Integrand size = 23, antiderivative size = 135 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {b d^2 n}{4 x^2}-\frac {b e^2 n x^{-2 (1-r)}}{4 (1-r)^2}-\frac {2 b d e n x^{-2+r}}{(2-r)^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}-\frac {2 d e x^{-2+r} \left (a+b \log \left (c x^n\right )\right )}{2-r} \]
-1/4*b*d^2*n/x^2-1/4*b*e^2*n/(1-r)^2/(x^(2-2*r))-2*b*d*e*n*x^(-2+r)/(2-r)^ 2-1/2*d^2*(a+b*ln(c*x^n))/x^2-1/2*e^2*(a+b*ln(c*x^n))/(1-r)/(x^(2-2*r))-2* d*e*x^(-2+r)*(a+b*ln(c*x^n))/(2-r)
Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.89 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {b n \left (-d^2-\frac {8 d e x^r}{(-2+r)^2}-\frac {e^2 x^{2 r}}{(-1+r)^2}\right )+a \left (-2 d^2+\frac {8 d e x^r}{-2+r}+\frac {2 e^2 x^{2 r}}{-1+r}\right )+2 b \left (-d^2+\frac {4 d e x^r}{-2+r}+\frac {e^2 x^{2 r}}{-1+r}\right ) \log \left (c x^n\right )}{4 x^2} \]
(b*n*(-d^2 - (8*d*e*x^r)/(-2 + r)^2 - (e^2*x^(2*r))/(-1 + r)^2) + a*(-2*d^ 2 + (8*d*e*x^r)/(-2 + r) + (2*e^2*x^(2*r))/(-1 + r)) + 2*b*(-d^2 + (4*d*e* x^r)/(-2 + r) + (e^2*x^(2*r))/(-1 + r))*Log[c*x^n])/(4*x^2)
Time = 0.43 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.23, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2772, 27, 1691, 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.
\(\displaystyle \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle -b n \int -\frac {4 d e (1-r) x^r+e^2 (2-r) x^{2 r}+d^2 (1-r) (2-r)}{2 \left (r^2-3 r+2\right ) x^3}dx-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {2 d e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b n \int \frac {4 d e (1-r) x^r+e^2 (2-r) x^{2 r}+d^2 (1-r) (2-r)}{x^3}dx}{2 \left (r^2-3 r+2\right )}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {2 d e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}\) |
\(\Big \downarrow \) 1691 |
\(\displaystyle \frac {b n \int \left (-4 d e (r-1) x^{r-3}-e^2 (r-2) x^{2 r-3}+\frac {d^2 (r-2) (r-1)}{x^3}\right )dx}{2 \left (r^2-3 r+2\right )}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {2 d e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {2 d e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}+\frac {b n \left (-\frac {d^2 (1-r) (2-r)}{2 x^2}-\frac {4 d e (1-r) x^{r-2}}{2-r}-\frac {e^2 (2-r) x^{-2 (1-r)}}{2 (1-r)}\right )}{2 \left (r^2-3 r+2\right )}\) |
(b*n*(-1/2*(d^2*(1 - r)*(2 - r))/x^2 - (e^2*(2 - r))/(2*(1 - r)*x^(2*(1 - r))) - (4*d*e*(1 - r)*x^(-2 + r))/(2 - r)))/(2*(2 - 3*r + r^2)) - (d^2*(a + b*Log[c*x^n]))/(2*x^2) - (e^2*(a + b*Log[c*x^n]))/(2*(1 - r)*x^(2*(1 - r ))) - (2*d*e*x^(-2 + r)*(a + b*Log[c*x^n]))/(2 - r)
3.4.83.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && IGtQ[p, 0] && !IntegerQ [Simplify[(m + 1)/n]]
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. \(473\) vs. \(2(127)=254\).
