Integrand size = 23, antiderivative size = 179 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {b d^3 n}{x}-\frac {3 b d^2 e n x^{-1+r}}{(1-r)^2}-\frac {3 b d e^2 n x^{-1+2 r}}{(1-2 r)^2}-\frac {b e^3 n x^{-1+3 r}}{(1-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 d^2 e x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac {3 d e^2 x^{-1+2 r} \left (a+b \log \left (c x^n\right )\right )}{1-2 r}-\frac {e^3 x^{-1+3 r} \left (a+b \log \left (c x^n\right )\right )}{1-3 r} \] Output:
-b*d^3*n/x-3*b*d^2*e*n*x^(-1+r)/(1-r)^2-3*b*d*e^2*n*x^(-1+2*r)/(1-2*r)^2-b *e^3*n*x^(-1+3*r)/(1-3*r)^2-d^3*(a+b*ln(c*x^n))/x-3*d^2*e*x^(-1+r)*(a+b*ln (c*x^n))/(1-r)-3*d*e^2*x^(-1+2*r)*(a+b*ln(c*x^n))/(1-2*r)-e^3*x^(-1+3*r)*( a+b*ln(c*x^n))/(1-3*r)
Time = 0.44 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {b n \left (-d^3-\frac {3 d^2 e x^r}{(-1+r)^2}-\frac {3 d e^2 x^{2 r}}{(1-2 r)^2}-\frac {e^3 x^{3 r}}{(1-3 r)^2}\right )+a \left (-d^3+\frac {3 d^2 e x^r}{-1+r}+\frac {3 d e^2 x^{2 r}}{-1+2 r}+\frac {e^3 x^{3 r}}{-1+3 r}\right )+b \left (-d^3+\frac {3 d^2 e x^r}{-1+r}+\frac {3 d e^2 x^{2 r}}{-1+2 r}+\frac {e^3 x^{3 r}}{-1+3 r}\right ) \log \left (c x^n\right )}{x} \] Input:
Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^2,x]
Output:
(b*n*(-d^3 - (3*d^2*e*x^r)/(-1 + r)^2 - (3*d*e^2*x^(2*r))/(1 - 2*r)^2 - (e ^3*x^(3*r))/(1 - 3*r)^2) + a*(-d^3 + (3*d^2*e*x^r)/(-1 + r) + (3*d*e^2*x^( 2*r))/(-1 + 2*r) + (e^3*x^(3*r))/(-1 + 3*r)) + b*(-d^3 + (3*d^2*e*x^r)/(-1 + r) + (3*d*e^2*x^(2*r))/(-1 + 2*r) + (e^3*x^(3*r))/(-1 + 3*r))*Log[c*x^n ])/x
Time = 0.67 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2772, 25, 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.
\(\displaystyle \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle -b n \int -\frac {\frac {3 d^2 e x^r}{1-r}+\frac {3 d e^2 x^{2 r}}{1-2 r}+\frac {e^3 x^{3 r}}{1-3 r}+d^3}{x^2}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 d^2 e x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac {3 d e^2 x^{2 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-2 r}-\frac {e^3 x^{3 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-3 r}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle b n \int \frac {\frac {3 d^2 e x^r}{1-r}+\frac {3 d e^2 x^{2 r}}{1-2 r}+\frac {e^3 x^{3 r}}{1-3 r}+d^3}{x^2}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 d^2 e x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac {3 d e^2 x^{2 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-2 r}-\frac {e^3 x^{3 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-3 r}\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle b n \int \left (-\frac {3 d^2 e x^{r-2}}{r-1}+\frac {3 d e^2 x^{2 (r-1)}}{1-2 r}-\frac {e^3 x^{3 r-2}}{3 r-1}+\frac {d^3}{x^2}\right )dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 d^2 e x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac {3 d e^2 x^{2 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-2 r}-\frac {e^3 x^{3 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-3 r}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 d^2 e x^{r-1} \left (a+b \log \left (c x^n\right )\right )}{1-r}-\frac {3 d e^2 x^{2 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-2 r}-\frac {e^3 x^{3 r-1} \left (a+b \log \left (c x^n\right )\right )}{1-3 r}+b n \left (-\frac {d^3}{x}-\frac {3 d^2 e x^{r-1}}{(1-r)^2}-\frac {3 d e^2 x^{2 r-1}}{(1-2 r)^2}-\frac {e^3 x^{3 r-1}}{(1-3 r)^2}\right )\) |
Input:
Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^2,x]
Output:
b*n*(-(d^3/x) - (3*d^2*e*x^(-1 + r))/(1 - r)^2 - (3*d*e^2*x^(-1 + 2*r))/(1 - 2*r)^2 - (e^3*x^(-1 + 3*r))/(1 - 3*r)^2) - (d^3*(a + b*Log[c*x^n]))/x - (3*d^2*e*x^(-1 + r)*(a + b*Log[c*x^n]))/(1 - r) - (3*d*e^2*x^(-1 + 2*r)*( a + b*Log[c*x^n]))/(1 - 2*r) - (e^3*x^(-1 + 3*r)*(a + b*Log[c*x^n]))/(1 - 3*r)
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. \(1039\) vs. \(2(179)=358\).
