Integrand size = 20, antiderivative size = 173 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=2 b^2 d^2 n^2 x+\frac {1}{2} b^2 d e n^2 x^2+\frac {2}{27} b^2 e^2 n^2 x^3+\frac {b^2 d^3 n^2 \log ^2(x)}{3 e}-2 b d^2 n x \left (a+b \log \left (c x^n\right )\right )-b d e n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} b e^2 n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 b d^3 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e} \]
2*b^2*d^2*n^2*x+1/2*b^2*d*e*n^2*x^2+2/27*b^2*e^2*n^2*x^3+1/3*b^2*d^3*n^2*l n(x)^2/e-2*b*d^2*n*x*(a+b*ln(c*x^n))-b*d*e*n*x^2*(a+b*ln(c*x^n))-2/9*b*e^2 *n*x^3*(a+b*ln(c*x^n))-2/3*b*d^3*n*ln(x)*(a+b*ln(c*x^n))/e+1/3*(e*x+d)^3*( a+b*ln(c*x^n))^2/e
Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.78 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2}{27} b e^2 n x^3 \left (-3 a+b n-3 b \log \left (c x^n\right )\right )+\frac {1}{2} b d e n x^2 \left (-2 a+b n-2 b \log \left (c x^n\right )\right )+d^2 x \left (a+b \log \left (c x^n\right )\right )^2+d e x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{3} e^2 x^3 \left (a+b \log \left (c x^n\right )\right )^2-2 b d^2 n x \left (a-b n+b \log \left (c x^n\right )\right ) \]
(2*b*e^2*n*x^3*(-3*a + b*n - 3*b*Log[c*x^n]))/27 + (b*d*e*n*x^2*(-2*a + b* n - 2*b*Log[c*x^n]))/2 + d^2*x*(a + b*Log[c*x^n])^2 + d*e*x^2*(a + b*Log[c *x^n])^2 + (e^2*x^3*(a + b*Log[c*x^n])^2)/3 - 2*b*d^2*n*x*(a - b*n + b*Log [c*x^n])
Time = 0.36 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2756, 2772, 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.
| \(\displaystyle \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {2 b n \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x}dx}{3 e}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {2 b n \left (-b n \int \left (\frac {\log (x) d^3}{x}+\frac {1}{6} e \left (18 d^2+9 e x d+2 e^2 x^2\right )\right )dx+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+3 d^2 e x \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )\right )}{3 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {2 b n \left (d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+3 d^2 e x \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {1}{2} d^3 \log ^2(x)+3 d^2 e x+\frac {3}{4} d e^2 x^2+\frac {e^3 x^3}{9}\right )\right )}{3 e}\) |
((d + e*x)^3*(a + b*Log[c*x^n])^2)/(3*e) - (2*b*n*(-(b*n*(3*d^2*e*x + (3*d *e^2*x^2)/4 + (e^3*x^3)/9 + (d^3*Log[x]^2)/2)) + 3*d^2*e*x*(a + b*Log[c*x^ n]) + (3*d*e^2*x^2*(a + b*Log[c*x^n]))/2 + (e^3*x^3*(a + b*Log[c*x^n]))/3 + d^3*Log[x]*(a + b*Log[c*x^n])))/(3*e)
3.1.86.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
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])
Time = 0.60 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.43
| method | result | size |
