3.404 \(\int \frac{(d+e x^r)^3 (a+b \log (c x^n))}{x^8} \, dx\)

Optimal. Leaf size=183 \[ -\frac{3 d^2 e x^{r-7} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac{3 d e^2 x^{2 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac{e^3 x^{3 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-3 r}-\frac{3 b d^2 e n x^{r-7}}{(7-r)^2}-\frac{b d^3 n}{49 x^7}-\frac{3 b d e^2 n x^{2 r-7}}{(7-2 r)^2}-\frac{b e^3 n x^{3 r-7}}{(7-3 r)^2} \]

[Out]

-(b*d^3*n)/(49*x^7) - (3*b*d^2*e*n*x^(-7 + r))/(7 - r)^2 - (3*b*d*e^2*n*x^(-7 + 2*r))/(7 - 2*r)^2 - (b*e^3*n*x
^(-7 + 3*r))/(7 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/(7*x^7) - (3*d^2*e*x^(-7 + r)*(a + b*Log[c*x^n]))/(7 - r)
- (3*d*e^2*x^(-7 + 2*r)*(a + b*Log[c*x^n]))/(7 - 2*r) - (e^3*x^(-7 + 3*r)*(a + b*Log[c*x^n]))/(7 - 3*r)

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Rubi [A]  time = 0.412631, 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} \[ -\frac{1}{7} \left (\frac{21 d^2 e x^{r-7}}{7-r}+\frac{d^3}{x^7}+\frac{21 d e^2 x^{2 r-7}}{7-2 r}+\frac{7 e^3 x^{3 r-7}}{7-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n x^{r-7}}{(7-r)^2}-\frac{b d^3 n}{49 x^7}-\frac{3 b d e^2 n x^{2 r-7}}{(7-2 r)^2}-\frac{b e^3 n x^{3 r-7}}{(7-3 r)^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^8,x]

[Out]

-(b*d^3*n)/(49*x^7) - (3*b*d^2*e*n*x^(-7 + r))/(7 - r)^2 - (3*b*d*e^2*n*x^(-7 + 2*r))/(7 - 2*r)^2 - (b*e^3*n*x
^(-7 + 3*r))/(7 - 3*r)^2 - ((d^3/x^7 + (21*d^2*e*x^(-7 + r))/(7 - r) + (21*d*e^2*x^(-7 + 2*r))/(7 - 2*r) + (7*
e^3*x^(-7 + 3*r))/(7 - 3*r))*(a + b*Log[c*x^n]))/7

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^8} \, dx &=-\frac{1}{7} \left (\frac{d^3}{x^7}+\frac{21 d^2 e x^{-7+r}}{7-r}+\frac{21 d e^2 x^{-7+2 r}}{7-2 r}+\frac{7 e^3 x^{-7+3 r}}{7-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-d^3+\frac{21 d^2 e x^r}{-7+r}+\frac{21 d e^2 x^{2 r}}{-7+2 r}+\frac{7 e^3 x^{3 r}}{-7+3 r}}{7 x^8} \, dx\\ &=-\frac{1}{7} \left (\frac{d^3}{x^7}+\frac{21 d^2 e x^{-7+r}}{7-r}+\frac{21 d e^2 x^{-7+2 r}}{7-2 r}+\frac{7 e^3 x^{-7+3 r}}{7-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{7} (b n) \int \frac{-d^3+\frac{21 d^2 e x^r}{-7+r}+\frac{21 d e^2 x^{2 r}}{-7+2 r}+\frac{7 e^3 x^{3 r}}{-7+3 r}}{x^8} \, dx\\ &=-\frac{1}{7} \left (\frac{d^3}{x^7}+\frac{21 d^2 e x^{-7+r}}{7-r}+\frac{21 d e^2 x^{-7+2 r}}{7-2 r}+\frac{7 e^3 x^{-7+3 r}}{7-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{7} (b n) \int \left (-\frac{d^3}{x^8}+\frac{21 d^2 e x^{-8+r}}{-7+r}+\frac{21 d e^2 x^{2 (-4+r)}}{-7+2 r}+\frac{7 e^3 x^{-8+3 r}}{-7+3 r}\right ) \, dx\\ &=-\frac{b d^3 n}{49 x^7}-\frac{3 b d^2 e n x^{-7+r}}{(7-r)^2}-\frac{3 b d e^2 n x^{-7+2 r}}{(7-2 r)^2}-\frac{b e^3 n x^{-7+3 r}}{(7-3 r)^2}-\frac{1}{7} \left (\frac{d^3}{x^7}+\frac{21 d^2 e x^{-7+r}}{7-r}+\frac{21 d e^2 x^{-7+2 r}}{7-2 r}+\frac{7 e^3 x^{-7+3 r}}{7-3 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.390751, size = 188, normalized size = 1.03 \[ \frac{7 a \left (\frac{21 d^2 e x^r}{r-7}-d^3+\frac{21 d e^2 x^{2 r}}{2 r-7}+\frac{7 e^3 x^{3 r}}{3 r-7}\right )+7 b \log \left (c x^n\right ) \left (\frac{21 d^2 e x^r}{r-7}-d^3+\frac{21 d e^2 x^{2 r}}{2 r-7}+\frac{7 e^3 x^{3 r}}{3 r-7}\right )+b n \left (-\frac{147 d^2 e x^r}{(r-7)^2}-d^3-\frac{147 d e^2 x^{2 r}}{(7-2 r)^2}-\frac{49 e^3 x^{3 r}}{(7-3 r)^2}\right )}{49 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^8,x]

