3.403 \(\int \frac{(d+e x^r)^3 (a+b \log (c x^n))}{x^6} \, dx\)

Optimal. Leaf size=183 \[ -\frac{3 d^2 e x^{r-5} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{3 d e^2 x^{2 r-5} \left (a+b \log \left (c x^n\right )\right )}{5-2 r}-\frac{e^3 x^{3 r-5} \left (a+b \log \left (c x^n\right )\right )}{5-3 r}-\frac{3 b d^2 e n x^{r-5}}{(5-r)^2}-\frac{b d^3 n}{25 x^5}-\frac{3 b d e^2 n x^{2 r-5}}{(5-2 r)^2}-\frac{b e^3 n x^{3 r-5}}{(5-3 r)^2} \]

[Out]

-(b*d^3*n)/(25*x^5) - (3*b*d^2*e*n*x^(-5 + r))/(5 - r)^2 - (3*b*d*e^2*n*x^(-5 + 2*r))/(5 - 2*r)^2 - (b*e^3*n*x
^(-5 + 3*r))/(5 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/(5*x^5) - (3*d^2*e*x^(-5 + r)*(a + b*Log[c*x^n]))/(5 - r)
- (3*d*e^2*x^(-5 + 2*r)*(a + b*Log[c*x^n]))/(5 - 2*r) - (e^3*x^(-5 + 3*r)*(a + b*Log[c*x^n]))/(5 - 3*r)

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Rubi [A]  time = 0.411522, antiderivative size = 155, normalized size of antiderivative = 0.85, 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} \[ -\frac{1}{5} \left (\frac{15 d^2 e x^{r-5}}{5-r}+\frac{d^3}{x^5}+\frac{15 d e^2 x^{2 r-5}}{5-2 r}+\frac{5 e^3 x^{3 r-5}}{5-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r-5}}{(5-r)^2}-\frac{b d^3 n}{25 x^5}-\frac{3 b d e^2 n x^{2 r-5}}{(5-2 r)^2}-\frac{b e^3 n x^{3 r-5}}{(5-3 r)^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-(b*d^3*n)/(25*x^5) - (3*b*d^2*e*n*x^(-5 + r))/(5 - r)^2 - (3*b*d*e^2*n*x^(-5 + 2*r))/(5 - 2*r)^2 - (b*e^3*n*x
^(-5 + 3*r))/(5 - 3*r)^2 - ((d^3/x^5 + (15*d^2*e*x^(-5 + r))/(5 - r) + (15*d*e^2*x^(-5 + 2*r))/(5 - 2*r) + (5*
e^3*x^(-5 + 3*r))/(5 - 3*r))*(a + b*Log[c*x^n]))/5

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^6} \, dx &=-\frac{1}{5} \left (\frac{d^3}{x^5}+\frac{15 d^2 e x^{-5+r}}{5-r}+\frac{15 d e^2 x^{-5+2 r}}{5-2 r}+\frac{5 e^3 x^{-5+3 r}}{5-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-d^3+\frac{15 d^2 e x^r}{-5+r}+\frac{15 d e^2 x^{2 r}}{-5+2 r}+\frac{5 e^3 x^{3 r}}{-5+3 r}}{5 x^6} \, dx\\ &=-\frac{1}{5} \left (\frac{d^3}{x^5}+\frac{15 d^2 e x^{-5+r}}{5-r}+\frac{15 d e^2 x^{-5+2 r}}{5-2 r}+\frac{5 e^3 x^{-5+3 r}}{5-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{5} (b n) \int \frac{-d^3+\frac{15 d^2 e x^r}{-5+r}+\frac{15 d e^2 x^{2 r}}{-5+2 r}+\frac{5 e^3 x^{3 r}}{-5+3 r}}{x^6} \, dx\\ &=-\frac{1}{5} \left (\frac{d^3}{x^5}+\frac{15 d^2 e x^{-5+r}}{5-r}+\frac{15 d e^2 x^{-5+2 r}}{5-2 r}+\frac{5 e^3 x^{-5+3 r}}{5-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{5} (b n) \int \left (-\frac{d^3}{x^6}+\frac{15 d^2 e x^{-6+r}}{-5+r}+\frac{15 d e^2 x^{2 (-3+r)}}{-5+2 r}+\frac{5 e^3 x^{3 (-2+r)}}{-5+3 r}\right ) \, dx\\ &=-\frac{b d^3 n}{25 x^5}-\frac{3 b d^2 e n x^{-5+r}}{(5-r)^2}-\frac{3 b d e^2 n x^{-5+2 r}}{(5-2 r)^2}-\frac{b e^3 n x^{-5+3 r}}{(5-3 r)^2}-\frac{1}{5} \left (\frac{d^3}{x^5}+\frac{15 d^2 e x^{-5+r}}{5-r}+\frac{15 d e^2 x^{-5+2 r}}{5-2 r}+\frac{5 e^3 x^{-5+3 r}}{5-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.40007, size = 187, normalized size = 1.02 \[ \frac{a \left (\frac{75 d^2 e x^r}{r-5}-5 d^3+\frac{75 d e^2 x^{2 r}}{2 r-5}+\frac{25 e^3 x^{3 r}}{3 r-5}\right )+5 b \log \left (c x^n\right ) \left (\frac{15 d^2 e x^r}{r-5}-d^3+\frac{15 d e^2 x^{2 r}}{2 r-5}+\frac{5 e^3 x^{3 r}}{3 r-5}\right )+b n \left (-\frac{75 d^2 e x^r}{(r-5)^2}-d^3-\frac{75 d e^2 x^{2 r}}{(5-2 r)^2}-\frac{25 e^3 x^{3 r}}{(5-3 r)^2}\right )}{25 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^6,x]