Time = 3.39 (sec) , antiderivative size = 1040, normalized size of antiderivative = 5.81
method | result | size |
parallelrisch | \(\text {Expression too large to display}\) | \(1040\) |
risch | \(\text {Expression too large to display}\) | \(4031\) |
Input:
int((d+e*x^r)^3*(a+b*ln(c*x^n))/x^2,x,method=_RETURNVERBOSE)
Output:
-(a*d^3+40*a*e^3*r^4*(x^r)^3+b*e^3*n*(x^r)^3-51*a*e^3*r^3*(x^r)^3+31*a*e^3 *r^2*(x^r)^3-9*a*e^3*r*(x^r)^3-12*a*e^3*r^5*(x^r)^3-180*b*d^2*e*n*r^3*x^r+ b*ln(c*x^n)*d^3+e^3*(x^r)^3*a+e^3*(x^r)^3*b*ln(c*x^n)+3*d^2*e*x^r*a+3*d*e^ 2*(x^r)^2*a+36*a*d^3*r^6-132*a*d^3*r^5+193*a*d^3*r^4-204*a*d*e^2*r^3*(x^r) ^2+114*a*d*e^2*r^2*(x^r)^2-30*a*d*e^2*r*(x^r)^2+3*d^2*e*x^r*b*ln(c*x^n)+3* d*e^2*(x^r)^2*b*ln(c*x^n)+36*b*d^3*n*r^6-132*b*d^3*n*r^5+193*b*d^3*n*r^4-1 44*b*d^3*n*r^3+58*b*d^3*n*r^2-12*b*d^3*n*r-144*a*d^3*r^3+58*a*d^3*r^2-12*a *d^3*r-291*a*d^2*e*r^3*x^r+141*a*d^2*e*r^2*x^r-33*a*d^2*e*r*x^r+13*b*e^3*n *r^2*(x^r)^3+108*b*d^2*e*n*r^4*x^r+66*b*d*e^2*n*r^2*(x^r)^2+111*b*d^2*e*n* r^2*x^r-24*b*d*e^2*n*r*(x^r)^2-30*b*d^2*e*n*r*x^r+27*b*d*e^2*n*r^4*(x^r)^2 -72*b*d*e^2*n*r^3*(x^r)^2+b*d^3*n+36*ln(c*x^n)*b*d^3*r^6-132*ln(c*x^n)*b*d ^3*r^5+193*ln(c*x^n)*b*d^3*r^4-144*ln(c*x^n)*b*d^3*r^3+58*ln(c*x^n)*b*d^3* r^2-12*ln(c*x^n)*b*d^3*r-6*b*e^3*n*r*(x^r)^3+3*b*d*e^2*n*(x^r)^2+3*b*d^2*e *n*x^r+4*b*e^3*n*r^4*(x^r)^3-12*b*e^3*n*r^3*(x^r)^3-54*a*d*e^2*r^5*(x^r)^2 +171*a*d*e^2*r^4*(x^r)^2-108*a*d^2*e*r^5*x^r+288*a*d^2*e*r^4*x^r-108*x^r*l n(c*x^n)*b*d^2*e*r^5+288*x^r*ln(c*x^n)*b*d^2*e*r^4-291*x^r*ln(c*x^n)*b*d^2 *e*r^3+141*x^r*ln(c*x^n)*b*d^2*e*r^2-33*x^r*ln(c*x^n)*b*d^2*e*r-9*(x^r)^3* ln(c*x^n)*b*e^3*r-12*(x^r)^3*ln(c*x^n)*b*e^3*r^5+40*(x^r)^3*ln(c*x^n)*b*e^ 3*r^4-51*(x^r)^3*ln(c*x^n)*b*e^3*r^3+31*(x^r)^3*ln(c*x^n)*b*e^3*r^2-54*(x^ r)^2*ln(c*x^n)*b*d*e^2*r^5+171*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-204*(x^r)^...
Leaf count of result is larger than twice the leaf count of optimal. 967 vs. \(2 (174) = 348\).