[Out]

(b*n*(-d^3 - (147*d^2*e*x^r)/(-7 + r)^2 - (147*d*e^2*x^(2*r))/(7 - 2*r)^2 - (49*e^3*x^(3*r))/(7 - 3*r)^2) + 7*
a*(-d^3 + (21*d^2*e*x^r)/(-7 + r) + (21*d*e^2*x^(2*r))/(-7 + 2*r) + (7*e^3*x^(3*r))/(-7 + 3*r)) + 7*b*(-d^3 +
(21*d^2*e*x^r)/(-7 + r) + (21*d*e^2*x^(2*r))/(-7 + 2*r) + (7*e^3*x^(3*r))/(-7 + 3*r))*Log[c*x^n])/(49*x^7)

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Maple [C]  time = 0.358, 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^8,x)

[Out]

-1/7*b*(-14*e^3*r^2*(x^r)^3-63*d*e^2*r^2*(x^r)^2+147*e^3*r*(x^r)^3+6*d^3*r^3-126*d^2*e*r^2*x^r+588*d*e^2*r*(x^
r)^2-343*e^3*(x^r)^3-77*d^3*r^2+735*d^2*e*r*x^r-1029*d*e^2*(x^r)^2+294*d^3*r-1029*d^2*e*x^r-343*d^3)/x^7/(-7+3
*r)/(-7+2*r)/(-7+r)*ln(x^n)-1/98*(1647086*a*d^3+1647086*a*e^3*(x^r)^3+1647086*ln(c)*b*d^3+72*b*d^3*n*r^6-1848*
b*d^3*n*r^5+18914*b*d^3*n*r^4+504*a*d^3*r^6-12936*a*d^3*r^5+132398*a*d^3*r^4-98784*I*Pi*b*d^2*e*r^4*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)*x^r+2646*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-1176*a*e^3*r^5
*(x^r)^3+27440*a*e^3*r^4*(x^r)^3+4941258*a*d*e^2*(x^r)^2+4941258*a*d^2*e*x^r+235298*b*e^3*n*(x^r)^3-244902*a*e
^3*r^3*(x^r)^3+1042034*a*e^3*r^2*(x^r)^3-2117682*a*e^3*r*(x^r)^3+1647086*ln(c)*b*e^3*(x^r)^3-588*I*Pi*b*e^3*r^
5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+13720*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+13720*I*Pi*b*e^
3*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+5292*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r+698691*I*Pi*b*d^2*e*r^3*csgn
(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-98784*b*d^3*n*r^3+278516*b*d^3*n*r^2-403368*b*d^3*n*r+504*ln(c)*b*d^3*r^6-
12936*ln(c)*b*d^3*r^5+132398*ln(c)*b*d^3*r^4-691488*ln(c)*b*d^3*r^3+1949612*ln(c)*b*d^3*r^2-2823576*ln(c)*b*d^
3*r-691488*a*d^3*r^3+1949612*a*d^3*r^2-2823576*a*d^3*r-7058940*a*d*e^2*r*(x^r)^2-1397382*a*d^2*e*r^3*x^r+47395
74*a*d^2*e*r^2*x^r-7764834*a*d^2*e*r*x^r+235298*b*d^3*n+66199*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+252*I
*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)-6468*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2-6468*I*Pi*b*d^3*r^5*cs
gn(I*c*x^n)^2*csgn(I*c)+27440*ln(c)*b*e^3*r^4*(x^r)^3-244902*ln(c)*b*e^3*r^3*(x^r)^3+1042034*ln(c)*b*e^3*r^2*(
x^r)^3-2117682*ln(c)*b*e^3*r*(x^r)^3+4941258*ln(c)*b*d^2*e*x^r+4941258*ln(c)*b*d*e^2*(x^r)^2+62426*b*e^3*n*r^2
*(x^r)^3-201684*b*e^3*n*r*(x^r)^3+705894*b*d*e^2*n*(x^r)^2+705894*b*d^2*e*n*x^r-979608*a*d*e^2*r^3*(x^r)^2+383
1996*a*d*e^2*r^2*(x^r)^2+392*b*e^3*n*r^4*(x^r)^3-8232*b*e^3*n*r^3*(x^r)^3-5292*a*d*e^2*r^5*(x^r)^2+117306*a*d*
e^2*r^4*(x^r)^2-10584*a*d^2*e*r^5*x^r+197568*a*d^2*e*r^4*x^r-1176*ln(c)*b*e^3*r^5*(x^r)^3+6468*I*Pi*b*d^3*r^5*
csgn(I*c*x^n)^3+823543*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+345744*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3-974806*I*P
i*b*d^3*r^2*csgn(I*c*x^n)^3+316932*b*d*e^2*n*r^2*(x^r)^2+533022*b*d^2*e*n*r^2*x^r+2369787*I*Pi*b*d^2*e*r^2*csg
n(I*x^n)*csgn(I*c*x^n)^2*x^r+2369787*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r-3529470*I*Pi*b*d*e^2*r*csg
n(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-823543*I*Pi*b*d^3*csgn(I*c*x^n)^3-252*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n
)*csgn(I*c)+6468*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-698691*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I
*c*x^n)^2*x^r-698691*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r+1915998*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(
I*c*x^n)^2*(x^r)^2+3882417*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-58653*I*Pi*b*d*e^2*r^4*csgn(
I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+3529470*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+5210
17*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+489804*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2-588*I*Pi*b
*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-1915998*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^
2-2369787*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+2646*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)
^2-974806*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3882417*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r+247062
9*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-2369787*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r-806736*b*d*e^2
*n*r*(x^r)^2-1008420*b*d^2*e*n*r*x^r+2646*b*d*e^2*n*r^4*(x^r)^2-49392*b*d*e^2*n*r^3*(x^r)^2+10584*b*d^2*e*n*r^
