Integrand size = 23, antiderivative size = 135 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {b d^2 n}{16 x^4}-\frac {b e^2 n x^{-2 (2-r)}}{4 (2-r)^2}-\frac {2 b d e n x^{-4+r}}{(4-r)^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}-\frac {2 d e x^{-4+r} \left (a+b \log \left (c x^n\right )\right )}{4-r} \]
-1/16*b*d^2*n/x^4-1/4*b*e^2*n/(2-r)^2/(x^(4-2*r))-2*b*d*e*n*x^(-4+r)/(4-r) ^2-1/4*d^2*(a+b*ln(c*x^n))/x^4-1/2*e^2*(a+b*ln(c*x^n))/(2-r)/(x^(4-2*r))-2 *d*e*x^(-4+r)*(a+b*ln(c*x^n))/(4-r)
Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\frac {b n \left (-d^2-\frac {32 d e x^r}{(-4+r)^2}-\frac {4 e^2 x^{2 r}}{(-2+r)^2}\right )+a \left (-4 d^2+\frac {32 d e x^r}{-4+r}+\frac {8 e^2 x^{2 r}}{-2+r}\right )+4 b \left (-d^2+\frac {8 d e x^r}{-4+r}+\frac {2 e^2 x^{2 r}}{-2+r}\right ) \log \left (c x^n\right )}{16 x^4} \]
(b*n*(-d^2 - (32*d*e*x^r)/(-4 + r)^2 - (4*e^2*x^(2*r))/(-2 + r)^2) + a*(-4 *d^2 + (32*d*e*x^r)/(-4 + r) + (8*e^2*x^(2*r))/(-2 + r)) + 4*b*(-d^2 + (8* d*e*x^r)/(-4 + r) + (2*e^2*x^(2*r))/(-2 + r))*Log[c*x^n])/(16*x^4)
Time = 0.42 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.21, 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.
\(\displaystyle \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle -b n \int -\frac {8 d e (2-r) x^r+2 e^2 (4-r) x^{2 r}+d^2 (2-r) (4-r)}{4 \left (r^2-6 r+8\right ) x^5}dx-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {2 d e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b n \int \frac {8 d e (2-r) x^r+2 e^2 (4-r) x^{2 r}+d^2 (2-r) (4-r)}{x^5}dx}{4 \left (r^2-6 r+8\right )}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {2 d e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}\) |
\(\Big \downarrow \) 1691 |
\(\displaystyle \frac {b n \int \left (-8 d e (r-2) x^{r-5}-2 e^2 (r-4) x^{2 r-5}+\frac {d^2 (r-4) (r-2)}{x^5}\right )dx}{4 \left (r^2-6 r+8\right )}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {2 d e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {2 d e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}+\frac {b n \left (-\frac {d^2 (2-r) (4-r)}{4 x^4}-\frac {8 d e (2-r) x^{r-4}}{4-r}-\frac {e^2 (4-r) x^{-2 (2-r)}}{2-r}\right )}{4 \left (r^2-6 r+8\right )}\) |
(b*n*(-1/4*(d^2*(2 - r)*(4 - r))/x^4 - (e^2*(4 - r))/((2 - r)*x^(2*(2 - r) )) - (8*d*e*(2 - r)*x^(-4 + r))/(4 - r)))/(4*(8 - 6*r + r^2)) - (d^2*(a + b*Log[c*x^n]))/(4*x^4) - (e^2*(a + b*Log[c*x^n]))/(2*(2 - r)*x^(2*(2 - r)) ) - (2*d*e*x^(-4 + r)*(a + b*Log[c*x^n]))/(4 - r)
3.4.84.3.1 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. \(474\) vs. \(2(127)=254\).