[Out]

(b*n*(-d^3 - (75*d^2*e*x^r)/(-5 + r)^2 - (75*d*e^2*x^(2*r))/(5 - 2*r)^2 - (25*e^3*x^(3*r))/(5 - 3*r)^2) + a*(-
5*d^3 + (75*d^2*e*x^r)/(-5 + r) + (75*d*e^2*x^(2*r))/(-5 + 2*r) + (25*e^3*x^(3*r))/(-5 + 3*r)) + 5*b*(-d^3 + (
15*d^2*e*x^r)/(-5 + r) + (15*d*e^2*x^(2*r))/(-5 + 2*r) + (5*e^3*x^(3*r))/(-5 + 3*r))*Log[c*x^n])/(25*x^5)

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Maple [C]  time = 0.363, size = 4031, normalized size = 22. \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^6,x)

[Out]

-1/5*b*(-10*e^3*r^2*(x^r)^3-45*d*e^2*r^2*(x^r)^2+75*e^3*r*(x^r)^3+6*d^3*r^3-90*d^2*e*r^2*x^r+300*d*e^2*r*(x^r)
^2-125*e^3*(x^r)^3-55*d^3*r^2+375*d^2*e*r*x^r-375*d*e^2*(x^r)^2+150*d^3*r-375*d^2*e*x^r-125*d^3)/x^5/(-5+3*r)/
(-5+2*r)/(-5+r)*ln(x^n)-1/50*(-140625*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+156250*a*d^3-356250*I*P
i*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-31875*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^
r)^3-31875*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-300*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^
r)^3+156250*a*e^3*(x^r)^3-187500*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2-90000*I*Pi*b*d^3*r^3*csgn(I*x^n)*csg
n(I*c*x^n)^2+156250*ln(c)*b*d^3+3300*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3+90000*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3+72*b*
d^3*n*r^6-1320*b*d^3*n*r^5+9650*b*d^3*n*r^4+360*a*d^3*r^6-6600*a*d^3*r^5+48250*a*d^3*r^4-181250*I*Pi*b*d^3*r^2
*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+5000*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+5000*I*Pi*b*e^3*r
^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-21375*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2+300*I*Pi*b*e^3*r^5*csgn(I*
c*x^n)^3*(x^r)^3-600*a*e^3*r^5*(x^r)^3+10000*a*e^3*r^4*(x^r)^3+468750*a*d*e^2*(x^r)^2+468750*a*d^2*e*x^r+31250
*b*e^3*n*(x^r)^3-63750*a*e^3*r^3*(x^r)^3+193750*a*e^3*r^2*(x^r)^3-281250*a*e^3*r*(x^r)^3+156250*ln(c)*b*e^3*(x
^r)^3-3300*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)-36000*b*d^3*n*r^3+72500*b*d^3*n*r^2-75000*b*d^3*n*r+360*ln
(c)*b*d^3*r^6-6600*ln(c)*b*d^3*r^5+48250*ln(c)*b*d^3*r^4-180000*ln(c)*b*d^3*r^3+362500*ln(c)*b*d^3*r^2-375000*
ln(c)*b*d^3*r-180000*a*d^3*r^3+362500*a*d^3*r^2-375000*a*d^3*r-937500*a*d*e^2*r*(x^r)^2-363750*a*d^2*e*r^3*x^r
+881250*a*d^2*e*r^2*x^r-1031250*a*d^2*e*r*x^r+31250*b*d^3*n+24125*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+2
4125*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)-234375*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+181250*I*Pi*b*d^3*r^2*cs
gn(I*x^n)*csgn(I*c*x^n)^2+181250*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)+180*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*c
sgn(I*c)-90000*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)-3300*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2+10000*
ln(c)*b*e^3*r^4*(x^r)^3-63750*ln(c)*b*e^3*r^3*(x^r)^3+193750*ln(c)*b*e^3*r^2*(x^r)^3-281250*ln(c)*b*e^3*r*(x^r
)^3+468750*ln(c)*b*d^2*e*x^r+468750*ln(c)*b*d*e^2*(x^r)^2+16250*b*e^3*n*r^2*(x^r)^3-37500*b*e^3*n*r*(x^r)^3+93
750*b*d*e^2*n*(x^r)^2+93750*b*d^2*e*n*x^r-255000*a*d*e^2*r^3*(x^r)^2+712500*a*d*e^2*r^2*(x^r)^2+200*b*e^3*n*r^
4*(x^r)^3-3000*b*e^3*n*r^3*(x^r)^3-2700*a*d*e^2*r^5*(x^r)^2+42750*a*d*e^2*r^4*(x^r)^2-5400*a*d^2*e*r^5*x^r+720
00*a*d^2*e*r^4*x^r-600*ln(c)*b*e^3*r^5*(x^r)^3+82500*b*d*e^2*n*r^2*(x^r)^2+138750*b*d^2*e*n*r^2*x^r-78125*I*Pi
*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+234375*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+234
375*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+234375*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+21375*I
*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+21375*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-9
6875*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-78125*I*Pi*b*d^3*csgn(I*c*x^n)^3+468750*I*Pi*b
*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+515625*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)
*x^r+181875*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-2700*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^2*csg
n(I*c)*x^r-2700*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-440625*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)*x^r-21375*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+127500*I*Pi*b*d*e^2*r^3*
csgn(I*c*x^n)^3*(x^r)^2-300*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+1350*I*Pi*b*d*e^2*r^5*csgn(I*c*x^
n)^3*(x^r)^2+78125*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2-150000*b*d*e^2*n*r*(x^r)^2-187500*b*d^2*e*n*r*x^r+13
50*b*d*e^2*n*r^4*(x^r)^2-18000*b*d*e^2*n*r^3*(x^r)^2+5400*b*d^2*e*n*r^4*x^r-45000*b*d^2*e*n*r^3*x^r-2700*ln(c)
*b*d*e^2*r^5*(x^r)^2+42750*ln(c)*b*d*e^2*r^4*(x^r)^2-5400*ln(c)*b*d^2*e*r^5*x^r+72000*ln(c)*b*d^2*e*r^4*x^r+96
875*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+515625*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r+31875*I*Pi*b*e^
3*r^3*csgn(I*c*x^n)^3*(x^r)^3+140625*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x^r)^3+1350*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)*(x^r)^2+2700*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-140625*I*Pi*b*e^3*
r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+468750*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-363750*ln(c)*b*d^2*e*r^3*x^r
+881250*ln(c)*b*d^2*e*r^2*x^r-1031250*ln(c)*b*d^2*e*r*x^r-255000*ln(c)*b*d*e^2*r^3*(x^r)^2+712500*ln(c)*b*d*e^
2*r^2*(x^r)^2-937500*ln(c)*b*d*e^2*r*(x^r)^2-36000*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+18
7500*I*Pi*b*d^3*r*csgn(I*c*x^n)^3+234375*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-234375*I*Pi*b*d*e^2*csgn(I
*c*x^n)^3*(x^r)^2-5000*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3+127500*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n
)*csgn(I*c)*(x^r)^2-187500*I*Pi*b*d^3*r*csgn(I*c*x^n)^2*csgn(I*c)-127500*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^2*csgn
(I*c)*(x^r)^2-24125*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-356250*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3
*(x^r)^2-440625*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r-181250*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3-78125*I*Pi*b*d^3*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+180*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2+2700*I*Pi*b*d^2*e*r^5*csgn(I*
c*x^n)^3*x^r+3300*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+181875*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x
^r+356250*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+356250*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^2*csgn(I*
c)*(x^r)^2-96875*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3+78125*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+7
8125*I*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-1350*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-13
50*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+187500*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+
90000*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-36000*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r+96875*I*Pi
*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-180*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-78125*I*
Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3+78125*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)-180*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3
-24125*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^3-468750*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-515625*I*Pi*b*d^
2*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-515625*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+36000*I*Pi*b*d^2*e*r
^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+36000*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^2*csgn(I*c)*x^r-468750*I*Pi*b*d*e^2*r*
csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-5000*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+140625*I*P
i*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+440625*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r+31
875*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-234375*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*c
sgn(I*c)*(x^r)^2-127500*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-234375*I*Pi*b*d^2*e*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)*x^r-181875*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r+440625*I*Pi*b*d^2*e*r^2*csgn(
I*x^n)*csgn(I*c*x^n)^2*x^r+300*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-181875*I*Pi*b*d^2*e*
r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r)/(-5+3*r)^2/x^5/(-5+2*r)^2/(-5+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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.49279, size = 2493, normalized size = 13.62 \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^6,x, algorithm="fricas")

[Out]

-1/25*(36*(b*d^3*n + 5*a*d^3)*r^6 - 660*(b*d^3*n + 5*a*d^3)*r^5 + 15625*b*d^3*n + 4825*(b*d^3*n + 5*a*d^3)*r^4
+ 78125*a*d^3 - 18000*(b*d^3*n + 5*a*d^3)*r^3 + 36250*(b*d^3*n + 5*a*d^3)*r^2 - 37500*(b*d^3*n + 5*a*d^3)*r -
25*(12*a*e^3*r^5 - 625*b*e^3*n - 4*(b*e^3*n + 50*a*e^3)*r^4 - 3125*a*e^3 + 15*(4*b*e^3*n + 85*a*e^3)*r^3 - 25
*(13*b*e^3*n + 155*a*e^3)*r^2 + 375*(2*b*e^3*n + 15*a*e^3)*r + (12*b*e^3*r^5 - 200*b*e^3*r^4 + 1275*b*e^3*r^3
- 3875*b*e^3*r^2 + 5625*b*e^3*r - 3125*b*e^3)*log(c) + (12*b*e^3*n*r^5 - 200*b*e^3*n*r^4 + 1275*b*e^3*n*r^3 -
3875*b*e^3*n*r^2 + 5625*b*e^3*n*r - 3125*b*e^3*n)*log(x))*x^(3*r) - 75*(18*a*d*e^2*r^5 - 625*b*d*e^2*n - 3*(3*
b*d*e^2*n + 95*a*d*e^2)*r^4 - 3125*a*d*e^2 + 20*(6*b*d*e^2*n + 85*a*d*e^2)*r^3 - 50*(11*b*d*e^2*n + 95*a*d*e^2
)*r^2 + 250*(4*b*d*e^2*n + 25*a*d*e^2)*r + (18*b*d*e^2*r^5 - 285*b*d*e^2*r^4 + 1700*b*d*e^2*r^3 - 4750*b*d*e^2
*r^2 + 6250*b*d*e^2*r - 3125*b*d*e^2)*log(c) + (18*b*d*e^2*n*r^5 - 285*b*d*e^2*n*r^4 + 1700*b*d*e^2*n*r^3 - 47
50*b*d*e^2*n*r^2 + 6250*b*d*e^2*n*r - 3125*b*d*e^2*n)*log(x))*x^(2*r) - 75*(36*a*d^2*e*r^5 - 625*b*d^2*e*n - 1
2*(3*b*d^2*e*n + 40*a*d^2*e)*r^4 - 3125*a*d^2*e + 25*(12*b*d^2*e*n + 97*a*d^2*e)*r^3 - 25*(37*b*d^2*e*n + 235*
a*d^2*e)*r^2 + 625*(2*b*d^2*e*n + 11*a*d^2*e)*r + (36*b*d^2*e*r^5 - 480*b*d^2*e*r^4 + 2425*b*d^2*e*r^3 - 5875*
b*d^2*e*r^2 + 6875*b*d^2*e*r - 3125*b*d^2*e)*log(c) + (36*b*d^2*e*n*r^5 - 480*b*d^2*e*n*r^4 + 2425*b*d^2*e*n*r
^3 - 5875*b*d^2*e*n*r^2 + 6875*b*d^2*e*n*r - 3125*b*d^2*e*n)*log(x))*x^r + 5*(36*b*d^3*r^6 - 660*b*d^3*r^5 + 4
825*b*d^3*r^4 - 18000*b*d^3*r^3 + 36250*b*d^3*r^2 - 37500*b*d^3*r + 15625*b*d^3)*log(c) + 5*(36*b*d^3*n*r^6 -
660*b*d^3*n*r^5 + 4825*b*d^3*n*r^4 - 18000*b*d^3*n*r^3 + 36250*b*d^3*n*r^2 - 37500*b*d^3*n*r + 15625*b*d^3*n)*
log(x))/((36*r^6 - 660*r^5 + 4825*r^4 - 18000*r^3 + 36250*r^2 - 37500*r + 15625)*x^5)

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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**6,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^{6}}\,{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^6,x, algorithm="giac")

[Out]

integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^6, x)