| parallelrisch | \(\frac {b^{2} \ln \left (c \,x^{n}\right )^{2} e^{2} x^{3}}{3}-\frac {2 \ln \left (c \,x^{n}\right ) x^{3} n \,e^{2} b^{2}}{9}+\frac {2 b^{2} e^{2} n^{2} x^{3}}{27}+\frac {2 a b \ln \left (c \,x^{n}\right ) e^{2} x^{3}}{3}-\frac {2 b n a \,e^{2} x^{3}}{9}+b^{2} \ln \left (c \,x^{n}\right )^{2} d e \,x^{2}-x^{2} \ln \left (c \,x^{n}\right ) b^{2} d e n +\frac {b^{2} d e \,n^{2} x^{2}}{2}+\frac {a^{2} e^{2} x^{3}}{3}+2 a b \ln \left (c \,x^{n}\right ) d e \,x^{2}-b n a d e \,x^{2}+x \,b^{2} \ln \left (c \,x^{n}\right )^{2} d^{2}-2 x \ln \left (c \,x^{n}\right ) b^{2} d^{2} n +2 b^{2} d^{2} n^{2} x +a^{2} d e \,x^{2}+2 x a b \ln \left (c \,x^{n}\right ) d^{2}-2 b n a \,d^{2} x +x \,a^{2} d^{2}\) | \(247\) |
| risch | \(\text {Expression too large to display}\) | \(2565\) |
1/3*b^2*ln(c*x^n)^2*e^2*x^3-2/9*ln(c*x^n)*x^3*n*e^2*b^2+2/27*b^2*e^2*n^2*x ^3+2/3*a*b*ln(c*x^n)*e^2*x^3-2/9*b*n*a*e^2*x^3+b^2*ln(c*x^n)^2*d*e*x^2-x^2 *ln(c*x^n)*b^2*d*e*n+1/2*b^2*d*e*n^2*x^2+1/3*a^2*e^2*x^3+2*a*b*ln(c*x^n)*d *e*x^2-b*n*a*d*e*x^2+x*b^2*ln(c*x^n)^2*d^2-2*x*ln(c*x^n)*b^2*d^2*n+2*b^2*d ^2*n^2*x+a^2*d*e*x^2+2*x*a*b*ln(c*x^n)*d^2-2*b*n*a*d^2*x+x*a^2*d^2
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (161) = 322\).
Time = 0.27 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.01 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{27} \, {\left (2 \, b^{2} e^{2} n^{2} - 6 \, a b e^{2} n + 9 \, a^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b^{2} d e n^{2} - 2 \, a b d e n + 2 \, a^{2} d e\right )} x^{2} + \frac {1}{3} \, {\left (b^{2} e^{2} x^{3} + 3 \, b^{2} d e x^{2} + 3 \, b^{2} d^{2} x\right )} \log \left (c\right )^{2} + \frac {1}{3} \, {\left (b^{2} e^{2} n^{2} x^{3} + 3 \, b^{2} d e n^{2} x^{2} + 3 \, b^{2} d^{2} n^{2} x\right )} \log \left (x\right )^{2} + {\left (2 \, b^{2} d^{2} n^{2} - 2 \, a b d^{2} n + a^{2} d^{2}\right )} x - \frac {1}{9} \, {\left (2 \, {\left (b^{2} e^{2} n - 3 \, a b e^{2}\right )} x^{3} + 9 \, {\left (b^{2} d e n - 2 \, a b d e\right )} x^{2} + 18 \, {\left (b^{2} d^{2} n - a b d^{2}\right )} x\right )} \log \left (c\right ) - \frac {1}{9} \, {\left (2 \, {\left (b^{2} e^{2} n^{2} - 3 \, a b e^{2} n\right )} x^{3} + 9 \, {\left (b^{2} d e n^{2} - 2 \, a b d e n\right )} x^{2} + 18 \, {\left (b^{2} d^{2} n^{2} - a b d^{2} n\right )} x - 6 \, {\left (b^{2} e^{2} n x^{3} + 3 \, b^{2} d e n x^{2} + 3 \, b^{2} d^{2} n x\right )} \log \left (c\right )\right )} \log \left (x\right ) \]
1/27*(2*b^2*e^2*n^2 - 6*a*b*e^2*n + 9*a^2*e^2)*x^3 + 1/2*(b^2*d*e*n^2 - 2* a*b*d*e*n + 2*a^2*d*e)*x^2 + 1/3*(b^2*e^2*x^3 + 3*b^2*d*e*x^2 + 3*b^2*d^2* x)*log(c)^2 + 1/3*(b^2*e^2*n^2*x^3 + 3*b^2*d*e*n^2*x^2 + 3*b^2*d^2*n^2*x)* log(x)^2 + (2*b^2*d^2*n^2 - 2*a*b*d^2*n + a^2*d^2)*x - 1/9*(2*(b^2*e^2*n - 3*a*b*e^2)*x^3 + 9*(b^2*d*e*n - 2*a*b*d*e)*x^2 + 18*(b^2*d^2*n - a*b*d^2) *x)*log(c) - 1/9*(2*(b^2*e^2*n^2 - 3*a*b*e^2*n)*x^3 + 9*(b^2*d*e*n^2 - 2*a *b*d*e*n)*x^2 + 18*(b^2*d^2*n^2 - a*b*d^2*n)*x - 6*(b^2*e^2*n*x^3 + 3*b^2* d*e*n*x^2 + 3*b^2*d^2*n*x)*log(c))*log(x)
