3.396 \(\int \frac{(d+e x^r)^3 (a+b \log (c x^n))}{x^3} \, dx\)
Optimal. Leaf size=191 \[ -\frac{3 d^2 e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{3 d e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}-\frac{e^3 x^{3 r-2} \left (a+b \log \left (c x^n\right )\right )}{2-3 r}-\frac{3 b d^2 e n x^{r-2}}{(2-r)^2}-\frac{b d^3 n}{4 x^2}-\frac{3 b d e^2 n x^{-2 (1-r)}}{4 (1-r)^2}-\frac{b e^3 n x^{3 r-2}}{(2-3 r)^2} \]
[Out]
-(b*d^3*n)/(4*x^2) - (3*b*d*e^2*n)/(4*(1 - r)^2*x^(2*(1 - r))) - (3*b*d^2*e*n*x^(-2 + r))/(2 - r)^2 - (b*e^3*n
*x^(-2 + 3*r))/(2 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/(2*x^2) - (3*d*e^2*(a + b*Log[c*x^n]))/(2*(1 - r)*x^(2*(
1 - r))) - (3*d^2*e*x^(-2 + r)*(a + b*Log[c*x^n]))/(2 - r) - (e^3*x^(-2 + 3*r)*(a + b*Log[c*x^n]))/(2 - 3*r)
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Rubi [A] time = 0.407696, 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} \[ -\frac{1}{2} \left (\frac{6 d^2 e x^{r-2}}{2-r}+\frac{d^3}{x^2}+\frac{3 d e^2 x^{-2 (1-r)}}{1-r}+\frac{2 e^3 x^{3 r-2}}{2-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r-2}}{(2-r)^2}-\frac{b d^3 n}{4 x^2}-\frac{3 b d e^2 n x^{-2 (1-r)}}{4 (1-r)^2}-\frac{b e^3 n x^{3 r-2}}{(2-3 r)^2} \]
Antiderivative was successfully verified.
[In]
Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^3,x]
[Out]
-(b*d^3*n)/(4*x^2) - (3*b*d*e^2*n)/(4*(1 - r)^2*x^(2*(1 - r))) - (3*b*d^2*e*n*x^(-2 + r))/(2 - r)^2 - (b*e^3*n
*x^(-2 + 3*r))/(2 - 3*r)^2 - ((d^3/x^2 + (3*d*e^2)/((1 - r)*x^(2*(1 - r))) + (6*d^2*e*x^(-2 + r))/(2 - r) + (2
*e^3*x^(-2 + 3*r))/(2 - 3*r))*(a + b*Log[c*x^n]))/2
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^3} \, dx &=-\frac{1}{2} \left (\frac{d^3}{x^2}+\frac{3 d e^2 x^{-2 (1-r)}}{1-r}+\frac{6 d^2 e x^{-2+r}}{2-r}+\frac{2 e^3 x^{-2+3 r}}{2-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-d^3+\frac{6 d^2 e x^r}{-2+r}+\frac{3 d e^2 x^{2 r}}{-1+r}+\frac{2 e^3 x^{3 r}}{-2+3 r}}{2 x^3} \, dx\\ &=-\frac{1}{2} \left (\frac{d^3}{x^2}+\frac{3 d e^2 x^{-2 (1-r)}}{1-r}+\frac{6 d^2 e x^{-2+r}}{2-r}+\frac{2 e^3 x^{-2+3 r}}{2-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int \frac{-d^3+\frac{6 d^2 e x^r}{-2+r}+\frac{3 d e^2 x^{2 r}}{-1+r}+\frac{2 e^3 x^{3 r}}{-2+3 r}}{x^3} \, dx\\ &=-\frac{1}{2} \left (\frac{d^3}{x^2}+\frac{3 d e^2 x^{-2 (1-r)}}{1-r}+\frac{6 d^2 e x^{-2+r}}{2-r}+\frac{2 e^3 x^{-2+3 r}}{2-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int \left (-\frac{d^3}{x^3}+\frac{6 d^2 e x^{-3+r}}{-2+r}+\frac{2 e^3 x^{3 (-1+r)}}{-2+3 r}+\frac{3 d e^2 x^{-3+2 r}}{-1+r}\right ) \, dx\\ &=-\frac{b d^3 n}{4 x^2}-\frac{3 b d e^2 n x^{-2 (1-r)}}{4 (1-r)^2}-\frac{3 b d^2 e n x^{-2+r}}{(2-r)^2}-\frac{b e^3 n x^{-2+3 r}}{(2-3 r)^2}-\frac{1}{2} \left (\frac{d^3}{x^2}+\frac{3 d e^2 x^{-2 (1-r)}}{1-r}+\frac{6 d^2 e x^{-2+r}}{2-r}+\frac{2 e^3 x^{-2+3 r}}{2-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.392199, size = 181, normalized size = 0.95 \[ \frac{a \left (\frac{12 d^2 e x^r}{r-2}-2 d^3+\frac{6 d e^2 x^{2 r}}{r-1}+\frac{4 e^3 x^{3 r}}{3 r-2}\right )+2 b \log \left (c x^n\right ) \left (\frac{6 d^2 e x^r}{r-2}-d^3+\frac{3 d e^2 x^{2 r}}{r-1}+\frac{2 e^3 x^{3 r}}{3 r-2}\right )+b n \left (-\frac{12 d^2 e x^r}{(r-2)^2}-d^3-\frac{3 d e^2 x^{2 r}}{(r-1)^2}-\frac{4 e^3 x^{3 r}}{(2-3 r)^2}\right )}{4 x^2} \]
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^3,x]
[Out]
(b*n*(-d^3 - (12*d^2*e*x^r)/(-2 + r)^2 - (3*d*e^2*x^(2*r))/(-1 + r)^2 - (4*e^3*x^(3*r))/(2 - 3*r)^2) + a*(-2*d