4*x^r-123480*b*d^2*e*n*r^3*x^r-5292*ln(c)*b*d*e^2*r^5*(x^r)^2+117306*ln(c)*b*d*e^2*r^4*(x^r)^2-10584*ln(c)*b*d
^2*e*r^5*x^r+197568*ln(c)*b*d^2*e*r^4*x^r+2470629*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-521017*I*Pi*b
*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3+823543*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+823543*I*Pi*b*e^3*csgn(
I*c*x^n)^2*csgn(I*c)*(x^r)^3-122451*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-1397382*ln(c)*b*d^2*e*r^3
*x^r+4739574*ln(c)*b*d^2*e*r^2*x^r-7764834*ln(c)*b*d^2*e*r*x^r-979608*ln(c)*b*d*e^2*r^3*(x^r)^2+3831996*ln(c)*
b*d*e^2*r^2*(x^r)^2-7058940*ln(c)*b*d*e^2*r*(x^r)^2+2470629*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+24706
29*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-122451*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+588*I*
Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3-13720*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3+1411788*I*Pi*b*d^3*r*csgn(I*
c*x^n)^3-823543*I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3+823543*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)-252*I*Pi*b*d^3*
r^6*csgn(I*c*x^n)^3+66199*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)-2470629*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+12
2451*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+3529470*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-823543*I*Pi*b*e^3*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-3529470*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+698691*I*P
i*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r-1915998*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-1058841*I*Pi*b*e^3*r*csgn(I
*x^n)*csgn(I*c*x^n)^2*(x^r)^3-1058841*I*Pi*b*e^3*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+1058841*I*Pi*b*e^3*r*csgn
(I*c*x^n)^3*(x^r)^3-2470629*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2-3882417*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x
^n)^2*x^r-3882417*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r-2470629*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*
csgn(I*c)*(x^r)^2-58653*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2-98784*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r+52
1017*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-1411788*I*Pi*b*d^3*r*csgn(I*c*x^n)^2*csgn(I*c)+974806*
I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+974806*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)+5292*I*Pi*b*d^2*e*r
^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+489804*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2
-823543*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+252*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2-5292*I*P
i*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+58653*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-48980
4*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-1411788*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2-345744*I
*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-345744*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)+1411788*I*Pi*b*d^3*r
*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+345744*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-2470629*I*Pi*b*
d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+98784*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+98784*I*P
i*b*d^2*e*r^4*csgn(I*c*x^n)^2*csgn(I*c)*x^r-2646*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-66199*I*
Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+58653*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+588*
I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-521017*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)*(x^r)^3-66199*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^3+1058841*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(
x^r)^3+122451*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-489804*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*c
sgn(I*c*x^n)^2*(x^r)^2-5292*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^2*csgn(I*c)*x^r+1915998*I*Pi*b*d*e^2*r^2*csgn(I*c*x
^n)^2*csgn(I*c)*(x^r)^2-2646*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-13720*I*Pi*b*e^3*r^4*csgn(I*x^
n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3)/(-7+3*r)^2/x^7/(-7+2*r)^2/(-7+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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.47732, size = 2554, normalized size = 13.96 \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^8,x, algorithm="fricas")