Time = 0.26 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.65 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=a^{2} d^{2} x + a^{2} d e x^{2} + \frac {a^{2} e^{2} x^{3}}{3} - 2 a b d^{2} n x + 2 a b d^{2} x \log {\left (c x^{n} \right )} - a b d e n x^{2} + 2 a b d e x^{2} \log {\left (c x^{n} \right )} - \frac {2 a b e^{2} n x^{3}}{9} + \frac {2 a b e^{2} x^{3} \log {\left (c x^{n} \right )}}{3} + 2 b^{2} d^{2} n^{2} x - 2 b^{2} d^{2} n x \log {\left (c x^{n} \right )} + b^{2} d^{2} x \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} d e n^{2} x^{2}}{2} - b^{2} d e n x^{2} \log {\left (c x^{n} \right )} + b^{2} d e x^{2} \log {\left (c x^{n} \right )}^{2} + \frac {2 b^{2} e^{2} n^{2} x^{3}}{27} - \frac {2 b^{2} e^{2} n x^{3} \log {\left (c x^{n} \right )}}{9} + \frac {b^{2} e^{2} x^{3} \log {\left (c x^{n} \right )}^{2}}{3} \]
a**2*d**2*x + a**2*d*e*x**2 + a**2*e**2*x**3/3 - 2*a*b*d**2*n*x + 2*a*b*d* *2*x*log(c*x**n) - a*b*d*e*n*x**2 + 2*a*b*d*e*x**2*log(c*x**n) - 2*a*b*e** 2*n*x**3/9 + 2*a*b*e**2*x**3*log(c*x**n)/3 + 2*b**2*d**2*n**2*x - 2*b**2*d **2*n*x*log(c*x**n) + b**2*d**2*x*log(c*x**n)**2 + b**2*d*e*n**2*x**2/2 - b**2*d*e*n*x**2*log(c*x**n) + b**2*d*e*x**2*log(c*x**n)**2 + 2*b**2*e**2*n **2*x**3/27 - 2*b**2*e**2*n*x**3*log(c*x**n)/9 + b**2*e**2*x**3*log(c*x**n )**2/3
Time = 0.21 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.36 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{3} \, b^{2} e^{2} x^{3} \log \left (c x^{n}\right )^{2} - \frac {2}{9} \, a b e^{2} n x^{3} + \frac {2}{3} \, a b e^{2} x^{3} \log \left (c x^{n}\right ) + b^{2} d e x^{2} \log \left (c x^{n}\right )^{2} - a b d e n x^{2} + \frac {1}{3} \, a^{2} e^{2} x^{3} + 2 \, a b d e x^{2} \log \left (c x^{n}\right ) + b^{2} d^{2} x \log \left (c x^{n}\right )^{2} - 2 \, a b d^{2} n x + a^{2} d e x^{2} + 2 \, a b d^{2} x \log \left (c x^{n}\right ) + 2 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} b^{2} d^{2} + \frac {1}{2} \, {\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} d e + \frac {2}{27} \, {\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} b^{2} e^{2} + a^{2} d^{2} x \]
1/3*b^2*e^2*x^3*log(c*x^n)^2 - 2/9*a*b*e^2*n*x^3 + 2/3*a*b*e^2*x^3*log(c*x ^n) + b^2*d*e*x^2*log(c*x^n)^2 - a*b*d*e*n*x^2 + 1/3*a^2*e^2*x^3 + 2*a*b*d *e*x^2*log(c*x^n) + b^2*d^2*x*log(c*x^n)^2 - 2*a*b*d^2*n*x + a^2*d*e*x^2 + 2*a*b*d^2*x*log(c*x^n) + 2*(n^2*x - n*x*log(c*x^n))*b^2*d^2 + 1/2*(n^2*x^ 2 - 2*n*x^2*log(c*x^n))*b^2*d*e + 2/27*(n^2*x^3 - 3*n*x^3*log(c*x^n))*b^2* e^2 + a^2*d^2*x
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (161) = 322\).