^3 + (12*d^2*e*x^r)/(-2 + r) + (6*d*e^2*x^(2*r))/(-1 + r) + (4*e^3*x^(3*r))/(-2 + 3*r)) + 2*b*(-d^3 + (6*d^2*e
*x^r)/(-2 + r) + (3*d*e^2*x^(2*r))/(-1 + r) + (2*e^3*x^(3*r))/(-2 + 3*r))*Log[c*x^n])/(4*x^2)
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Maple [C] time = 0.339, 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^3,x)
[Out]
-1/2*b*(-2*e^3*r^2*(x^r)^3-9*d*e^2*r^2*(x^r)^2+6*e^3*r*(x^r)^3+3*d^3*r^3-18*d^2*e*r^2*x^r+24*d*e^2*r*(x^r)^2-4
*e^3*(x^r)^3-11*d^3*r^2+30*d^2*e*r*x^r-12*d*e^2*(x^r)^2+12*d^3*r-12*d^2*e*x^r-4*d^3)/x^2/(-2+3*r)/(-1+r)/(-2+r
)*ln(x^n)-1/4*(582*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+32*a*d^3+408*I*Pi*b*d*e^2*r^3*csgn
(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-288*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-288*I*Pi*b*d^3*r^3*csgn
(I*c*x^n)^2*csgn(I*c)+32*a*e^3*(x^r)^3+32*ln(c)*b*d^3-72*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-72*I
*Pi*b*e^3*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+9*b*d^3*n*r^6-66*b*d^3*n*r^5+193*b*d^3*n*r^4+18*a*d^3*r^6-132*a*
d^3*r^5+386*a*d^3*r^4+6*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3-96*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2-96*
I*Pi*b*d^3*r*csgn(I*c*x^n)^2*csgn(I*c)-12*a*e^3*r^5*(x^r)^3+80*a*e^3*r^4*(x^r)^3+96*a*d*e^2*(x^r)^2+96*a*d^2*e
*x^r+16*b*e^3*n*(x^r)^3-204*a*e^3*r^3*(x^r)^3+248*a*e^3*r^2*(x^r)^3-144*a*e^3*r*(x^r)^3+32*ln(c)*b*e^3*(x^r)^3
-288*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-288*b*d^3*n*r^3+232*b*d^3*n*r^2-96*b*d^3*n*r+18*
ln(c)*b*d^3*r^6-132*ln(c)*b*d^3*r^5+386*ln(c)*b*d^3*r^4-576*ln(c)*b*d^3*r^3+464*ln(c)*b*d^3*r^2-192*ln(c)*b*d^
3*r-576*a*d^3*r^3+464*a*d^3*r^2-192*a*d^3*r-480*a*d*e^2*r*(x^r)^2-1164*a*d^2*e*r^3*x^r+1128*a*d^2*e*r^2*x^r-52
8*a*d^2*e*r*x^r+6*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+16*b*d^3*n-54*I*Pi*b*d^2*e*r^5*cs
gn(I*x^n)*csgn(I*c*x^n)^2*x^r+9*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2-124*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*
(x^r)^3+16*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+16*I*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-408*
I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+264*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r-6*I*Pi*b*e^3*r^5*csg
n(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-6*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+27*I*Pi*b*d*e^2*r^5*csgn(I
*c*x^n)^3*(x^r)^2-582*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r-16*I*Pi*b*d^3*csgn(I*c*x^n)^3+80*ln(c)*b*
e^3*r^4*(x^r)^3-204*ln(c)*b*e^3*r^3*(x^r)^3+248*ln(c)*b*e^3*r^2*(x^r)^3-144*ln(c)*b*e^3*r*(x^r)^3+96*ln(c)*b*d
^2*e*x^r+96*ln(c)*b*d*e^2*(x^r)^2+52*b*e^3*n*r^2*(x^r)^3-48*b*e^3*n*r*(x^r)^3+48*b*d*e^2*n*(x^r)^2+48*b*d^2*e*
n*x^r-816*a*d*e^2*r^3*(x^r)^2+912*a*d*e^2*r^2*(x^r)^2+4*b*e^3*n*r^4*(x^r)^3-24*b*e^3*n*r^3*(x^r)^3-54*a*d*e^2*
r^5*(x^r)^2+342*a*d*e^2*r^4*(x^r)^2-108*a*d^2*e*r^5*x^r+576*a*d^2*e*r^4*x^r-12*ln(c)*b*e^3*r^5*(x^r)^3+264*b*d
*e^2*n*r^2*(x^r)^2+444*b*d^2*e*n*r^2*x^r-193*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-456*I*Pi*b*d*e
^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-564*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r-16*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n
)*csgn(I*c)*(x^r)^3+48*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+48*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I
*c)*(x^r)^2+48*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-240*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^