Time = 1.15 (sec) , antiderivative size = 474, normalized size of antiderivative = 3.51
method | result | size |
parallelrisch | \(-\frac {8 b \ln \left (c \,x^{n}\right ) d^{2}+8 b d e n \,x^{r}+13 b \,d^{2} n \,r^{2}+16 d e \,x^{r} a -12 b \,d^{2} n r +16 d e \,x^{r} b \ln \left (c \,x^{n}\right )+2 a \,d^{2} r^{4}-12 a \,d^{2} r^{3}-8 a d e \,r^{3} x^{r}+4 b \,d^{2} n +8 a \,d^{2}+2 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{4}-12 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{3}+26 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{2}-24 \ln \left (c \,x^{n}\right ) b \,d^{2} r +b \,d^{2} n \,r^{4}-6 b \,d^{2} n \,r^{3}-16 b d e n r \,x^{r}+26 a \,d^{2} r^{2}-24 a \,d^{2} r +10 a \,e^{2} r^{2} x^{2 r}-16 a \,e^{2} r \,x^{2 r}+4 b \,e^{2} n \,x^{2 r}-2 a \,e^{2} r^{3} x^{2 r}+8 e^{2} x^{2 r} b \ln \left (c \,x^{n}\right )+8 b d e n \,r^{2} x^{r}+32 a d e \,r^{2} x^{r}-40 a d e r \,x^{r}-4 b \,e^{2} n r \,x^{2 r}-2 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{3}+10 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{2}-16 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r +b \,e^{2} n \,r^{2} x^{2 r}+8 e^{2} x^{2 r} a -8 x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{3}+32 x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{2}-40 x^{r} \ln \left (c \,x^{n}\right ) b d e r}{4 x^{2} \left (-1+r \right )^{2} \left (r^{2}-4 r +4\right )}\) | \(474\) |
risch | \(\text {Expression too large to display}\) | \(1923\) |
-1/4/x^2*(8*b*ln(c*x^n)*d^2-4*b*e^2*n*r*(x^r)^2+8*b*d*e*n*x^r+10*a*e^2*r^2 *(x^r)^2-16*a*e^2*r*(x^r)^2+4*b*e^2*n*(x^r)^2-2*a*e^2*r^3*(x^r)^2+8*e^2*(x ^r)^2*a+13*b*d^2*n*r^2+16*d*e*x^r*a+8*e^2*(x^r)^2*b*ln(c*x^n)-12*b*d^2*n*r +16*d*e*x^r*b*ln(c*x^n)+2*a*d^2*r^4-12*a*d^2*r^3-8*a*d*e*r^3*x^r+4*b*d^2*n +8*a*d^2+2*ln(c*x^n)*b*d^2*r^4-12*ln(c*x^n)*b*d^2*r^3+26*ln(c*x^n)*b*d^2*r ^2-24*ln(c*x^n)*b*d^2*r+b*d^2*n*r^4-6*b*d^2*n*r^3-16*b*d*e*n*r*x^r+26*a*d^ 2*r^2-24*a*d^2*r-2*(x^r)^2*ln(c*x^n)*b*e^2*r^3+10*(x^r)^2*ln(c*x^n)*b*e^2* r^2-16*(x^r)^2*ln(c*x^n)*b*e^2*r+8*b*d*e*n*r^2*x^r+32*a*d*e*r^2*x^r-40*a*d *e*r*x^r+b*e^2*n*r^2*(x^r)^2-8*x^r*ln(c*x^n)*b*d*e*r^3+32*x^r*ln(c*x^n)*b* d*e*r^2-40*x^r*ln(c*x^n)*b*d*e*r)/(-1+r)^2/(r^2-4*r+4)
Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (119) = 238\).
Time = 0.30 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.39 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {{\left (b d^{2} n + 2 \, a d^{2}\right )} r^{4} + 4 \, b d^{2} n - 6 \, {\left (b d^{2} n + 2 \, a d^{2}\right )} r^{3} + 8 \, a d^{2} + 13 \, {\left (b d^{2} n + 2 \, a d^{2}\right )} r^{2} - 12 \, {\left (b d^{2} n + 2 \, a d^{2}\right )} r - {\left (2 \, a e^{2} r^{3} - 4 \, b e^{2} n - 8 \, a e^{2} - {\left (b e^{2} n + 10 \, a e^{2}\right )} r^{2} + 4 \, {\left (b e^{2} n + 4 \, a e^{2}\right )} r + 2 \, {\left (b e^{2} r^{3} - 5 \, b e^{2} r^{2} + 8 \, b e^{2} r - 4 \, b e^{2}\right )} \log \left (c\right ) + 2 \, {\left (b e^{2} n r^{3} - 5 \, b e^{2} n r^{2} + 8 \, b e^{2} n r - 4 \, b e^{2} n\right )} \log \left (x\right )\right )} x^{2 \, r} - 8 \, {\left (a d e r^{3} - b d e n - 2 \, a d e - {\left (b d e n + 4 \, a d e\right )} r^{2} + {\left (2 \, b d e n + 5 \, a d e\right )} r + {\left (b d e r^{3} - 4 \, b d e r^{2} + 5 \, b d e r - 2 \, b d e\right )} \log \left (c\right ) + {\left (b d e n r^{3} - 4 \, b d e n r^{2} + 5 \, b d e n r - 2 \, b d e n\right )} \log \left (x\right )\right )} x^{r} + 2 \, {\left (b d^{2} r^{4} - 6 \, b d^{2} r^{3} + 13 \, b d^{2} r^{2} - 12 \, b d^{2} r + 4 \, b d^{2}\right )} \log \left (c\right ) + 2 \, {\left (b d^{2} n r^{4} - 6 \, b d^{2} n r^{3} + 13 \, b d^{2} n r^{2} - 12 \, b d^{2} n r + 4 \, b d^{2} n\right )} \log \left (x\right )}{4 \, {\left (r^{4} - 6 \, r^{3} + 13 \, r^{2} - 12 \, r + 4\right )} x^{2}} \]
-1/4*((b*d^2*n + 2*a*d^2)*r^4 + 4*b*d^2*n - 6*(b*d^2*n + 2*a*d^2)*r^3 + 8* a*d^2 + 13*(b*d^2*n + 2*a*d^2)*r^2 - 12*(b*d^2*n + 2*a*d^2)*r - (2*a*e^2*r ^3 - 4*b*e^2*n - 8*a*e^2 - (b*e^2*n + 10*a*e^2)*r^2 + 4*(b*e^2*n + 4*a*e^2 )*r + 2*(b*e^2*r^3 - 5*b*e^2*r^2 + 8*b*e^2*r - 4*b*e^2)*log(c) + 2*(b*e^2* n*r^3 - 5*b*e^2*n*r^2 + 8*b*e^2*n*r - 4*b*e^2*n)*log(x))*x^(2*r) - 8*(a*d* e*r^3 - b*d*e*n - 2*a*d*e - (b*d*e*n + 4*a*d*e)*r^2 + (2*b*d*e*n + 5*a*d*e )*r + (b*d*e*r^3 - 4*b*d*e*r^2 + 5*b*d*e*r - 2*b*d*e)*log(c) + (b*d*e*n*r^ 3 - 4*b*d*e*n*r^2 + 5*b*d*e*n*r - 2*b*d*e*n)*log(x))*x^r + 2*(b*d^2*r^4 - 6*b*d^2*r^3 + 13*b*d^2*r^2 - 12*b*d^2*r + 4*b*d^2)*log(c) + 2*(b*d^2*n*r^4 - 6*b*d^2*n*r^3 + 13*b*d^2*n*r^2 - 12*b*d^2*n*r + 4*b*d^2*n)*log(x))/((r^ 4 - 6*r^3 + 13*r^2 - 12*r + 4)*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 2118 vs. \(2 (119) = 238\).
Time = 3.30 (sec) , antiderivative size = 2118, normalized size of antiderivative = 15.69 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Too large to display} \]
Piecewise((-a*d**2/(2*x**2) - 2*a*d*e/x + a*e**2*log(x) + b*d**2*(-n/(4*x* *2) - log(c*x**n)/(2*x**2)) + 2*b*d*e*(-n/x - log(c*x**n)/x) - b*e**2*Piec ewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)), Eq(r, 1) ), (-a*d**2/(2*x**2) + 2*a*d*e*log(c*x**n)/n + a*e**2*x**2/2 - b*d**2*n/(4 *x**2) - b*d**2*log(c*x**n)/(2*x**2) + b*d*e*log(c*x**n)**2/n - b*e**2*n*x **2/4 + b*e**2*x**2*log(c*x**n)/2, Eq(r, 2)), (-2*a*d**2*r**4/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 12*a*d**2*r**3/(4* r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 26*a*d**2 *r**2/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 24*a*d**2*r/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x* *2) - 8*a*d**2/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16 *x**2) + 8*a*d*e*r**3*x**r/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48 *r*x**2 + 16*x**2) - 32*a*d*e*r**2*x**r/(4*r**4*x**2 - 24*r**3*x**2 + 52*r **2*x**2 - 48*r*x**2 + 16*x**2) + 40*a*d*e*r*x**r/(4*r**4*x**2 - 24*r**3*x **2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 16*a*d*e*x**r/(4*r**4*x**2 - 2 4*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 2*a*e**2*r**3*x**(2*r) /(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 10*a* e**2*r**2*x**(2*r)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 16*a*e**2*r*x**(2*r)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x* *2 - 48*r*x**2 + 16*x**2) - 8*a*e**2*x**(2*r)/(4*r**4*x**2 - 24*r**3*x*...
Exception generated. \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Exception raised: ValueError} \]
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-3>0)', see `assume?` for more details)Is
\[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{2} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \]