3.402 \(\int \frac{(d+e x^r)^3 (a+b \log (c x^n))}{x^4} \, dx\)

Optimal. Leaf size=191 \[ -\frac{3 d^2 e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{3 d e^2 x^{2 r-3} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac{e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac{3 b d^2 e n x^{r-3}}{(3-r)^2}-\frac{b d^3 n}{9 x^3}-\frac{3 b d e^2 n x^{2 r-3}}{(3-2 r)^2}-\frac{b e^3 n x^{-3 (1-r)}}{9 (1-r)^2} \]

[Out]

-(b*d^3*n)/(9*x^3) - (b*e^3*n)/(9*(1 - r)^2*x^(3*(1 - 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 - (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)

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Rubi [A]  time = 0.39323, antiderivative size = 160, 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} \[ -\frac{1}{3} \left (\frac{9 d^2 e x^{r-3}}{3-r}+\frac{d^3}{x^3}+\frac{9 d e^2 x^{2 r-3}}{3-2 r}+\frac{e^3 x^{-3 (1-r)}}{1-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r-3}}{(3-r)^2}-\frac{b d^3 n}{9 x^3}-\frac{3 b d e^2 n x^{2 r-3}}{(3-2 r)^2}-\frac{b e^3 n x^{-3 (1-r)}}{9 (1-r)^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^4,x]

[Out]

-(b*d^3*n)/(9*x^3) - (b*e^3*n)/(9*(1 - r)^2*x^(3*(1 - 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 - ((d^3/x^3 + e^3/((1 - r)*x^(3*(1 - r))) + (9*d^2*e*x^(-3 + r))/(3 - r) + (9*d*e^2
*x^(-3 + 2*r))/(3 - 2*r))*(a + b*Log[c*x^n]))/3

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^4} \, dx &=-\frac{1}{3} \left (\frac{d^3}{x^3}+\frac{e^3 x^{-3 (1-r)}}{1-r}+\frac{9 d^2 e x^{-3+r}}{3-r}+\frac{9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-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}}{3 x^4} \, dx\\ &=-\frac{1}{3} \left (\frac{d^3}{x^3}+\frac{e^3 x^{-3 (1-r)}}{1-r}+\frac{9 d^2 e x^{-3+r}}{3-r}+\frac{9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} (b n) \int \frac{-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}}{x^4} \, dx\\ &=-\frac{1}{3} \left (\frac{d^3}{x^3}+\frac{e^3 x^{-3 (1-r)}}{1-r}+\frac{9 d^2 e x^{-3+r}}{3-r}+\frac{9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} (b n) \int \left (-\frac{d^3}{x^4}+\frac{9 d^2 e x^{-4+r}}{-3+r}+\frac{9 d e^2 x^{2 (-2+r)}}{-3+2 r}+\frac{e^3 x^{-4+3 r}}{-1+r}\right ) \, 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{1}{3} \left (\frac{d^3}{x^3}+\frac{e^3 x^{-3 (1-r)}}{1-r}+\frac{9 d^2 e x^{-3+r}}{3-r}+\frac{9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.375765, size = 180, normalized size = 0.94 \[ \frac{3 a \left (\frac{9 d^2 e x^r}{r-3}-d^3+\frac{9 d e^2 x^{2 r}}{2 r-3}+\frac{e^3 x^{3 r}}{r-1}\right )+3 b \log \left (c x^n\right ) \left (\frac{9 d^2 e x^r}{r-3}-d^3+\frac{9 d e^2 x^{2 r}}{2 r-3}+\frac{e^3 x^{3 r}}{r-1}\right )+b n \left (-\frac{27 d^2 e x^r}{(r-3)^2}-d^3-\frac{27 d e^2 x^{2 r}}{(3-2 r)^2}-\frac{e^3 x^{3 r}}{(r-1)^2}\right )}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^4,x]

[Out]

(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)

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Maple [C]  time = 0.352, 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^4,x)

[Out]

-1/3*b*(-2*e^3*r^2*(x^r)^3-9*d*e^2*r^2*(x^r)^2+9*e^3*r*(x^r)^3+2*d^3*r^3-18*d^2*e*r^2*x^r+36*d*e^2*r*(x^r)^2-9
*e^3*(x^r)^3-11*d^3*r^2+45*d^2*e*r*x^r-27*d*e^2*(x^r)^2+18*d^3*r-27*d^2*e*x^r-9*d^3)/x^3/(-1+r)/(-3+2*r)/(-3+r
)*ln(x^n)-1/18*(486*a*d^3+486*a*e^3*(x^r)^3+486*ln(c)*b*d^3+972*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*
c)+1296*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+8*b*d^3*n*r^6-88*b*d^3*n*r^5+386*b*d^3*n*r^4+24*a*d
^3*r^6-264*a*d^3*r^5+1158*a*d^3*r^4+459*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+729*I*Pi*b*e^3*r*csgn(I*c*x^n)^
3*(x^r)^3-24*a*e^3*r^5*(x^r)^3+240*a*e^3*r^4*(x^r)^3+1458*a*d*e^2*(x^r)^2+1458*a*d^2*e*x^r+162*b*e^3*n*(x^r)^3
-918*a*e^3*r^3*(x^r)^3+1674*a*e^3*r^2*(x^r)^3-1458*a*e^3*r*(x^r)^3+486*ln(c)*b*e^3*(x^r)^3-12*I*Pi*b*e^3*r^5*c
sgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+108*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r+12*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*
(x^r)^3-864*b*d^3*n*r^3+1044*b*d^3*n*r^2-648*b*d^3*n*r+24*ln(c)*b*d^3*r^6-264*ln(c)*b*d^3*r^5+1158*ln(c)*b*d^3
*r^4-2592*ln(c)*b*d^3*r^3+3132*ln(c)*b*d^3*r^2-1944*ln(c)*b*d^3*r-2592*a*d^3*r^3+3132*a*d^3*r^2-1944*a*d^3*r-4
860*a*d*e^2*r*(x^r)^2-5238*a*d^2*e*r^3*x^r+7614*a*d^2*e*r^2*x^r-5346*a*d^2*e*r*x^r+162*b*d^3*n+972*I*Pi*b*d^3*
r*csgn(I*c*x^n)^3+240*ln(c)*b*e^3*r^4*(x^r)^3-918*ln(c)*b*e^3*r^3*(x^r)^3+1674*ln(c)*b*e^3*r^2*(x^r)^3-1458*ln
(c)*b*e^3*r*(x^r)^3+1458*ln(c)*b*d^2*e*x^r+1458*ln(c)*b*d*e^2*(x^r)^2+234*b*e^3*n*r^2*(x^r)^3-324*b*e^3*n*r*(x
^r)^3+486*b*d*e^2*n*(x^r)^2+486*b*d^2*e*n*x^r-3672*a*d*e^2*r^3*(x^r)^2+6156*a*d*e^2*r^2*(x^r)^2+8*b*e^3*n*r^4*
(x^r)^3-72*b*e^3*n*r^3*(x^r)^3-108*a*d*e^2*r^5*(x^r)^2+1026*a*d*e^2*r^4*(x^r)^2-216*a*d^2*e*r^5*x^r+1728*a*d^2
*e*r^4*x^r-24*ln(c)*b*e^3*r^5*(x^r)^3+2673*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r+1188*b*d*e^2*n*r^2*(x^r)^2+1998*
b*d^2*e*n*r^2*x^r-837*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3+243*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^
3+243*I*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-243*I*Pi*b*d^3*csgn(I*c*x^n)^3-579*I*Pi*b*d^3*r^4*csgn(I*c*
x^n)^3+243*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+729*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3
+459*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-1836*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n
)^2*(x^r)^2-2619*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r-864*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r+837*I
*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+837*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-729*I*P
i*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2-120*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3-243*I*Pi*b*d^3*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)+12*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2+729*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(
x^r)^2+729*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+729*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-729
*I*Pi*b*e^3*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-1296*b*d*e^2*n*r*(x^r)^2-1620*b*d^2*e*n*r*x^r+54*b*d*e^2*n*r^4
*(x^r)^2-432*b*d*e^2*n*r^3*(x^r)^2+216*b*d^2*e*n*r^4*x^r-1080*b*d^2*e*n*r^3*x^r-108*ln(c)*b*d*e^2*r^5*(x^r)^2+
1026*ln(c)*b*d*e^2*r^4*(x^r)^2-216*ln(c)*b*d^2*e*r^5*x^r+1728*ln(c)*b*d^2*e*r^4*x^r-1566*I*Pi*b*d^3*r^2*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)-243*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-459*I*Pi*b*e^3*r^3*cs
gn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-459*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-5238*ln(c)*b*d^2*e*r^3*
x^r+7614*ln(c)*b*d^2*e*r^2*x^r-5346*ln(c)*b*d^2*e*r*x^r-3672*ln(c)*b*d*e^2*r^3*(x^r)^2+6156*ln(c)*b*d*e^2*r^2*
(x^r)^2-4860*ln(c)*b*d*e^2*r*(x^r)^2-729*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-108*I*Pi*b*d^2*e*r^5
*csgn(I*c*x^n)^2*csgn(I*c)*x^r-120*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-108*I*Pi*b*d^2*e
*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+513*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-837*I*Pi*b*e^3*r^2
*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-972*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2-1296*I*Pi*b*d^3*r^3*
csgn(I*x^n)*csgn(I*c*x^n)^2-1296*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)-132*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(
I*c*x^n)^2-132*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)+1836*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2-12*I*Pi*
b*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+54*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2+1296*I*Pi*b*d^3*r^3*cs
gn(I*c*x^n)^3+729*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+120*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x
^r)^3+120*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-513*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2-12*I*P
i*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2619*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+
579*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+579*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)-2619*I*Pi*b*d^2*e*
r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+1566*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+1566*I*Pi*b*d^3*r^2*csgn(I
*c*x^n)^2*csgn(I*c)+12*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)-1566*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3-243*I*Pi*b
*e^3*csgn(I*c*x^n)^3*(x^r)^3+243*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)-12*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3-729*I*
Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+12*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)
^3+864*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-54*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+
864*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^2*csgn(I*c)*x^r-729*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+54*I*Pi*b*d*e^2*r^5*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-3078*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-38
07*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-972*I*Pi*b*d^3*r*csgn(I*c*x^n)^2*csgn(I*c)-579*I*P
i*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-3078*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-3807*I*Pi*b*d^2*
e*r^2*csgn(I*c*x^n)^3*x^r+2430*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-864*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*
c*x^n)*csgn(I*c)*x^r+108*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-513*I*Pi*b*d*e^2*r^4*csgn(I*
x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+1836*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+132*I*P
i*b*d^3*r^5*csgn(I*c*x^n)^3-2430*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+3807*I*Pi*b*d^2*e*r^2*csgn
(I*c*x^n)^2*csgn(I*c)*x^r-2430*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-729*I*Pi*b*d*e^2*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)*(x^r)^2-1836*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+132*I*Pi*b*d^3*r^5*csgn
(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+2619*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r+3807*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*cs
gn(I*c*x^n)^2*x^r-54*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+513*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csg
n(I*c*x^n)^2*(x^r)^2-2673*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+3078*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csg
n(I*c*x^n)^2*(x^r)^2-2673*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+3078*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^2*c
sgn(I*c)*(x^r)^2+2430*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+2673*I*Pi*b*d^2*e*r*csgn(I*x^
n)*csgn(I*c*x^n)*csgn(I*c)*x^r)/(-1+r)^2/x^3/(-3+2*r)^2/(-3+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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.47249, size = 2310, normalized size = 12.09 \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^4,x, algorithm="fricas")

[Out]

-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^3*r^2 - 324*b*d^3*r + 81*b*d^3)*log(c) + 3*(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 + 81*b*d^3*n)*log(x))/((4*r^
6 - 44*r^5 + 193*r^4 - 432*r^3 + 522*r^2 - 324*r + 81)*x^3)

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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**4,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^{4}}\,{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^4,x, algorithm="giac")

[Out]

integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^4, x)