3.405 \(\int \frac{(d+e x^r)^3 (a+b \log (c x^n))}{x^{10}} \, dx\)
Optimal. Leaf size=191 \[ -\frac{3 d^2 e x^{r-9} \left (a+b \log \left (c x^n\right )\right )}{9-r}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac{3 d e^2 x^{2 r-9} \left (a+b \log \left (c x^n\right )\right )}{9-2 r}-\frac{e^3 x^{-3 (3-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (3-r)}-\frac{3 b d^2 e n x^{r-9}}{(9-r)^2}-\frac{b d^3 n}{81 x^9}-\frac{3 b d e^2 n x^{2 r-9}}{(9-2 r)^2}-\frac{b e^3 n x^{-3 (3-r)}}{9 (3-r)^2} \]
[Out]
-(b*d^3*n)/(81*x^9) - (b*e^3*n)/(9*(3 - r)^2*x^(3*(3 - r))) - (3*b*d^2*e*n*x^(-9 + r))/(9 - r)^2 - (3*b*d*e^2*
n*x^(-9 + 2*r))/(9 - 2*r)^2 - (d^3*(a + b*Log[c*x^n]))/(9*x^9) - (e^3*(a + b*Log[c*x^n]))/(3*(3 - r)*x^(3*(3 -
r))) - (3*d^2*e*x^(-9 + r)*(a + b*Log[c*x^n]))/(9 - r) - (3*d*e^2*x^(-9 + 2*r)*(a + b*Log[c*x^n]))/(9 - 2*r)
________________________________________________________________________________________
Rubi [A] time = 0.418994, antiderivative size = 161, normalized size of antiderivative =
0.84, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules
used = {270, 2334, 12, 14} \[ -\frac{1}{9} \left (\frac{27 d^2 e x^{r-9}}{9-r}+\frac{d^3}{x^9}+\frac{27 d e^2 x^{2 r-9}}{9-2 r}+\frac{3 e^3 x^{-3 (3-r)}}{3-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r-9}}{(9-r)^2}-\frac{b d^3 n}{81 x^9}-\frac{3 b d e^2 n x^{2 r-9}}{(9-2 r)^2}-\frac{b e^3 n x^{-3 (3-r)}}{9 (3-r)^2} \]
Antiderivative was successfully verified.
[In]
Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^10,x]
[Out]
-(b*d^3*n)/(81*x^9) - (b*e^3*n)/(9*(3 - r)^2*x^(3*(3 - r))) - (3*b*d^2*e*n*x^(-9 + r))/(9 - r)^2 - (3*b*d*e^2*
n*x^(-9 + 2*r))/(9 - 2*r)^2 - ((d^3/x^9 + (3*e^3)/((3 - r)*x^(3*(3 - r))) + (27*d^2*e*x^(-9 + r))/(9 - r) + (2
7*d*e^2*x^(-9 + 2*r))/(9 - 2*r))*(a + b*Log[c*x^n]))/9
Rule 270
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Rule 2334
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[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])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 14
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]]
Rubi steps
\begin{align*} \int \frac{\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx &=-\frac{1}{9} \left (\frac{d^3}{x^9}+\frac{3 e^3 x^{-3 (3-r)}}{3-r}+\frac{27 d^2 e x^{-9+r}}{9-r}+\frac{27 d e^2 x^{-9+2 r}}{9-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-d^3+\frac{27 d^2 e x^r}{-9+r}+\frac{27 d e^2 x^{2 r}}{-9+2 r}+\frac{3 e^3 x^{3 r}}{-3+r}}{9 x^{10}} \, dx\\ &=-\frac{1}{9} \left (\frac{d^3}{x^9}+\frac{3 e^3 x^{-3 (3-r)}}{3-r}+\frac{27 d^2 e x^{-9+r}}{9-r}+\frac{27 d e^2 x^{-9+2 r}}{9-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} (b n) \int \frac{-d^3+\frac{27 d^2 e x^r}{-9+r}+\frac{27 d e^2 x^{2 r}}{-9+2 r}+\frac{3 e^3 x^{3 r}}{-3+r}}{x^{10}} \, dx\\ &=-\frac{1}{9} \left (\frac{d^3}{x^9}+\frac{3 e^3 x^{-3 (3-r)}}{3-r}+\frac{27 d^2 e x^{-9+r}}{9-r}+\frac{27 d e^2 x^{-9+2 r}}{9-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} (b n) \int \left (-\frac{d^3}{x^{10}}+\frac{27 d^2 e x^{-10+r}}{-9+r}+\frac{27 d e^2 x^{2 (-5+r)}}{-9+2 r}+\frac{3 e^3 x^{-10+3 r}}{-3+r}\right ) \, dx\\ &=-\frac{b d^3 n}{81 x^9}-\frac{b e^3 n x^{-3 (3-r)}}{9 (3-r)^2}-\frac{3 b d^2 e n x^{-9+r}}{(9-r)^2}-\frac{3 b d e^2 n x^{-9+2 r}}{(9-2 r)^2}-\frac{1}{9} \left (\frac{d^3}{x^9}+\frac{3 e^3 x^{-3 (3-r)}}{3-r}+\frac{27 d^2 e x^{-9+r}}{9-r}+\frac{27 d e^2 x^{-9+2 r}}{9-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.389352, size = 182, normalized size = 0.95 \[ \frac{9 a \left (\frac{27 d^2 e x^r}{r-9}-d^3+\frac{27 d e^2 x^{2 r}}{2 r-9}+\frac{3 e^3 x^{3 r}}{r-3}\right )+9 b \log \left (c x^n\right ) \left (\frac{27 d^2 e x^r}{r-9}-d^3+\frac{27 d e^2 x^{2 r}}{2 r-9}+\frac{3 e^3 x^{3 r}}{r-3}\right )+b n \left (-\frac{243 d^2 e x^r}{(r-9)^2}-d^3-\frac{243 d e^2 x^{2 r}}{(9-2 r)^2}-\frac{9 e^3 x^{3 r}}{(r-3)^2}\right )}{81 x^9} \]
