Integrand size = 23, antiderivative size = 191 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {b d^3 n}{9 x^3}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2}-\frac {3 b d^2 e n x^{-3+r}}{(3-r)^2}-\frac {3 b d e^2 n x^{-3+2 r}}{(3-2 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {3 d^2 e x^{-3+r} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{-3+2 r} \left (a+b \log \left (c x^n\right )\right )}{3-2 r} \] Output:
-1/9*b*d^3*n/x^3-1/9*b*e^3*n/(1-r)^2/(x^(3-3*r))-3*b*d^2*e*n*x^(-3+r)/(3-r )^2-3*b*d*e^2*n*x^(-3+2*r)/(3-2*r)^2-1/3*d^3*(a+b*ln(c*x^n))/x^3-1/3*e^3*( a+b*ln(c*x^n))/(1-r)/(x^(3-3*r))-3*d^2*e*x^(-3+r)*(a+b*ln(c*x^n))/(3-r)-3* d*e^2*x^(-3+2*r)*(a+b*ln(c*x^n))/(3-2*r)
Time = 0.47 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\frac {b n \left (-d^3-\frac {27 d^2 e x^r}{(-3+r)^2}-\frac {27 d e^2 x^{2 r}}{(3-2 r)^2}-\frac {e^3 x^{3 r}}{(-1+r)^2}\right )+3 a \left (-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}\right )+3 b \left (-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}\right ) \log \left (c x^n\right )}{9 x^3} \] Input:
Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^4,x]
Output:
(b*n*(-d^3 - (27*d^2*e*x^r)/(-3 + r)^2 - (27*d*e^2*x^(2*r))/(3 - 2*r)^2 - (e^3*x^(3*r))/(-1 + r)^2) + 3*a*(-d^3 + (9*d^2*e*x^r)/(-3 + r) + (9*d*e^2* x^(2*r))/(-3 + 2*r) + (e^3*x^(3*r))/(-1 + r)) + 3*b*(-d^3 + (9*d^2*e*x^r)/ (-3 + r) + (9*d*e^2*x^(2*r))/(-3 + 2*r) + (e^3*x^(3*r))/(-1 + r))*Log[c*x^ n])/(9*x^3)
Time = 0.66 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2772, 27, 2010, 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.
\(\displaystyle \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle -b n \int -\frac {\frac {9 d^2 e x^r}{3-r}+\frac {9 d e^2 x^{2 r}}{3-2 r}+\frac {e^3 x^{3 r}}{1-r}+d^3}{3 x^4}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {3 d^2 e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{2 r-3} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} b n \int \frac {\frac {9 d^2 e x^r}{3-r}+\frac {9 d e^2 x^{2 r}}{3-2 r}+\frac {e^3 x^{3 r}}{1-r}+d^3}{x^4}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {3 d^2 e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{2 r-3} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {1}{3} b n \int \left (-\frac {9 d^2 e x^{r-4}}{r-3}+\frac {9 d e^2 x^{2 (r-2)}}{3-2 r}-\frac {e^3 x^{3 r-4}}{r-1}+\frac {d^3}{x^4}\right )dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {3 d^2 e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{2 r-3} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {3 d^2 e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{2 r-3} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}+\frac {1}{3} b n \left (-\frac {d^3}{3 x^3}-\frac {9 d^2 e x^{r-3}}{(3-r)^2}-\frac {9 d e^2 x^{2 r-3}}{(3-2 r)^2}-\frac {e^3 x^{-3 (1-r)}}{3 (1-r)^2}\right )\) |
Input:
Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^4,x]
Output:
(b*n*(-1/3*d^3/x^3 - e^3/(3*(1 - r)^2*x^(3*(1 - r))) - (9*d^2*e*x^(-3 + r) )/(3 - r)^2 - (9*d*e^2*x^(-3 + 2*r))/(3 - 2*r)^2))/3 - (d^3*(a + b*Log[c*x ^n]))/(3*x^3) - (e^3*(a + b*Log[c*x^n]))/(3*(1 - r)*x^(3*(1 - r))) - (3*d^ 2*e*x^(-3 + r)*(a + b*Log[c*x^n]))/(3 - r) - (3*d*e^2*x^(-3 + 2*r)*(a + b* Log[c*x^n]))/(3 - 2*r)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
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. \(1043\) vs. \(2(183)=366\).