Time = 0.10 (sec) , antiderivative size = 967, normalized size of antiderivative = 5.40 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx =\text {Too large to display} \] Input:
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^2,x, algorithm="fricas")
Output:
-(36*(b*d^3*n + a*d^3)*r^6 - 132*(b*d^3*n + a*d^3)*r^5 + b*d^3*n + 193*(b* d^3*n + a*d^3)*r^4 + a*d^3 - 144*(b*d^3*n + a*d^3)*r^3 + 58*(b*d^3*n + a*d ^3)*r^2 - 12*(b*d^3*n + a*d^3)*r - (12*a*e^3*r^5 - b*e^3*n - 4*(b*e^3*n + 10*a*e^3)*r^4 - a*e^3 + 3*(4*b*e^3*n + 17*a*e^3)*r^3 - (13*b*e^3*n + 31*a* e^3)*r^2 + 3*(2*b*e^3*n + 3*a*e^3)*r + (12*b*e^3*r^5 - 40*b*e^3*r^4 + 51*b *e^3*r^3 - 31*b*e^3*r^2 + 9*b*e^3*r - b*e^3)*log(c) + (12*b*e^3*n*r^5 - 40 *b*e^3*n*r^4 + 51*b*e^3*n*r^3 - 31*b*e^3*n*r^2 + 9*b*e^3*n*r - b*e^3*n)*lo g(x))*x^(3*r) - 3*(18*a*d*e^2*r^5 - b*d*e^2*n - 3*(3*b*d*e^2*n + 19*a*d*e^ 2)*r^4 - a*d*e^2 + 4*(6*b*d*e^2*n + 17*a*d*e^2)*r^3 - 2*(11*b*d*e^2*n + 19 *a*d*e^2)*r^2 + 2*(4*b*d*e^2*n + 5*a*d*e^2)*r + (18*b*d*e^2*r^5 - 57*b*d*e ^2*r^4 + 68*b*d*e^2*r^3 - 38*b*d*e^2*r^2 + 10*b*d*e^2*r - b*d*e^2)*log(c) + (18*b*d*e^2*n*r^5 - 57*b*d*e^2*n*r^4 + 68*b*d*e^2*n*r^3 - 38*b*d*e^2*n*r ^2 + 10*b*d*e^2*n*r - b*d*e^2*n)*log(x))*x^(2*r) - 3*(36*a*d^2*e*r^5 - b*d ^2*e*n - 12*(3*b*d^2*e*n + 8*a*d^2*e)*r^4 - a*d^2*e + (60*b*d^2*e*n + 97*a *d^2*e)*r^3 - (37*b*d^2*e*n + 47*a*d^2*e)*r^2 + (10*b*d^2*e*n + 11*a*d^2*e )*r + (36*b*d^2*e*r^5 - 96*b*d^2*e*r^4 + 97*b*d^2*e*r^3 - 47*b*d^2*e*r^2 + 11*b*d^2*e*r - b*d^2*e)*log(c) + (36*b*d^2*e*n*r^5 - 96*b*d^2*e*n*r^4 + 9 7*b*d^2*e*n*r^3 - 47*b*d^2*e*n*r^2 + 11*b*d^2*e*n*r - b*d^2*e*n)*log(x))*x ^r + (36*b*d^3*r^6 - 132*b*d^3*r^5 + 193*b*d^3*r^4 - 144*b*d^3*r^3 + 58*b* d^3*r^2 - 12*b*d^3*r + b*d^3)*log(c) + (36*b*d^3*n*r^6 - 132*b*d^3*n*r^...