r)^2-240*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-102*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)
^3-102*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-171*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*
c)*(x^r)^2-27*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-456*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n
)*csgn(I*c)*(x^r)^2-564*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-264*I*Pi*b*d^2*e*r*csgn(I*x^n
)*csgn(I*c*x^n)^2*x^r-264*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+124*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^2*csgn
(I*c)*(x^r)^3-232*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+408*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r
)^2-192*b*d*e^2*n*r*(x^r)^2-240*b*d^2*e*n*r*x^r+27*b*d*e^2*n*r^4*(x^r)^2-144*b*d*e^2*n*r^3*(x^r)^2+108*b*d^2*e
*n*r^4*x^r-360*b*d^2*e*n*r^3*x^r-54*ln(c)*b*d*e^2*r^5*(x^r)^2+342*ln(c)*b*d*e^2*r^4*(x^r)^2-108*ln(c)*b*d^2*e*
r^5*x^r+576*ln(c)*b*d^2*e*r^4*x^r+72*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+102*I*Pi*b*e^3*r
^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-582*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+102*I*Pi*b
*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+72*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x^r)^3+54*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x
^r-16*I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3+16*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)-9*I*Pi*b*d^3*r^6*csgn(I*c*x^n
)^3-193*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^3+96*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1164*ln(c)*b*d^2*e*
r^3*x^r+1128*ln(c)*b*d^2*e*r^2*x^r-528*ln(c)*b*d^2*e*r*x^r-816*ln(c)*b*d*e^2*r^3*(x^r)^2+912*ln(c)*b*d*e^2*r^2
*(x^r)^2-480*ln(c)*b*d*e^2*r*(x^r)^2+54*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-27*I*Pi*b*d*e
^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-66*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2-66*I*Pi*b*d^3*r^5*csg
n(I*c*x^n)^2*csgn(I*c)-48*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+288*I*Pi*b*d^2*e*r^4*csgn(I*x^n
)*csgn(I*c*x^n)^2*x^r+288*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^2*csgn(I*c)*x^r+240*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x
^r)^2-408*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-40*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)*(x^r)^3+171*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+171*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^2*c
sgn(I*c)*(x^r)^2+232*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+232*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)+9
*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)+48*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+40*I*Pi*b*e^3*r^4*csgn
(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+40*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+27*I*Pi*b*d*e^2*r^5*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-124*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+456*I*Pi*
b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+66*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3-171*I*Pi*b*d*e^2*r^4*csgn(I*
c*x^n)^3*(x^r)^2-288*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r+124*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)
^3-16*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+582*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r+96*I*Pi*b*d^3*r*