[Out]

-1/49*(36*(b*d^3*n + 7*a*d^3)*r^6 - 924*(b*d^3*n + 7*a*d^3)*r^5 + 117649*b*d^3*n + 9457*(b*d^3*n + 7*a*d^3)*r^
4 + 823543*a*d^3 - 49392*(b*d^3*n + 7*a*d^3)*r^3 + 139258*(b*d^3*n + 7*a*d^3)*r^2 - 201684*(b*d^3*n + 7*a*d^3)
*r - 49*(12*a*e^3*r^5 - 2401*b*e^3*n - 4*(b*e^3*n + 70*a*e^3)*r^4 - 16807*a*e^3 + 21*(4*b*e^3*n + 119*a*e^3)*r
^3 - 49*(13*b*e^3*n + 217*a*e^3)*r^2 + 1029*(2*b*e^3*n + 21*a*e^3)*r + (12*b*e^3*r^5 - 280*b*e^3*r^4 + 2499*b*
e^3*r^3 - 10633*b*e^3*r^2 + 21609*b*e^3*r - 16807*b*e^3)*log(c) + (12*b*e^3*n*r^5 - 280*b*e^3*n*r^4 + 2499*b*e
^3*n*r^3 - 10633*b*e^3*n*r^2 + 21609*b*e^3*n*r - 16807*b*e^3*n)*log(x))*x^(3*r) - 147*(18*a*d*e^2*r^5 - 2401*b
*d*e^2*n - 3*(3*b*d*e^2*n + 133*a*d*e^2)*r^4 - 16807*a*d*e^2 + 28*(6*b*d*e^2*n + 119*a*d*e^2)*r^3 - 98*(11*b*d
*e^2*n + 133*a*d*e^2)*r^2 + 686*(4*b*d*e^2*n + 35*a*d*e^2)*r + (18*b*d*e^2*r^5 - 399*b*d*e^2*r^4 + 3332*b*d*e^
2*r^3 - 13034*b*d*e^2*r^2 + 24010*b*d*e^2*r - 16807*b*d*e^2)*log(c) + (18*b*d*e^2*n*r^5 - 399*b*d*e^2*n*r^4 +
3332*b*d*e^2*n*r^3 - 13034*b*d*e^2*n*r^2 + 24010*b*d*e^2*n*r - 16807*b*d*e^2*n)*log(x))*x^(2*r) - 147*(36*a*d^
2*e*r^5 - 2401*b*d^2*e*n - 12*(3*b*d^2*e*n + 56*a*d^2*e)*r^4 - 16807*a*d^2*e + 7*(60*b*d^2*e*n + 679*a*d^2*e)*
r^3 - 49*(37*b*d^2*e*n + 329*a*d^2*e)*r^2 + 343*(10*b*d^2*e*n + 77*a*d^2*e)*r + (36*b*d^2*e*r^5 - 672*b*d^2*e*
r^4 + 4753*b*d^2*e*r^3 - 16121*b*d^2*e*r^2 + 26411*b*d^2*e*r - 16807*b*d^2*e)*log(c) + (36*b*d^2*e*n*r^5 - 672
*b*d^2*e*n*r^4 + 4753*b*d^2*e*n*r^3 - 16121*b*d^2*e*n*r^2 + 26411*b*d^2*e*n*r - 16807*b*d^2*e*n)*log(x))*x^r +
7*(36*b*d^3*r^6 - 924*b*d^3*r^5 + 9457*b*d^3*r^4 - 49392*b*d^3*r^3 + 139258*b*d^3*r^2 - 201684*b*d^3*r + 1176
49*b*d^3)*log(c) + 7*(36*b*d^3*n*r^6 - 924*b*d^3*n*r^5 + 9457*b*d^3*n*r^4 - 49392*b*d^3*n*r^3 + 139258*b*d^3*n
*r^2 - 201684*b*d^3*n*r + 117649*b*d^3*n)*log(x))/((36*r^6 - 924*r^5 + 9457*r^4 - 49392*r^3 + 139258*r^2 - 201
684*r + 117649)*x^7)

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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**8,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^{8}}\,{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^8,x, algorithm="giac")

[Out]

integrate((e*x^r + d)^3*(b*log(c*x^n) + a)/x^8, x)