Time = 0.36 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.23 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{3} \, b^{2} e^{2} n^{2} x^{3} \log \left (x\right )^{2} - \frac {2}{9} \, b^{2} e^{2} n^{2} x^{3} \log \left (x\right ) + \frac {2}{3} \, b^{2} e^{2} n x^{3} \log \left (c\right ) \log \left (x\right ) + b^{2} d e n^{2} x^{2} \log \left (x\right )^{2} + \frac {2}{27} \, b^{2} e^{2} n^{2} x^{3} - \frac {2}{9} \, b^{2} e^{2} n x^{3} \log \left (c\right ) + \frac {1}{3} \, b^{2} e^{2} x^{3} \log \left (c\right )^{2} - b^{2} d e n^{2} x^{2} \log \left (x\right ) + \frac {2}{3} \, a b e^{2} n x^{3} \log \left (x\right ) + 2 \, b^{2} d e n x^{2} \log \left (c\right ) \log \left (x\right ) + b^{2} d^{2} n^{2} x \log \left (x\right )^{2} + \frac {1}{2} \, b^{2} d e n^{2} x^{2} - \frac {2}{9} \, a b e^{2} n x^{3} - b^{2} d e n x^{2} \log \left (c\right ) + \frac {2}{3} \, a b e^{2} x^{3} \log \left (c\right ) + b^{2} d e x^{2} \log \left (c\right )^{2} - 2 \, b^{2} d^{2} n^{2} x \log \left (x\right ) + 2 \, a b d e n x^{2} \log \left (x\right ) + 2 \, b^{2} d^{2} n x \log \left (c\right ) \log \left (x\right ) + 2 \, b^{2} d^{2} n^{2} x - a b d e n x^{2} + \frac {1}{3} \, a^{2} e^{2} x^{3} - 2 \, b^{2} d^{2} n x \log \left (c\right ) + 2 \, a b d e x^{2} \log \left (c\right ) + b^{2} d^{2} x \log \left (c\right )^{2} + 2 \, a b d^{2} n x \log \left (x\right ) - 2 \, a b d^{2} n x + a^{2} d e x^{2} + 2 \, a b d^{2} x \log \left (c\right ) + a^{2} d^{2} x \]
1/3*b^2*e^2*n^2*x^3*log(x)^2 - 2/9*b^2*e^2*n^2*x^3*log(x) + 2/3*b^2*e^2*n* x^3*log(c)*log(x) + b^2*d*e*n^2*x^2*log(x)^2 + 2/27*b^2*e^2*n^2*x^3 - 2/9* b^2*e^2*n*x^3*log(c) + 1/3*b^2*e^2*x^3*log(c)^2 - b^2*d*e*n^2*x^2*log(x) + 2/3*a*b*e^2*n*x^3*log(x) + 2*b^2*d*e*n*x^2*log(c)*log(x) + b^2*d^2*n^2*x* log(x)^2 + 1/2*b^2*d*e*n^2*x^2 - 2/9*a*b*e^2*n*x^3 - b^2*d*e*n*x^2*log(c) + 2/3*a*b*e^2*x^3*log(c) + b^2*d*e*x^2*log(c)^2 - 2*b^2*d^2*n^2*x*log(x) + 2*a*b*d*e*n*x^2*log(x) + 2*b^2*d^2*n*x*log(c)*log(x) + 2*b^2*d^2*n^2*x - a*b*d*e*n*x^2 + 1/3*a^2*e^2*x^3 - 2*b^2*d^2*n*x*log(c) + 2*a*b*d*e*x^2*log (c) + b^2*d^2*x*log(c)^2 + 2*a*b*d^2*n*x*log(x) - 2*a*b*d^2*n*x + a^2*d*e* x^2 + 2*a*b*d^2*x*log(c) + a^2*d^2*x
Time = 0.44 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.96 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx={\ln \left (c\,x^n\right )}^2\,\left (b^2\,d^2\,x+b^2\,d\,e\,x^2+\frac {b^2\,e^2\,x^3}{3}\right )+\ln \left (c\,x^n\right )\,\left (2\,b\,\left (a-b\,n\right )\,d^2\,x+b\,\left (2\,a-b\,n\right )\,d\,e\,x^2+\frac {2\,b\,\left (3\,a-b\,n\right )\,e^2\,x^3}{9}\right )+d^2\,x\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right )+\frac {e^2\,x^3\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{27}+\frac {d\,e\,x^2\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{2} \]