3.397 \(\int \frac{(d+e x^r)^3 (a+b \log (c x^n))}{x^5} \, dx\)
Optimal. Leaf size=191 \[ -\frac{3 d^2 e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac{3 d e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}-\frac{e^3 x^{3 r-4} \left (a+b \log \left (c x^n\right )\right )}{4-3 r}-\frac{3 b d^2 e n x^{r-4}}{(4-r)^2}-\frac{b d^3 n}{16 x^4}-\frac{3 b d e^2 n x^{-2 (2-r)}}{4 (2-r)^2}-\frac{b e^3 n x^{3 r-4}}{(4-3 r)^2} \]
[Out]
-(b*d^3*n)/(16*x^4) - (3*b*d*e^2*n)/(4*(2 - r)^2*x^(2*(2 - r))) - (3*b*d^2*e*n*x^(-4 + r))/(4 - r)^2 - (b*e^3*
n*x^(-4 + 3*r))/(4 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/(4*x^4) - (3*d*e^2*(a + b*Log[c*x^n]))/(2*(2 - r)*x^(2*
(2 - r))) - (3*d^2*e*x^(-4 + r)*(a + b*Log[c*x^n]))/(4 - r) - (e^3*x^(-4 + 3*r)*(a + b*Log[c*x^n]))/(4 - 3*r)
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Rubi [A] time = 0.39653, 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} \[ -\frac{1}{4} \left (\frac{12 d^2 e x^{r-4}}{4-r}+\frac{d^3}{x^4}+\frac{6 d e^2 x^{-2 (2-r)}}{2-r}+\frac{4 e^3 x^{3 r-4}}{4-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r-4}}{(4-r)^2}-\frac{b d^3 n}{16 x^4}-\frac{3 b d e^2 n x^{-2 (2-r)}}{4 (2-r)^2}-\frac{b e^3 n x^{3 r-4}}{(4-3 r)^2} \]
Antiderivative was successfully verified.
[In]
Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^5,x]
[Out]
-(b*d^3*n)/(16*x^4) - (3*b*d*e^2*n)/(4*(2 - r)^2*x^(2*(2 - r))) - (3*b*d^2*e*n*x^(-4 + r))/(4 - r)^2 - (b*e^3*
n*x^(-4 + 3*r))/(4 - 3*r)^2 - ((d^3/x^4 + (6*d*e^2)/((2 - r)*x^(2*(2 - r))) + (12*d^2*e*x^(-4 + r))/(4 - r) +
(4*e^3*x^(-4 + 3*r))/(4 - 3*r))*(a + b*Log[c*x^n]))/4
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^5} \, dx &=-\frac{1}{4} \left (\frac{d^3}{x^4}+\frac{6 d e^2 x^{-2 (2-r)}}{2-r}+\frac{12 d^2 e x^{-4+r}}{4-r}+\frac{4 e^3 x^{-4+3 r}}{4-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-d^3+\frac{12 d^2 e x^r}{-4+r}+\frac{6 d e^2 x^{2 r}}{-2+r}+\frac{4 e^3 x^{3 r}}{-4+3 r}}{4 x^5} \, dx\\ &=-\frac{1}{4} \left (\frac{d^3}{x^4}+\frac{6 d e^2 x^{-2 (2-r)}}{2-r}+\frac{12 d^2 e x^{-4+r}}{4-r}+\frac{4 e^3 x^{-4+3 r}}{4-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} (b n) \int \frac{-d^3+\frac{12 d^2 e x^r}{-4+r}+\frac{6 d e^2 x^{2 r}}{-2+r}+\frac{4 e^3 x^{3 r}}{-4+3 r}}{x^5} \, dx\\ &=-\frac{1}{4} \left (\frac{d^3}{x^4}+\frac{6 d e^2 x^{-2 (2-r)}}{2-r}+\frac{12 d^2 e x^{-4+r}}{4-r}+\frac{4 e^3 x^{-4+3 r}}{4-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} (b n) \int \left (-\frac{d^3}{x^5}+\frac{12 d^2 e x^{-5+r}}{-4+r}+\frac{6 d e^2 x^{-5+2 r}}{-2+r}+\frac{4 e^3 x^{-5+3 r}}{-4+3 r}\right ) \, dx\\ &=-\frac{b d^3 n}{16 x^4}-\frac{3 b d e^2 n x^{-2 (2-r)}}{4 (2-r)^2}-\frac{3 b d^2 e n x^{-4+r}}{(4-r)^2}-\frac{b e^3 n x^{-4+3 r}}{(4-3 r)^2}-\frac{1}{4} \left (\frac{d^3}{x^4}+\frac{6 d e^2 x^{-2 (2-r)}}{2-r}+\frac{12 d^2 e x^{-4+r}}{4-r}+\frac{4 e^3 x^{-4+3 r}}{4-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.389925, size = 181, normalized size = 0.95 \[ \frac{a \left (\frac{48 d^2 e x^r}{r-4}-4 d^3+\frac{24 d e^2 x^{2 r}}{r-2}+\frac{16 e^3 x^{3 r}}{3 r-4}\right )+4 b \log \left (c x^n\right ) \left (\frac{12 d^2 e x^r}{r-4}-d^3+\frac{6 d e^2 x^{2 r}}{r-2}+\frac{4 e^3 x^{3 r}}{3 r-4}\right )+b n \left (-\frac{48 d^2 e x^r}{(r-4)^2}-d^3-\frac{12 d e^2 x^{2 r}}{(r-2)^2}-\frac{16 e^3 x^{3 r}}{(4-3 r)^2}\right )}{16 x^4} \]
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^5,x]
[Out]
(b*n*(-d^3 - (48*d^2*e*x^r)/(-4 + r)^2 - (12*d*e^2*x^(2*r))/(-2 + r)^2 - (16*e^3*x^(3*r))/(4 - 3*r)^2) + a*(-4