csgn(I*c*x^n)^3+66*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+288*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)+456*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+564*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*
c*x^n)^2*x^r+564*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r-48*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)*(x^r)^2+193*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+193*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)-48*I
*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+16*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2-232*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3-9
*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-54*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^2*csgn(I*c)*x^r+288*I*Pi
*b*d^3*r^3*csgn(I*c*x^n)^3-48*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2-40*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3+2
40*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+264*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)*x^r)/(-2+3*r)^2/x^2/(-1+r)^2/(-2+r)^2
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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^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.49595, size = 2287, normalized size = 11.97 \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^3,x, algorithm="fricas")
[Out]
-1/4*(9*(b*d^3*n + 2*a*d^3)*r^6 - 66*(b*d^3*n + 2*a*d^3)*r^5 + 16*b*d^3*n + 193*(b*d^3*n + 2*a*d^3)*r^4 + 32*a
*d^3 - 288*(b*d^3*n + 2*a*d^3)*r^3 + 232*(b*d^3*n + 2*a*d^3)*r^2 - 96*(b*d^3*n + 2*a*d^3)*r - 4*(3*a*e^3*r^5 -
4*b*e^3*n - (b*e^3*n + 20*a*e^3)*r^4 - 8*a*e^3 + 3*(2*b*e^3*n + 17*a*e^3)*r^3 - (13*b*e^3*n + 62*a*e^3)*r^2 +
12*(b*e^3*n + 3*a*e^3)*r + (3*b*e^3*r^5 - 20*b*e^3*r^4 + 51*b*e^3*r^3 - 62*b*e^3*r^2 + 36*b*e^3*r - 8*b*e^3)*
log(c) + (3*b*e^3*n*r^5 - 20*b*e^3*n*r^4 + 51*b*e^3*n*r^3 - 62*b*e^3*n*r^2 + 36*b*e^3*n*r - 8*b*e^3*n)*log(x))
*x^(3*r) - 3*(18*a*d*e^2*r^5 - 16*b*d*e^2*n - 3*(3*b*d*e^2*n + 38*a*d*e^2)*r^4 - 32*a*d*e^2 + 16*(3*b*d*e^2*n
+ 17*a*d*e^2)*r^3 - 8*(11*b*d*e^2*n + 38*a*d*e^2)*r^2 + 32*(2*b*d*e^2*n + 5*a*d*e^2)*r + 2*(9*b*d*e^2*r^5 - 57
*b*d*e^2*r^4 + 136*b*d*e^2*r^3 - 152*b*d*e^2*r^2 + 80*b*d*e^2*r - 16*b*d*e^2)*log(c) + 2*(9*b*d*e^2*n*r^5 - 57
*b*d*e^2*n*r^4 + 136*b*d*e^2*n*r^3 - 152*b*d*e^2*n*r^2 + 80*b*d*e^2*n*r - 16*b*d*e^2*n)*log(x))*x^(2*r) - 12*(
9*a*d^2*e*r^5 - 4*b*d^2*e*n - 3*(3*b*d^2*e*n + 16*a*d^2*e)*r^4 - 8*a*d^2*e + (30*b*d^2*e*n + 97*a*d^2*e)*r^3 -
(37*b*d^2*e*n + 94*a*d^2*e)*r^2 + 4*(5*b*d^2*e*n + 11*a*d^2*e)*r + (9*b*d^2*e*r^5 - 48*b*d^2*e*r^4 + 97*b*d^2
*e*r^3 - 94*b*d^2*e*r^2 + 44*b*d^2*e*r - 8*b*d^2*e)*log(c) + (9*b*d^2*e*n*r^5 - 48*b*d^2*e*n*r^4 + 97*b*d^2*e*
n*r^3 - 94*b*d^2*e*n*r^2 + 44*b*d^2*e*n*r - 8*b*d^2*e*n)*log(x))*x^r + 2*(9*b*d^3*r^6 - 66*b*d^3*r^5 + 193*b*d
^3*r^4 - 288*b*d^3*r^3 + 232*b*d^3*r^2 - 96*b*d^3*r + 16*b*d^3)*log(c) + 2*(9*b*d^3*n*r^6 - 66*b*d^3*n*r^5 + 1
93*b*d^3*n*r^4 - 288*b*d^3*n*r^3 + 232*b*d^3*n*r^2 - 96*b*d^3*n*r + 16*b*d^3*n)*log(x))/((9*r^6 - 66*r^5 + 193
*r^4 - 288*r^3 + 232*r^2 - 96*r + 16)*x^2)
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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**3,x)
[Out]
Timed out
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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^{3}}\,{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^3,x, algorithm="giac")
[Out]
integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^3, x)