Time = 10.83 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.80 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx =\text {Too large to display} \] Input:
integrate((d+e*x**r)**3*(a+b*ln(c*x**n))/x**2,x)
Output:
-a*d**3/x + 3*a*d**2*e*Piecewise((x**r/(r*x - x), Ne(r, 1)), (x**r*log(x)/ x, True)) + 3*a*d*e**2*Piecewise((x**(2*r)/(2*r*x - x), Ne(r, 1/2)), (x**( 2*r)*log(x)/x, True)) + a*e**3*Piecewise((x**(3*r)/(3*r*x - x), Ne(r, 1/3) ), (x**(3*r)*log(x)/x, True)) - b*d**3*n/x - b*d**3*log(c*x**n)/x - 3*b*d* *2*e*n*Piecewise((Piecewise((x**(r - 1)/(r - 1), Ne(r, 1)), (log(x), True) )/(r - 1), (r > -oo) & (r < oo) & Ne(r, 1)), (log(x)**2/2, True)) + 3*b*d* *2*e*Piecewise((x**(r - 1)/(r - 1), Ne(r, 1)), (log(x), True))*log(c*x**n) - 3*b*d*e**2*n*Piecewise((Piecewise((x**(2*r - 1)/(2*r - 1), Ne(r, 1/2)), (log(x), True))/(2*r - 1), (r > -oo) & (r < oo) & Ne(r, 1/2)), (log(x)**2 /2, True)) + 3*b*d*e**2*Piecewise((x**(2*r - 1)/(2*r - 1), Ne(r, 1/2)), (l og(x), True))*log(c*x**n) - b*e**3*n*Piecewise((Piecewise((x**(3*r - 1)/(3 *r - 1), Ne(r, 1/3)), (log(x), True))/(3*r - 1), (r > -oo) & (r < oo) & Ne (r, 1/3)), (log(x)**2/2, True)) + b*e**3*Piecewise((x**(3*r - 1)/(3*r - 1) , Ne(r, 1/3)), (log(x), True))*log(c*x**n)
Exception generated. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^2,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-2>0)', see `assume?` for more details)Is
\[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}} \,d x } \] Input:
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^2,x, algorithm="giac")
Output:
integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^2, x)
Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \] Input:
int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^2,x)
Output:
int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^2, x)
Time = 0.15 (sec) , antiderivative size = 1052, normalized size of antiderivative = 5.88 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx =\text {Too large to display} \] Input:
int((d+e*x^r)^3*(a+b*log(c*x^n))/x^2,x)
Output:
(12*x**(3*r)*log(x**n*c)*b*e**3*r**5 - 40*x**(3*r)*log(x**n*c)*b*e**3*r**4 + 51*x**(3*r)*log(x**n*c)*b*e**3*r**3 - 31*x**(3*r)*log(x**n*c)*b*e**3*r* *2 + 9*x**(3*r)*log(x**n*c)*b*e**3*r - x**(3*r)*log(x**n*c)*b*e**3 + 12*x* *(3*r)*a*e**3*r**5 - 40*x**(3*r)*a*e**3*r**4 + 51*x**(3*r)*a*e**3*r**3 - 3 1*x**(3*r)*a*e**3*r**2 + 9*x**(3*r)*a*e**3*r - x**(3*r)*a*e**3 - 4*x**(3*r )*b*e**3*n*r**4 + 12*x**(3*r)*b*e**3*n*r**3 - 13*x**(3*r)*b*e**3*n*r**2 + 6*x**(3*r)*b*e**3*n*r - x**(3*r)*b*e**3*n + 54*x**(2*r)*log(x**n*c)*b*d*e* *2*r**5 - 171*x**(2*r)*log(x**n*c)*b*d*e**2*r**4 + 204*x**(2*r)*log(x**n*c )*b*d*e**2*r**3 - 114*x**(2*r)*log(x**n*c)*b*d*e**2*r**2 + 30*x**(2*r)*log (x**n*c)*b*d*e**2*r - 3*x**(2*r)*log(x**n*c)*b*d*e**2 + 54*x**(2*r)*a*d*e* *2*r**5 - 171*x**(2*r)*a*d*e**2*r**4 + 204*x**(2*r)*a*d*e**2*r**3 - 114*x* *(2*r)*a*d*e**2*r**2 + 30*x**(2*r)*a*d*e**2*r - 3*x**(2*r)*a*d*e**2 - 27*x **(2*r)*b*d*e**2*n*r**4 + 72*x**(2*r)*b*d*e**2*n*r**3 - 66*x**(2*r)*b*d*e* *2*n*r**2 + 24*x**(2*r)*b*d*e**2*n*r - 3*x**(2*r)*b*d*e**2*n + 108*x**r*lo g(x**n*c)*b*d**2*e*r**5 - 288*x**r*log(x**n*c)*b*d**2*e*r**4 + 291*x**r*lo g(x**n*c)*b*d**2*e*r**3 - 141*x**r*log(x**n*c)*b*d**2*e*r**2 + 33*x**r*log (x**n*c)*b*d**2*e*r - 3*x**r*log(x**n*c)*b*d**2*e + 108*x**r*a*d**2*e*r**5 - 288*x**r*a*d**2*e*r**4 + 291*x**r*a*d**2*e*r**3 - 141*x**r*a*d**2*e*r** 2 + 33*x**r*a*d**2*e*r - 3*x**r*a*d**2*e - 108*x**r*b*d**2*e*n*r**4 + 180* x**r*b*d**2*e*n*r**3 - 111*x**r*b*d**2*e*n*r**2 + 30*x**r*b*d**2*e*n*r ...