)*x)*log(c) + 6*(2* b^2*d^2*n^2 + 6*a*b*d^2*n + 18*(b^2*e^2*n^2 + a*b*e^2*n)*x^2 + 9*(b^2*d*e* n^2 + 2*a*b*d*e*n)*x + 6*(3*b^2*e^2*n*x^2 + 3*b^2*d*e*n*x + b^2*d^2*n)*log (c))*log(x))/x^3
Time = 0.40 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.71 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=- \frac {a^{2} d^{2}}{3 x^{3}} - \frac {a^{2} d e}{x^{2}} - \frac {a^{2} e^{2}}{x} - \frac {2 a b d^{2} n}{9 x^{3}} - \frac {2 a b d^{2} \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {a b d e n}{x^{2}} - \frac {2 a b d e \log {\left (c x^{n} \right )}}{x^{2}} - \frac {2 a b e^{2} n}{x} - \frac {2 a b e^{2} \log {\left (c x^{n} \right )}}{x} - \frac {2 b^{2} d^{2} n^{2}}{27 x^{3}} - \frac {2 b^{2} d^{2} n \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {b^{2} d^{2} \log {\left (c x^{n} \right )}^{2}}{3 x^{3}} - \frac {b^{2} d e n^{2}}{2 x^{2}} - \frac {b^{2} d e n \log {\left (c x^{n} \right )}}{x^{2}} - \frac {b^{2} d e \log {\left (c x^{n} \right )}^{2}}{x^{2}} - \frac {2 b^{2} e^{2} n^{2}}{x} - \frac {2 b^{2} e^{2} n \log {\left (c x^{n} \right )}}{x} - \frac {b^{2} e^{2} \log {\left (c x^{n} \right )}^{2}}{x} \] Input:
integrate((e*x+d)**2*(a+b*ln(c*x**n))**2/x**4,x)
Output:
-a**2*d**2/(3*x**3) - a**2*d*e/x**2 - a**2*e**2/x - 2*a*b*d**2*n/(9*x**3) - 2*a*b*d**2*log(c*x**n)/(3*x**3) - a*b*d*e*n/x**2 - 2*a*b*d*e*log(c*x**n) /x**2 - 2*a*b*e**2*n/x - 2*a*b*e**2*log(c*x**n)/x - 2*b**2*d**2*n**2/(27*x **3) - 2*b**2*d**2*n*log(c*x**n)/(9*x**3) - b**2*d**2*log(c*x**n)**2/(3*x* *3) - b**2*d*e*n**2/(2*x**2) - b**2*d*e*n*log(c*x**n)/x**2 - b**2*d*e*log( c*x**n)**2/x**2 - 2*b**2*e**2*n**2/x - 2*b**2*e**2*n*log(c*x**n)/x - b**2* e**2*log(c*x**n)**2/x
Time = 0.04 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.49 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-2 \, b^{2} e^{2} {\left (\frac {n^{2}}{x} + \frac {n \log \left (c x^{n}\right )}{x}\right )} - \frac {1}{2} \, b^{2} d e {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {2}{27} \, b^{2} d^{2} {\left (\frac {n^{2}}{x^{3}} + \frac {3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {b^{2} e^{2} \log \left (c x^{n}\right )^{2}}{x} - \frac {2 \, a b e^{2} n}{x} - \frac {2 \, a b e^{2} \log \left (c x^{n}\right )}{x} - \frac {b^{2} d e \log \left (c x^{n}\right )^{2}}{x^{2}} - \frac {a b d e n}{x^{2}} - \frac {a^{2} e^{2}}{x} - \frac {2 \, a b d e \log \left (c x^{n}\right )}{x^{2}} - \frac {b^{2} d^{2} \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac {2 \, a b d^{2} n}{9 \, x^{3}} - \frac {a^{2} d e}{x^{2}} - \frac {2 \, a b d^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a^{2} d^{2}}{3 \, x^{3}} \] Input:
integrate((e*x+d)^2*(a+b*log(c*x^n))^2/x^4,x, algorithm="maxima")
Output:
-2*b^2*e^2*(n^2/x + n*log(c*x^n)/x) - 1/2*b^2*d*e*(n^2/x^2 + 2*n*log(c*x^n )/x^2) - 2/27*b^2*d^2*(n^2/x^3 + 3*n*log(c*x^n)/x^3) - b^2*e^2*log(c*x^n)^ 2/x - 2*a*b*e^2*n/x - 2*a*b*e^2*log(c*x^n)/x - b^2*d*e*log(c*x^n)^2/x^2 - a*b*d*e*n/x^2 - a^2*e^2/x - 2*a*b*d*e*log(c*x^n)/x^2 - 1/3*b^2*d^2*log(c*x ^n)^2/x^3 - 2/9*a*b*d^2*n/x^3 - a^2*d*e/x^2 - 2/3*a*b*d^2*log(c*x^n)/x^3 - 1/3*a^2*d^2/x^3
Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (160) = 320\).