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^10,x]
[Out]
(b*n*(-d^3 - (243*d^2*e*x^r)/(-9 + r)^2 - (243*d*e^2*x^(2*r))/(9 - 2*r)^2 - (9*e^3*x^(3*r))/(-3 + r)^2) + 9*a*
(-d^3 + (27*d^2*e*x^r)/(-9 + r) + (27*d*e^2*x^(2*r))/(-9 + 2*r) + (3*e^3*x^(3*r))/(-3 + r)) + 9*b*(-d^3 + (27*
d^2*e*x^r)/(-9 + r) + (27*d*e^2*x^(2*r))/(-9 + 2*r) + (3*e^3*x^(3*r))/(-3 + r))*Log[c*x^n])/(81*x^9)
________________________________________________________________________________________
Maple [C] time = 0.372, size = 4027, normalized size = 21.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d+e*x^r)^3*(a+b*ln(c*x^n))/x^10,x)
[Out]
-1/9*b*(-6*e^3*r^2*(x^r)^3-27*d*e^2*r^2*(x^r)^2+81*e^3*r*(x^r)^3+2*d^3*r^3-54*d^2*e*r^2*x^r+324*d*e^2*r*(x^r)^
2-243*e^3*(x^r)^3-33*d^3*r^2+405*d^2*e*r*x^r-729*d*e^2*(x^r)^2+162*d^3*r-729*d^2*e*x^r-243*d^3)/x^9/(-3+r)/(-9
+2*r)/(-9+r)*ln(x^n)-1/162*(1062882*a*d^3+212139*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+1062
882*a*e^3*(x^r)^3+1062882*ln(c)*b*d^3+531441*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x^r)^3-1594323*I*Pi*b*d*e^2*csgn(I*
c*x^n)^3*(x^r)^2+108*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3-3240*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3+8*b*d^
3*n*r^6-264*b*d^3*n*r^5+3474*b*d^3*n*r^4+72*a*d^3*r^6-2376*a*d^3*r^5+31266*a*d^3*r^4-216*a*e^3*r^5*(x^r)^3+648
0*a*e^3*r^4*(x^r)^3+3188646*a*d*e^2*(x^r)^2+3188646*a*d^2*e*x^r+118098*b*e^3*n*(x^r)^3-74358*a*e^3*r^3*(x^r)^3
+406782*a*e^3*r^2*(x^r)^3-1062882*a*e^3*r*(x^r)^3+1062882*ln(c)*b*e^3*(x^r)^3-531441*I*Pi*b*e^3*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)*(x^r)^3+1948617*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r+1594323*I*Pi*b*d*e^2*csgn(I*x^n)*csgn
(I*c*x^n)^2*(x^r)^2-531441*I*Pi*b*d^3*csgn(I*c*x^n)^3-23328*b*d^3*n*r^3+84564*b*d^3*n*r^2-157464*b*d^3*n*r+72*
ln(c)*b*d^3*r^6-2376*ln(c)*b*d^3*r^5+31266*ln(c)*b*d^3*r^4-209952*ln(c)*b*d^3*r^3+761076*ln(c)*b*d^3*r^2-14171
76*ln(c)*b*d^3*r-209952*a*d^3*r^3+761076*a*d^3*r^2-1417176*a*d^3*r-3542940*a*d*e^2*r*(x^r)^2-424278*a*d^2*e*r^
3*x^r+1850202*a*d^2*e*r^2*x^r-3897234*a*d^2*e*r*x^r+118098*b*d^3*n+6480*ln(c)*b*e^3*r^4*(x^r)^3-74358*ln(c)*b*
e^3*r^3*(x^r)^3+406782*ln(c)*b*e^3*r^2*(x^r)^3-1062882*ln(c)*b*e^3*r*(x^r)^3+3188646*ln(c)*b*d^2*e*x^r+3188646
*ln(c)*b*d*e^2*(x^r)^2+18954*b*e^3*n*r^2*(x^r)^3-78732*b*e^3*n*r*(x^r)^3+354294*b*d*e^2*n*(x^r)^2+354294*b*d^2
*e*n*x^r-297432*a*d*e^2*r^3*(x^r)^2+1495908*a*d*e^2*r^2*(x^r)^2+72*b*e^3*n*r^4*(x^r)^3-1944*b*e^3*n*r^3*(x^r)^
3-972*a*d*e^2*r^5*(x^r)^2+27702*a*d*e^2*r^4*(x^r)^2-1944*a*d^2*e*r^5*x^r+46656*a*d^2*e*r^4*x^r-216*ln(c)*b*e^3
*r^5*(x^r)^3+212139*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r-747954*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-53144
1*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+1188*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+962
28*b*d*e^2*n*r^2*(x^r)^2+161838*b*d^2*e*n*r^2*x^r+203391*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+1487
16*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2-108*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+104976*I*Pi*b
*d^3*r^3*csgn(I*c*x^n)^3-380538*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3+708588*I*Pi*b*d^3*r*csgn(I*c*x^n)^3-531441*I*Pi
*b*e^3*csgn(I*c*x^n)^3*(x^r)^3-148716*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+925101*I*Pi*b*d^2*e
*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+925101*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r-747954*I*Pi*b*d*e^2
*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-925101*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x