Time = 1.10 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.52
method | result | size |
parallelrisch | \(-\frac {256 b \ln \left (c \,x^{n}\right ) d^{2}+128 b d e n \,x^{r}+52 b \,d^{2} n \,r^{2}+512 d e \,x^{r} a -96 b \,d^{2} n r +512 d e \,x^{r} b \ln \left (c \,x^{n}\right )+4 a \,d^{2} r^{4}-48 a \,d^{2} r^{3}-32 a d e \,r^{3} x^{r}+64 b \,d^{2} n +256 a \,d^{2}+4 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{4}-48 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{3}+208 \ln \left (c \,x^{n}\right ) b \,d^{2} r^{2}-384 \ln \left (c \,x^{n}\right ) b \,d^{2} r +b \,d^{2} n \,r^{4}-12 b \,d^{2} n \,r^{3}-128 b d e n r \,x^{r}+208 a \,d^{2} r^{2}-384 a \,d^{2} r +80 a \,e^{2} r^{2} x^{2 r}-256 a \,e^{2} r \,x^{2 r}+64 b \,e^{2} n \,x^{2 r}-8 a \,e^{2} r^{3} x^{2 r}+256 e^{2} x^{2 r} b \ln \left (c \,x^{n}\right )+32 b d e n \,r^{2} x^{r}+256 a d e \,r^{2} x^{r}-640 a d e r \,x^{r}-32 b \,e^{2} n r \,x^{2 r}-8 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{3}+80 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{2}-256 x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r +4 b \,e^{2} n \,r^{2} x^{2 r}+256 e^{2} x^{2 r} a -32 x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{3}+256 x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{2}-640 x^{r} \ln \left (c \,x^{n}\right ) b d e r}{16 x^{4} \left (-2+r \right )^{2} \left (r^{2}-8 r +16\right )}\) | \(475\) |
risch | \(\text {Expression too large to display}\) | \(1924\) |
-1/16/x^4*(256*b*ln(c*x^n)*d^2-32*b*e^2*n*r*(x^r)^2+128*b*d*e*n*x^r+80*a*e ^2*r^2*(x^r)^2-256*a*e^2*r*(x^r)^2+64*b*e^2*n*(x^r)^2-8*a*e^2*r^3*(x^r)^2+ 256*e^2*(x^r)^2*a+52*b*d^2*n*r^2+512*d*e*x^r*a+256*e^2*(x^r)^2*b*ln(c*x^n) -96*b*d^2*n*r+512*d*e*x^r*b*ln(c*x^n)+4*a*d^2*r^4-48*a*d^2*r^3-32*a*d*e*r^ 3*x^r+64*b*d^2*n+256*a*d^2+4*ln(c*x^n)*b*d^2*r^4-48*ln(c*x^n)*b*d^2*r^3+20 8*ln(c*x^n)*b*d^2*r^2-384*ln(c*x^n)*b*d^2*r+b*d^2*n*r^4-12*b*d^2*n*r^3-128 *b*d*e*n*r*x^r+208*a*d^2*r^2-384*a*d^2*r-8*(x^r)^2*ln(c*x^n)*b*e^2*r^3+80* (x^r)^2*ln(c*x^n)*b*e^2*r^2-256*(x^r)^2*ln(c*x^n)*b*e^2*r+32*b*d*e*n*r^2*x ^r+256*a*d*e*r^2*x^r-640*a*d*e*r*x^r+4*b*e^2*n*r^2*(x^r)^2-32*x^r*ln(c*x^n )*b*d*e*r^3+256*x^r*ln(c*x^n)*b*d*e*r^2-640*x^r*ln(c*x^n)*b*d*e*r)/(-2+r)^ 2/(r^2-8*r+16)
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 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {{\left (b d^{2} n + 4 \, a d^{2}\right )} r^{4} + 64 \, b d^{2} n - 12 \, {\left (b d^{2} n + 4 \, a d^{2}\right )} r^{3} + 256 \, a d^{2} + 52 \, {\left (b d^{2} n + 4 \, a d^{2}\right )} r^{2} - 96 \, {\left (b d^{2} n + 4 \, a d^{2}\right )} r - 4 \, {\left (2 \, a e^{2} r^{3} - 16 \, b e^{2} n - 64 \, a e^{2} - {\left (b e^{2} n + 20 \, a e^{2}\right )} r^{2} + 8 \, {\left (b e^{2} n + 8 \, a e^{2}\right )} r + 2 \, {\left (b e^{2} r^{3} - 10 \, b e^{2} r^{2} + 32 \, b e^{2} r - 32 \, b e^{2}\right )} \log \left (c\right ) + 2 \, {\left (b e^{2} n r^{3} - 10 \, b e^{2} n r^{2} + 32 \, b e^{2} n r - 32 \, b e^{2} n\right )} \log \left (x\right )\right )} x^{2 \, r} - 32 \, {\left (a d e r^{3} - 4 \, b d e n - 16 \, a d e - {\left (b d e n + 8 \, a d e\right )} r^{2} + 4 \, {\left (b d e n + 5 \, a d e\right )} r + {\left (b d e r^{3} - 8 \, b d e r^{2} + 20 \, b d e r - 16 \, b d e\right )} \log \left (c\right ) + {\left (b d e n r^{3} - 8 \, b d e n r^{2} + 20 \, b d e n r - 16 \, b d e n\right )} \log \left (x\right )\right )} x^{r} + 4 \, {\left (b d^{2} r^{4} - 12 \, b d^{2} r^{3} + 52 \, b d^{2} r^{2} - 96 \, b d^{2} r + 64 \, b d^{2}\right )} \log \left (c\right ) + 4 \, {\left (b d^{2} n r^{4} - 12 \, b d^{2} n r^{3} + 52 \, b d^{2} n r^{2} - 96 \, b d^{2} n r + 64 \, b d^{2} n\right )} \log \left (x\right )}{16 \, {\left (r^{4} - 12 \, r^{3} + 52 \, r^{2} - 96 \, r + 64\right )} x^{4}} \]