*d^3 + (48*d^2*e*x^r)/(-4 + r) + (24*d*e^2*x^(2*r))/(-2 + r) + (16*e^3*x^(3*r))/(-4 + 3*r)) + 4*b*(-d^3 + (12*
d^2*e*x^r)/(-4 + r) + (6*d*e^2*x^(2*r))/(-2 + r) + (4*e^3*x^(3*r))/(-4 + 3*r))*Log[c*x^n])/(16*x^4)
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Maple [C] time = 0.347, 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^5,x)
[Out]
-1/4*b*(-4*e^3*r^2*(x^r)^3-18*d*e^2*r^2*(x^r)^2+24*e^3*r*(x^r)^3+3*d^3*r^3-36*d^2*e*r^2*x^r+96*d*e^2*r*(x^r)^2
-32*e^3*(x^r)^3-22*d^3*r^2+120*d^2*e*r*x^r-96*d*e^2*(x^r)^2+48*d^3*r-96*d^2*e*x^r-32*d^3)/x^4/(-4+3*r)/(-2+r)/
(-4+r)*ln(x^n)-1/16*(4096*a*d^3+24*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-108*I*Pi*b*d*e^2
*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+6144*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+264*I*Pi*b*d^3*
r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+4096*a*e^3*(x^r)^3+7424*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+742
4*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)+18*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)+4096*ln(c)*b*d^3-1544*I
*Pi*b*d^3*r^4*csgn(I*c*x^n)^3+2048*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+9*b*d^3*n*r^6-132*b*d^3*n*r^5+772*b*
d^3*n*r^4+6528*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2+36*a*d^3*r^6-528*a*d^3*r^5+3088*a*d^3*r^4-48*a*e^3*r^5
*(x^r)^3+640*a*e^3*r^4*(x^r)^3+12288*a*d*e^2*(x^r)^2+12288*a*d^2*e*x^r+1024*b*e^3*n*(x^r)^3-3264*a*e^3*r^3*(x^
r)^3+7936*a*e^3*r^2*(x^r)^3-9216*a*e^3*r*(x^r)^3+4096*ln(c)*b*e^3*(x^r)^3-2304*b*d^3*n*r^3+3712*b*d^3*n*r^2-30
72*b*d^3*n*r+36*ln(c)*b*d^3*r^6-528*ln(c)*b*d^3*r^5+3088*ln(c)*b*d^3*r^4-9216*ln(c)*b*d^3*r^3+14848*ln(c)*b*d^
3*r^2-12288*ln(c)*b*d^3*r-9216*a*d^3*r^3+14848*a*d^3*r^2-12288*a*d^3*r-30720*a*d*e^2*r*(x^r)^2-18624*a*d^2*e*r
^3*x^r+36096*a*d^2*e*r^2*x^r-33792*a*d^2*e*r*x^r+1024*b*d^3*n-15360*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(
x^r)^2-16896*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+1544*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+15
44*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)-6144*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+640*ln(c)*b*e^3*r^4*(x^r)^3-
3264*ln(c)*b*e^3*r^3*(x^r)^3+7936*ln(c)*b*e^3*r^2*(x^r)^3-9216*ln(c)*b*e^3*r*(x^r)^3+12288*ln(c)*b*d^2*e*x^r+1
2288*ln(c)*b*d*e^2*(x^r)^2+832*b*e^3*n*r^2*(x^r)^3-1536*b*e^3*n*r*(x^r)^3+3072*b*d*e^2*n*(x^r)^2+3072*b*d^2*e*
n*x^r-13056*a*d*e^2*r^3*(x^r)^2+29184*a*d*e^2*r^2*(x^r)^2+16*b*e^3*n*r^4*(x^r)^3-192*b*e^3*n*r^3*(x^r)^3-216*a
*d*e^2*r^5*(x^r)^2+2736*a*d*e^2*r^4*(x^r)^2-432*a*d^2*e*r^5*x^r+4608*a*d^2*e*r^4*x^r-48*ln(c)*b*e^3*r^5*(x^r)^
3+4224*b*d*e^2*n*r^2*(x^r)^2+7104*b*d^2*e*n*r^2*x^r+4608*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r
)^3-216*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^2*csgn(I*c)*x^r+1632*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)
*(x^r)^3+24*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3-2048*I*Pi*b*d^3*csgn(I*c*x^n)^3-6528*I*Pi*b*d*e^2*r^3*csgn(
I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+3968*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+3968*I*Pi*b*e^3*r^2*csg
n(I*c*x^n)^2*csgn(I*c)*(x^r)^3-1632*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-24*I*Pi*b*e^3*r^5*csgn(I*
x^n)*csgn(I*c*x^n)^2*(x^r)^3-24*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-1632*I*Pi*b*e^3*r^3*csgn(I*x^
n)*csgn(I*c*x^n)^2*(x^r)^3+14592*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+14592*I*Pi*b*d*e^2*r^2*c
sgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+18048*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+18048*I*Pi*b*d^2*e*r^2