^r+486*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+972*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)*x^r+148716*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-37179*I*Pi*b*e^3*r^3*cs
gn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-108*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+3240*I*Pi*b*e^3*r^4*csg
n(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+13851*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+108*I*Pi*b*e^3*r^5*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-203391*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-2
12139*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+3240*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-3
6*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+531441*I*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-31492
8*b*d*e^2*n*r*(x^r)^2-393660*b*d^2*e*n*r*x^r+486*b*d*e^2*n*r^4*(x^r)^2-11664*b*d*e^2*n*r^3*(x^r)^2+1944*b*d^2*
e*n*r^4*x^r-29160*b*d^2*e*n*r^3*x^r-972*ln(c)*b*d*e^2*r^5*(x^r)^2+27702*ln(c)*b*d*e^2*r^4*(x^r)^2-1944*ln(c)*b
*d^2*e*r^5*x^r+46656*ln(c)*b*d^2*e*r^4*x^r-15633*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+708588*I*P
i*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+104976*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-104976
*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2+36*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2-104976*I*Pi*b*d^3*r^
3*csgn(I*c*x^n)^2*csgn(I*c)-1188*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2+1771470*I*Pi*b*d*e^2*r*csgn(I*x^n)
*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+1948617*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-13851*I*Pi*b*d
*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-424278*ln(c)*b*d^2*e*r^3*x^r+1850202*ln(c)*b*d^2*e*r^2*x^
r-3897234*ln(c)*b*d^2*e*r*x^r-297432*ln(c)*b*d*e^2*r^3*(x^r)^2+1495908*ln(c)*b*d*e^2*r^2*(x^r)^2-3542940*ln(c)
*b*d*e^2*r*(x^r)^2-486*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-3240*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn
(I*c*x^n)*csgn(I*c)*(x^r)^3-972*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^2*csgn(I*c)*x^r+531441*I*Pi*b*d^3*csgn(I*c*x^n)
^2*csgn(I*c)+1188*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3-15633*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^3+531441*I*Pi*b*d^3*csgn(I
*x^n)*csgn(I*c*x^n)^2-531441*I*Pi*b*e^3*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-925101*I*Pi*b*d^2*e*r^2*csgn(I*c*x
^n)^3*x^r+1771470*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-23328*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)*x^r-23328*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r+203391*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3
-1594323*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+23328*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)
^2*x^r-148716*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-1771470*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x
^n)^2*(x^r)^2-1771470*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-1948617*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn
(I*c*x^n)^2*x^r-1594323*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+37179*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+1594323*
I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+1594323*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+1594323*I*
Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-37179*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-1188*I*Pi*b*
d^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)+15633*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+36*I*Pi*b*d^3*r^6*csgn(I*c*
x^n)^2*csgn(I*c)+15633*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)+13851*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^
n)^2*(x^r)^2+747954*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+486*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*