-1/16*((b*d^2*n + 4*a*d^2)*r^4 + 64*b*d^2*n - 12*(b*d^2*n + 4*a*d^2)*r^3 + 256*a*d^2 + 52*(b*d^2*n + 4*a*d^2)*r^2 - 96*(b*d^2*n + 4*a*d^2)*r - 4*(2* a*e^2*r^3 - 16*b*e^2*n - 64*a*e^2 - (b*e^2*n + 20*a*e^2)*r^2 + 8*(b*e^2*n + 8*a*e^2)*r + 2*(b*e^2*r^3 - 10*b*e^2*r^2 + 32*b*e^2*r - 32*b*e^2)*log(c) + 2*(b*e^2*n*r^3 - 10*b*e^2*n*r^2 + 32*b*e^2*n*r - 32*b*e^2*n)*log(x))*x^ (2*r) - 32*(a*d*e*r^3 - 4*b*d*e*n - 16*a*d*e - (b*d*e*n + 8*a*d*e)*r^2 + 4 *(b*d*e*n + 5*a*d*e)*r + (b*d*e*r^3 - 8*b*d*e*r^2 + 20*b*d*e*r - 16*b*d*e) *log(c) + (b*d*e*n*r^3 - 8*b*d*e*n*r^2 + 20*b*d*e*n*r - 16*b*d*e*n)*log(x) )*x^r + 4*(b*d^2*r^4 - 12*b*d^2*r^3 + 52*b*d^2*r^2 - 96*b*d^2*r + 64*b*d^2 )*log(c) + 4*(b*d^2*n*r^4 - 12*b*d^2*n*r^3 + 52*b*d^2*n*r^2 - 96*b*d^2*n*r + 64*b*d^2*n)*log(x))/((r^4 - 12*r^3 + 52*r^2 - 96*r + 64)*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 2127 vs. \(2 (119) = 238\).
Time = 4.82 (sec) , antiderivative size = 2127, normalized size of antiderivative = 15.76 \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\text {Too large to display} \]
Piecewise((-a*d**2/(4*x**4) - a*d*e/x**2 + a*e**2*log(x) + b*d**2*(-n/(16* x**4) - log(c*x**n)/(4*x**4)) + 2*b*d*e*(-n/(4*x**2) - log(c*x**n)/(2*x**2 )) - b*e**2*Piecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)), Eq(r, 2)), (-a*d**2/(4*x**4) + 2*a*d*e*log(c*x**n)/n + a*e**2*x**4 /4 - b*d**2*n/(16*x**4) - b*d**2*log(c*x**n)/(4*x**4) + b*d*e*log(c*x**n)* *2/n - b*e**2*n*x**4/16 + b*e**2*x**4*log(c*x**n)/4, Eq(r, 4)), (-4*a*d**2 *r**4/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x **4) + 48*a*d**2*r**3/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536 *r*x**4 + 1024*x**4) - 208*a*d**2*r**2/(16*r**4*x**4 - 192*r**3*x**4 + 832 *r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 384*a*d**2*r/(16*r**4*x**4 - 192*r **3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 256*a*d**2/(16*r**4* x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 32*a*d*e *r**3*x**r/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1 024*x**4) - 256*a*d*e*r**2*x**r/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x **4 - 1536*r*x**4 + 1024*x**4) + 640*a*d*e*r*x**r/(16*r**4*x**4 - 192*r**3 *x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) - 512*a*d*e*x**r/(16*r**4 *x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 8*a*e** 2*r**3*x**(2*r)/(16*r**4*x**4 - 192*r**3*x**4 + 832*r**2*x**4 - 1536*r*x** 4 + 1024*x**4) - 80*a*e**2*r**2*x**(2*r)/(16*r**4*x**4 - 192*r**3*x**4 + 8 32*r**2*x**4 - 1536*r*x**4 + 1024*x**4) + 256*a*e**2*r*x**(2*r)/(16*r**...
Exception generated. \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\text {Exception raised: ValueError} \]
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-5>0)', see `assume?` for more details)Is
\[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{2} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\int \frac {{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^5} \,d x \]