Time = 3.42 (sec) , antiderivative size = 1044, normalized size of antiderivative = 5.47
method | result | size |
parallelrisch | \(\text {Expression too large to display}\) | \(1044\) |
risch | \(\text {Expression too large to display}\) | \(4027\) |
Input:
int((d+e*x^r)^3*(a+b*ln(c*x^n))/x^4,x,method=_RETURNVERBOSE)
Output:
-1/9*(243*a*d^3+120*a*e^3*r^4*(x^r)^3+81*b*e^3*n*(x^r)^3-459*a*e^3*r^3*(x^ r)^3+837*a*e^3*r^2*(x^r)^3-729*a*e^3*r*(x^r)^3-12*a*e^3*r^5*(x^r)^3-540*b* d^2*e*n*r^3*x^r+243*b*ln(c*x^n)*d^3+243*e^3*(x^r)^3*a+243*e^3*(x^r)^3*b*ln (c*x^n)+729*d^2*e*x^r*a+729*d*e^2*(x^r)^2*a+12*a*d^3*r^6-132*a*d^3*r^5+579 *a*d^3*r^4-1836*a*d*e^2*r^3*(x^r)^2+3078*a*d*e^2*r^2*(x^r)^2-2430*a*d*e^2* r*(x^r)^2+729*d^2*e*x^r*b*ln(c*x^n)+729*d*e^2*(x^r)^2*b*ln(c*x^n)+4*b*d^3* n*r^6-44*b*d^3*n*r^5+193*b*d^3*n*r^4-432*b*d^3*n*r^3+522*b*d^3*n*r^2-324*b *d^3*n*r-1296*a*d^3*r^3+1566*a*d^3*r^2-972*a*d^3*r-2619*a*d^2*e*r^3*x^r+38 07*a*d^2*e*r^2*x^r-2673*a*d^2*e*r*x^r+117*b*e^3*n*r^2*(x^r)^3+108*b*d^2*e* n*r^4*x^r+594*b*d*e^2*n*r^2*(x^r)^2+999*b*d^2*e*n*r^2*x^r-648*b*d*e^2*n*r* (x^r)^2-810*b*d^2*e*n*r*x^r+27*b*d*e^2*n*r^4*(x^r)^2-216*b*d*e^2*n*r^3*(x^ r)^2+81*b*d^3*n+12*ln(c*x^n)*b*d^3*r^6-132*ln(c*x^n)*b*d^3*r^5+579*ln(c*x^ n)*b*d^3*r^4-1296*ln(c*x^n)*b*d^3*r^3+1566*ln(c*x^n)*b*d^3*r^2-972*ln(c*x^ n)*b*d^3*r-162*b*e^3*n*r*(x^r)^3+243*b*d*e^2*n*(x^r)^2+243*b*d^2*e*n*x^r+4 *b*e^3*n*r^4*(x^r)^3-36*b*e^3*n*r^3*(x^r)^3-54*a*d*e^2*r^5*(x^r)^2+513*a*d *e^2*r^4*(x^r)^2-108*a*d^2*e*r^5*x^r+864*a*d^2*e*r^4*x^r-108*x^r*ln(c*x^n) *b*d^2*e*r^5+864*x^r*ln(c*x^n)*b*d^2*e*r^4-2619*x^r*ln(c*x^n)*b*d^2*e*r^3+ 3807*x^r*ln(c*x^n)*b*d^2*e*r^2-2673*x^r*ln(c*x^n)*b*d^2*e*r-729*(x^r)^3*ln (c*x^n)*b*e^3*r-12*(x^r)^3*ln(c*x^n)*b*e^3*r^5+120*(x^r)^3*ln(c*x^n)*b*e^3 *r^4-459*(x^r)^3*ln(c*x^n)*b*e^3*r^3+837*(x^r)^3*ln(c*x^n)*b*e^3*r^2-54...
Leaf count of result is larger than twice the leaf count of optimal. 980 vs. \(2 (175) = 350\).