Time = 0.13 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.11 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-\frac {{\left (3 \, b^{2} e^{2} n^{2} x^{2} + 3 \, b^{2} d e n^{2} x + b^{2} d^{2} n^{2}\right )} \log \left (x\right )^{2}}{3 \, x^{3}} - \frac {{\left (18 \, b^{2} e^{2} n^{2} x^{2} + 18 \, b^{2} e^{2} n x^{2} \log \left (c\right ) + 9 \, b^{2} d e n^{2} x + 18 \, a b e^{2} n x^{2} + 18 \, b^{2} d e n x \log \left (c\right ) + 2 \, b^{2} d^{2} n^{2} + 18 \, a b d e n x + 6 \, b^{2} d^{2} n \log \left (c\right ) + 6 \, a b d^{2} n\right )} \log \left (x\right )}{9 \, x^{3}} - \frac {108 \, b^{2} e^{2} n^{2} x^{2} + 108 \, b^{2} e^{2} n x^{2} \log \left (c\right ) + 54 \, b^{2} e^{2} x^{2} \log \left (c\right )^{2} + 27 \, b^{2} d e n^{2} x + 108 \, a b e^{2} n x^{2} + 54 \, b^{2} d e n x \log \left (c\right ) + 108 \, a b e^{2} x^{2} \log \left (c\right ) + 54 \, b^{2} d e x \log \left (c\right )^{2} + 4 \, b^{2} d^{2} n^{2} + 54 \, a b d e n x + 54 \, a^{2} e^{2} x^{2} + 12 \, b^{2} d^{2} n \log \left (c\right ) + 108 \, a b d e x \log \left (c\right ) + 18 \, b^{2} d^{2} \log \left (c\right )^{2} + 12 \, a b d^{2} n + 54 \, a^{2} d e x + 36 \, a b d^{2} \log \left (c\right ) + 18 \, a^{2} d^{2}}{54 \, x^{3}} \] Input:
integrate((e*x+d)^2*(a+b*log(c*x^n))^2/x^4,x, algorithm="giac")
Output:
-1/3*(3*b^2*e^2*n^2*x^2 + 3*b^2*d*e*n^2*x + b^2*d^2*n^2)*log(x)^2/x^3 - 1/ 9*(18*b^2*e^2*n^2*x^2 + 18*b^2*e^2*n*x^2*log(c) + 9*b^2*d*e*n^2*x + 18*a*b *e^2*n*x^2 + 18*b^2*d*e*n*x*log(c) + 2*b^2*d^2*n^2 + 18*a*b*d*e*n*x + 6*b^ 2*d^2*n*log(c) + 6*a*b*d^2*n)*log(x)/x^3 - 1/54*(108*b^2*e^2*n^2*x^2 + 108 *b^2*e^2*n*x^2*log(c) + 54*b^2*e^2*x^2*log(c)^2 + 27*b^2*d*e*n^2*x + 108*a *b*e^2*n*x^2 + 54*b^2*d*e*n*x*log(c) + 108*a*b*e^2*x^2*log(c) + 54*b^2*d*e *x*log(c)^2 + 4*b^2*d^2*n^2 + 54*a*b*d*e*n*x + 54*a^2*e^2*x^2 + 12*b^2*d^2 *n*log(c) + 108*a*b*d*e*x*log(c) + 18*b^2*d^2*log(c)^2 + 12*a*b*d^2*n + 54 *a^2*d*e*x + 36*a*b*d^2*log(c) + 18*a^2*d^2)/x^3