(x^r)^2-380538*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+972*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r-138
51*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2-531441*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-203391*I*Pi*
b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3+531441*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-708588*I*Pi*b*d^3*r*cs
gn(I*c*x^n)^2*csgn(I*c)+380538*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+380538*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^
2*csgn(I*c)-708588*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2+23328*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^2*csgn(I*c)*x
^r-486*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-1948617*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x
^r-1594323*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-36*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3-972*I*Pi
*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-212139*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r+747954*I*Pi
*b*d*e^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+531441*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3
+37179*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3)/(-3+r)^2/x^9/(-9+2*r)^2/(-9+r)^2
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^10,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 1.4512, size = 2476, normalized size = 12.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^10,x, algorithm="fricas")
[Out]
-1/81*(4*(b*d^3*n + 9*a*d^3)*r^6 - 132*(b*d^3*n + 9*a*d^3)*r^5 + 59049*b*d^3*n + 1737*(b*d^3*n + 9*a*d^3)*r^4
+ 531441*a*d^3 - 11664*(b*d^3*n + 9*a*d^3)*r^3 + 42282*(b*d^3*n + 9*a*d^3)*r^2 - 78732*(b*d^3*n + 9*a*d^3)*r -
9*(12*a*e^3*r^5 - 6561*b*e^3*n - 4*(b*e^3*n + 90*a*e^3)*r^4 - 59049*a*e^3 + 27*(4*b*e^3*n + 153*a*e^3)*r^3 -
81*(13*b*e^3*n + 279*a*e^3)*r^2 + 2187*(2*b*e^3*n + 27*a*e^3)*r + 3*(4*b*e^3*r^5 - 120*b*e^3*r^4 + 1377*b*e^3*
r^3 - 7533*b*e^3*r^2 + 19683*b*e^3*r - 19683*b*e^3)*log(c) + 3*(4*b*e^3*n*r^5 - 120*b*e^3*n*r^4 + 1377*b*e^3*n
*r^3 - 7533*b*e^3*n*r^2 + 19683*b*e^3*n*r - 19683*b*e^3*n)*log(x))*x^(3*r) - 243*(2*a*d*e^2*r^5 - 729*b*d*e^2*
n - (b*d*e^2*n + 57*a*d*e^2)*r^4 - 6561*a*d*e^2 + 12*(2*b*d*e^2*n + 51*a*d*e^2)*r^3 - 18*(11*b*d*e^2*n + 171*a
*d*e^2)*r^2 + 162*(4*b*d*e^2*n + 45*a*d*e^2)*r + (2*b*d*e^2*r^5 - 57*b*d*e^2*r^4 + 612*b*d*e^2*r^3 - 3078*b*d*
e^2*r^2 + 7290*b*d*e^2*r - 6561*b*d*e^2)*log(c) + (2*b*d*e^2*n*r^5 - 57*b*d*e^2*n*r^4 + 612*b*d*e^2*n*r^3 - 30
78*b*d*e^2*n*r^2 + 7290*b*d*e^2*n*r - 6561*b*d*e^2*n)*log(x))*x^(2*r) - 243*(4*a*d^2*e*r^5 - 729*b*d^2*e*n - 4
*(b*d^2*e*n + 24*a*d^2*e)*r^4 - 6561*a*d^2*e + 3*(20*b*d^2*e*n + 291*a*d^2*e)*r^3 - 9*(37*b*d^2*e*n + 423*a*d^
2*e)*r^2 + 81*(10*b*d^2*e*n + 99*a*d^2*e)*r + (4*b*d^2*e*r^5 - 96*b*d^2*e*r^4 + 873*b*d^2*e*r^3 - 3807*b*d^2*e
*r^2 + 8019*b*d^2*e*r - 6561*b*d^2*e)*log(c) + (4*b*d^2*e*n*r^5 - 96*b*d^2*e*n*r^4 + 873*b*d^2*e*n*r^3 - 3807*
b*d^2*e*n*r^2 + 8019*b*d^2*e*n*r - 6561*b*d^2*e*n)*log(x))*x^r + 9*(4*b*d^3*r^6 - 132*b*d^3*r^5 + 1737*b*d^3*r
^4 - 11664*b*d^3*r^3 + 42282*b*d^3*r^2 - 78732*b*d^3*r + 59049*b*d^3)*log(c) + 9*(4*b*d^3*n*r^6 - 132*b*d^3*n*
r^5 + 1737*b*d^3*n*r^4 - 11664*b*d^3*n*r^3 + 42282*b*d^3*n*r^2 - 78732*b*d^3*n*r + 59049*b*d^3*n)*log(x))/((4*
r^6 - 132*r^5 + 1737*r^4 - 11664*r^3 + 42282*r^2 - 78732*r + 59049)*x^9)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d+e*x**r)**3*(a+b*ln(c*x**n))/x**10,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{r} + d\right )}^{3}{\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d+e*x^r)^3*(a+b*log(c*x^n))/x^10,x, algorithm="giac")
[Out]
integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^10, x)