Time = 0.09 (sec) , antiderivative size = 980, normalized size of antiderivative = 5.13 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx =\text {Too large to display} \] Input:
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^4,x, algorithm="fricas")
Output:
-1/9*(4*(b*d^3*n + 3*a*d^3)*r^6 - 44*(b*d^3*n + 3*a*d^3)*r^5 + 81*b*d^3*n + 193*(b*d^3*n + 3*a*d^3)*r^4 + 243*a*d^3 - 432*(b*d^3*n + 3*a*d^3)*r^3 + 522*(b*d^3*n + 3*a*d^3)*r^2 - 324*(b*d^3*n + 3*a*d^3)*r - (12*a*e^3*r^5 - 81*b*e^3*n - 4*(b*e^3*n + 30*a*e^3)*r^4 - 243*a*e^3 + 9*(4*b*e^3*n + 51*a* e^3)*r^3 - 9*(13*b*e^3*n + 93*a*e^3)*r^2 + 81*(2*b*e^3*n + 9*a*e^3)*r + 3* (4*b*e^3*r^5 - 40*b*e^3*r^4 + 153*b*e^3*r^3 - 279*b*e^3*r^2 + 243*b*e^3*r - 81*b*e^3)*log(c) + 3*(4*b*e^3*n*r^5 - 40*b*e^3*n*r^4 + 153*b*e^3*n*r^3 - 279*b*e^3*n*r^2 + 243*b*e^3*n*r - 81*b*e^3*n)*log(x))*x^(3*r) - 27*(2*a*d *e^2*r^5 - 9*b*d*e^2*n - (b*d*e^2*n + 19*a*d*e^2)*r^4 - 27*a*d*e^2 + 4*(2* b*d*e^2*n + 17*a*d*e^2)*r^3 - 2*(11*b*d*e^2*n + 57*a*d*e^2)*r^2 + 6*(4*b*d *e^2*n + 15*a*d*e^2)*r + (2*b*d*e^2*r^5 - 19*b*d*e^2*r^4 + 68*b*d*e^2*r^3 - 114*b*d*e^2*r^2 + 90*b*d*e^2*r - 27*b*d*e^2)*log(c) + (2*b*d*e^2*n*r^5 - 19*b*d*e^2*n*r^4 + 68*b*d*e^2*n*r^3 - 114*b*d*e^2*n*r^2 + 90*b*d*e^2*n*r - 27*b*d*e^2*n)*log(x))*x^(2*r) - 27*(4*a*d^2*e*r^5 - 9*b*d^2*e*n - 4*(b*d ^2*e*n + 8*a*d^2*e)*r^4 - 27*a*d^2*e + (20*b*d^2*e*n + 97*a*d^2*e)*r^3 - ( 37*b*d^2*e*n + 141*a*d^2*e)*r^2 + 3*(10*b*d^2*e*n + 33*a*d^2*e)*r + (4*b*d ^2*e*r^5 - 32*b*d^2*e*r^4 + 97*b*d^2*e*r^3 - 141*b*d^2*e*r^2 + 99*b*d^2*e* r - 27*b*d^2*e)*log(c) + (4*b*d^2*e*n*r^5 - 32*b*d^2*e*n*r^4 + 97*b*d^2*e* n*r^3 - 141*b*d^2*e*n*r^2 + 99*b*d^2*e*n*r - 27*b*d^2*e*n)*log(x))*x^r + 3 *(4*b*d^3*r^6 - 44*b*d^3*r^5 + 193*b*d^3*r^4 - 432*b*d^3*r^3 + 522*b*d^...