*csgn(I*c*x^n)^2*csgn(I*c)*x^r+1368*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+1368*I*Pi*b*d*e^2*r^4
*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-3968*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-15360*I*Pi*
b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+6144*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+6144*I*Pi*b*
d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+6144*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+320*I*Pi*b*e^3*r^4*csgn(
I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+108*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2+15360*I*Pi*b*d*e^2*r*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)*(x^r)^2+16896*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-6144*b*d*e^2*n*r*(
x^r)^2-7680*b*d^2*e*n*r*x^r+108*b*d*e^2*n*r^4*(x^r)^2-1152*b*d*e^2*n*r^3*(x^r)^2+432*b*d^2*e*n*r^4*x^r-2880*b*
d^2*e*n*r^3*x^r-216*ln(c)*b*d*e^2*r^5*(x^r)^2+2736*ln(c)*b*d*e^2*r^4*(x^r)^2-432*ln(c)*b*d^2*e*r^5*x^r+4608*ln
(c)*b*d^2*e*r^4*x^r-108*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+4608*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(
x^r)^3-16896*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+1632*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3-2048*I*P
i*b*e^3*csgn(I*c*x^n)^3*(x^r)^3+2048*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)-18*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3+16
896*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r-18624*ln(c)*b*d^2*e*r^3*x^r+36096*ln(c)*b*d^2*e*r^2*x^r-33792*ln(c)*b*d
^2*e*r*x^r-13056*ln(c)*b*d*e^2*r^3*(x^r)^2+29184*ln(c)*b*d*e^2*r^2*(x^r)^2-30720*ln(c)*b*d*e^2*r*(x^r)^2-7424*
I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3-4608*I*Pi*b*e^3*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+15360*I*Pi*b*d*e^2*r*csgn(I
*c*x^n)^3*(x^r)^2-264*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2-6144*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2
-216*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-6528*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-
6144*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-6144*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)*x^r+6528*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+9312*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c)*x^r-18*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1544*I*Pi*b*d^3*r^4*csgn(I*x
^n)*csgn(I*c*x^n)*csgn(I*c)-14592*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-2048*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*
c*x^n)*csgn(I*c)-6144*I*Pi*b*d^3*r*csgn(I*c*x^n)^2*csgn(I*c)+2304*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2
*x^r+2304*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^2*csgn(I*c)*x^r-320*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c
)*(x^r)^3-264*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)-18048*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r-2048*I*Pi*b*
e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+6144*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-6144*I*P
i*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2-320*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3-9312*I*Pi*b*d^2*e*r^3*csgn(I*x^n)
*csgn(I*c*x^n)^2*x^r+4608*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3+6144*I*Pi*b*d^3*r*csgn(I*c*x^n)^3+108*I*Pi*b*d*e^2*r^
5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+216*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+18*