Time = 27.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-\frac {x\,\left (9\,d\,e\,a^2+9\,d\,e\,a\,b\,n+\frac {9\,d\,e\,b^2\,n^2}{2}\right )+x^2\,\left (9\,a^2\,e^2+18\,a\,b\,e^2\,n+18\,b^2\,e^2\,n^2\right )+3\,a^2\,d^2+\frac {2\,b^2\,d^2\,n^2}{3}+2\,a\,b\,d^2\,n}{9\,x^3}-\frac {{\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,d^2}{3}+b^2\,d\,e\,x+b^2\,e^2\,x^2\right )}{x^3}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {2\,b\,\left (3\,a+b\,n\right )\,d^2}{3}+3\,b\,\left (2\,a+b\,n\right )\,d\,e\,x+6\,b\,\left (a+b\,n\right )\,e^2\,x^2\right )}{3\,x^3} \] Input:
int(((a + b*log(c*x^n))^2*(d + e*x)^2)/x^4,x)
Output:
- (x*(9*a^2*d*e + (9*b^2*d*e*n^2)/2 + 9*a*b*d*e*n) + x^2*(9*a^2*e^2 + 18*b ^2*e^2*n^2 + 18*a*b*e^2*n) + 3*a^2*d^2 + (2*b^2*d^2*n^2)/3 + 2*a*b*d^2*n)/ (9*x^3) - (log(c*x^n)^2*((b^2*d^2)/3 + b^2*e^2*x^2 + b^2*d*e*x))/x^3 - (lo g(c*x^n)*((2*b*d^2*(3*a + b*n))/3 + 6*b*e^2*x^2*(a + b*n) + 3*b*d*e*x*(2*a + b*n)))/(3*x^3)
Time = 0.16 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.41 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=\frac {-18 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d^{2}-54 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d e x -54 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e^{2} x^{2}-36 \,\mathrm {log}\left (x^{n} c \right ) a b \,d^{2}-108 \,\mathrm {log}\left (x^{n} c \right ) a b d e x -108 \,\mathrm {log}\left (x^{n} c \right ) a b \,e^{2} x^{2}-12 \,\mathrm {log}\left (x^{n} c \right ) b^{2} d^{2} n -54 \,\mathrm {log}\left (x^{n} c \right ) b^{2} d e n x -108 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e^{2} n \,x^{2}-18 a^{2} d^{2}-54 a^{2} d e x -54 a^{2} e^{2} x^{2}-12 a b \,d^{2} n -54 a b d e n x -108 a b \,e^{2} n \,x^{2}-4 b^{2} d^{2} n^{2}-27 b^{2} d e \,n^{2} x -108 b^{2} e^{2} n^{2} x^{2}}{54 x^{3}} \] Input:
int((e*x+d)^2*(a+b*log(c*x^n))^2/x^4,x)
Output:
( - 18*log(x**n*c)**2*b**2*d**2 - 54*log(x**n*c)**2*b**2*d*e*x - 54*log(x* *n*c)**2*b**2*e**2*x**2 - 36*log(x**n*c)*a*b*d**2 - 108*log(x**n*c)*a*b*d* e*x - 108*log(x**n*c)*a*b*e**2*x**2 - 12*log(x**n*c)*b**2*d**2*n - 54*log( x**n*c)*b**2*d*e*n*x - 108*log(x**n*c)*b**2*e**2*n*x**2 - 18*a**2*d**2 - 5 4*a**2*d*e*x - 54*a**2*e**2*x**2 - 12*a*b*d**2*n - 54*a*b*d*e*n*x - 108*a* b*e**2*n*x**2 - 4*b**2*d**2*n**2 - 27*b**2*d*e*n**2*x - 108*b**2*e**2*n**2 *x**2)/(54*x**3)