Time = 48.33 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.82 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx =\text {Too large to display} \] Input:
integrate((d+e*x**r)**3*(a+b*ln(c*x**n))/x**4,x)
Output:
-a*d**3/(3*x**3) + 3*a*d**2*e*Piecewise((x**r/(r*x**3 - 3*x**3), Ne(r, 3)) , (x**r*log(x)/x**3, True)) + 3*a*d*e**2*Piecewise((x**(2*r)/(2*r*x**3 - 3 *x**3), Ne(r, 3/2)), (x**(2*r)*log(x)/x**3, True)) + a*e**3*Piecewise((x** (3*r)/(3*r*x**3 - 3*x**3), Ne(r, 1)), (x**(3*r)*log(x)/x**3, True)) - b*d* *3*n/(9*x**3) - b*d**3*log(c*x**n)/(3*x**3) - 3*b*d**2*e*n*Piecewise((Piec ewise((x**(r - 3)/(r - 3), Ne(r, 3)), (log(x), True))/(r - 3), (r > -oo) & (r < oo) & Ne(r, 3)), (log(x)**2/2, True)) + 3*b*d**2*e*Piecewise((x**(r - 3)/(r - 3), Ne(r, 3)), (log(x), True))*log(c*x**n) - 3*b*d*e**2*n*Piecew ise((Piecewise((x**(2*r - 3)/(2*r - 3), Ne(r, 3/2)), (log(x), True))/(2*r - 3), (r > -oo) & (r < oo) & Ne(r, 3/2)), (log(x)**2/2, True)) + 3*b*d*e** 2*Piecewise((x**(2*r - 3)/(2*r - 3), Ne(r, 3/2)), (log(x), True))*log(c*x* *n) - b*e**3*n*Piecewise((Piecewise((x**(3*r - 3)/(3*r - 3), Ne(r, 1)), (l og(x), True))/(3*r - 3), (r > -oo) & (r < oo) & Ne(r, 1)), (log(x)**2/2, T rue)) + b*e**3*Piecewise((x**(3*r - 3)/(3*r - 3), Ne(r, 1)), (log(x), True ))*log(c*x**n)
Exception generated. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^4,x, algorithm="maxima")
Output:
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-4>0)', see `assume?` for more details)Is
\[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{4}} \,d x } \] Input:
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^4,x, algorithm="giac")
Output:
integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^4, x)
Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \] Input:
int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^4,x)
Output:
int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^4, x)
Time = 0.16 (sec) , antiderivative size = 1053, normalized size of antiderivative = 5.51 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx =\text {Too large to display} \] Input:
int((d+e*x^r)^3*(a+b*log(c*x^n))/x^4,x)
Output:
(12*x**(3*r)*log(x**n*c)*b*e**3*r**5 - 120*x**(3*r)*log(x**n*c)*b*e**3*r** 4 + 459*x**(3*r)*log(x**n*c)*b*e**3*r**3 - 837*x**(3*r)*log(x**n*c)*b*e**3 *r**2 + 729*x**(3*r)*log(x**n*c)*b*e**3*r - 243*x**(3*r)*log(x**n*c)*b*e** 3 + 12*x**(3*r)*a*e**3*r**5 - 120*x**(3*r)*a*e**3*r**4 + 459*x**(3*r)*a*e* *3*r**3 - 837*x**(3*r)*a*e**3*r**2 + 729*x**(3*r)*a*e**3*r - 243*x**(3*r)* a*e**3 - 4*x**(3*r)*b*e**3*n*r**4 + 36*x**(3*r)*b*e**3*n*r**3 - 117*x**(3* r)*b*e**3*n*r**2 + 162*x**(3*r)*b*e**3*n*r - 81*x**(3*r)*b*e**3*n + 54*x** (2*r)*log(x**n*c)*b*d*e**2*r**5 - 513*x**(2*r)*log(x**n*c)*b*d*e**2*r**4 + 1836*x**(2*r)*log(x**n*c)*b*d*e**2*r**3 - 3078*x**(2*r)*log(x**n*c)*b*d*e **2*r**2 + 2430*x**(2*r)*log(x**n*c)*b*d*e**2*r - 729*x**(2*r)*log(x**n*c) *b*d*e**2 + 54*x**(2*r)*a*d*e**2*r**5 - 513*x**(2*r)*a*d*e**2*r**4 + 1836* x**(2*r)*a*d*e**2*r**3 - 3078*x**(2*r)*a*d*e**2*r**2 + 2430*x**(2*r)*a*d*e **2*r - 729*x**(2*r)*a*d*e**2 - 27*x**(2*r)*b*d*e**2*n*r**4 + 216*x**(2*r) *b*d*e**2*n*r**3 - 594*x**(2*r)*b*d*e**2*n*r**2 + 648*x**(2*r)*b*d*e**2*n* r - 243*x**(2*r)*b*d*e**2*n + 108*x**r*log(x**n*c)*b*d**2*e*r**5 - 864*x** r*log(x**n*c)*b*d**2*e*r**4 + 2619*x**r*log(x**n*c)*b*d**2*e*r**3 - 3807*x **r*log(x**n*c)*b*d**2*e*r**2 + 2673*x**r*log(x**n*c)*b*d**2*e*r - 729*x** r*log(x**n*c)*b*d**2*e + 108*x**r*a*d**2*e*r**5 - 864*x**r*a*d**2*e*r**4 + 2619*x**r*a*d**2*e*r**3 - 3807*x**r*a*d**2*e*r**2 + 2673*x**r*a*d**2*e*r - 729*x**r*a*d**2*e - 108*x**r*b*d**2*e*n*r**4 + 540*x**r*b*d**2*e*n*r*...