I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2-3968*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3+2048*I*Pi*b*e^3*csgn(I*
x^n)*csgn(I*c*x^n)^2*(x^r)^3+2048*I*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-14592*I*Pi*b*d*e^2*r^2*csgn(I*x
^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+216*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r+320*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^
2*csgn(I*c)*(x^r)^3-1368*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2-2304*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r-18
048*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-1368*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*c
sgn(I*c)*(x^r)^2-2304*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+4608*I*Pi*b*d^3*r^3*csgn(I*x^n)
*csgn(I*c*x^n)*csgn(I*c)-7424*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-9312*I*Pi*b*d^2*e*r^3*csgn(I*
c*x^n)^2*csgn(I*c)*x^r+9312*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r-4608*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2
*(x^r)^3+264*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3-4608*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-4608*I*Pi*b*d^3*r^
3*csgn(I*c*x^n)^2*csgn(I*c))/(-4+3*r)^2/x^4/(-2+r)^2/(-4+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^5,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.52473, size = 2395, normalized size = 12.54 \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^5,x, algorithm="fricas")
[Out]
-1/16*(9*(b*d^3*n + 4*a*d^3)*r^6 - 132*(b*d^3*n + 4*a*d^3)*r^5 + 1024*b*d^3*n + 772*(b*d^3*n + 4*a*d^3)*r^4 +
4096*a*d^3 - 2304*(b*d^3*n + 4*a*d^3)*r^3 + 3712*(b*d^3*n + 4*a*d^3)*r^2 - 3072*(b*d^3*n + 4*a*d^3)*r - 16*(3*
a*e^3*r^5 - 64*b*e^3*n - (b*e^3*n + 40*a*e^3)*r^4 - 256*a*e^3 + 12*(b*e^3*n + 17*a*e^3)*r^3 - 4*(13*b*e^3*n +
124*a*e^3)*r^2 + 96*(b*e^3*n + 6*a*e^3)*r + (3*b*e^3*r^5 - 40*b*e^3*r^4 + 204*b*e^3*r^3 - 496*b*e^3*r^2 + 576*
b*e^3*r - 256*b*e^3)*log(c) + (3*b*e^3*n*r^5 - 40*b*e^3*n*r^4 + 204*b*e^3*n*r^3 - 496*b*e^3*n*r^2 + 576*b*e^3*
n*r - 256*b*e^3*n)*log(x))*x^(3*r) - 12*(18*a*d*e^2*r^5 - 256*b*d*e^2*n - 3*(3*b*d*e^2*n + 76*a*d*e^2)*r^4 - 1
024*a*d*e^2 + 32*(3*b*d*e^2*n + 34*a*d*e^2)*r^3 - 32*(11*b*d*e^2*n + 76*a*d*e^2)*r^2 + 512*(b*d*e^2*n + 5*a*d*
e^2)*r + 2*(9*b*d*e^2*r^5 - 114*b*d*e^2*r^4 + 544*b*d*e^2*r^3 - 1216*b*d*e^2*r^2 + 1280*b*d*e^2*r - 512*b*d*e^
2)*log(c) + 2*(9*b*d*e^2*n*r^5 - 114*b*d*e^2*n*r^4 + 544*b*d*e^2*n*r^3 - 1216*b*d*e^2*n*r^2 + 1280*b*d*e^2*n*r
- 512*b*d*e^2*n)*log(x))*x^(2*r) - 48*(9*a*d^2*e*r^5 - 64*b*d^2*e*n - 3*(3*b*d^2*e*n + 32*a*d^2*e)*r^4 - 256*
a*d^2*e + 4*(15*b*d^2*e*n + 97*a*d^2*e)*r^3 - 4*(37*b*d^2*e*n + 188*a*d^2*e)*r^2 + 32*(5*b*d^2*e*n + 22*a*d^2*
e)*r + (9*b*d^2*e*r^5 - 96*b*d^2*e*r^4 + 388*b*d^2*e*r^3 - 752*b*d^2*e*r^2 + 704*b*d^2*e*r - 256*b*d^2*e)*log(
c) + (9*b*d^2*e*n*r^5 - 96*b*d^2*e*n*r^4 + 388*b*d^2*e*n*r^3 - 752*b*d^2*e*n*r^2 + 704*b*d^2*e*n*r - 256*b*d^2
*e*n)*log(x))*x^r + 4*(9*b*d^3*r^6 - 132*b*d^3*r^5 + 772*b*d^3*r^4 - 2304*b*d^3*r^3 + 3712*b*d^3*r^2 - 3072*b*
d^3*r + 1024*b*d^3)*log(c) + 4*(9*b*d^3*n*r^6 - 132*b*d^3*n*r^5 + 772*b*d^3*n*r^4 - 2304*b*d^3*n*r^3 + 3712*b*
d^3*n*r^2 - 3072*b*d^3*n*r + 1024*b*d^3*n)*log(x))/((9*r^6 - 132*r^5 + 772*r^4 - 2304*r^3 + 3712*r^2 - 3072*r
+ 1024)*x^4)
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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**5,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^{5}}\,{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^5,x, algorithm="giac")
[Out]
integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^5, x)