3.5.4 \(\int \frac {(d+e x^r)^3 (a+b \log (c x^n))}{x^8} \, dx\) [404]
Optimal. Leaf size=183 \[ -\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n x^{-7+r}}{(7-r)^2}-\frac {3 b d e^2 n x^{-7+2 r}}{(7-2 r)^2}-\frac {b e^3 n x^{-7+3 r}}{(7-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{-7+3 r} \left (a+b \log \left (c x^n\right )\right )}{7-3 r} \]
[Out]
-1/49*b*d^3*n/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)^2-1
/7*d^3*(a+b*ln(c*x^n))/x^7-3*d^2*e*x^(-7+r)*(a+b*ln(c*x^n))/(7-r)-3*d*e^2*x^(-7+2*r)*(a+b*ln(c*x^n))/(7-2*r)-e
^3*x^(-7+3*r)*(a+b*ln(c*x^n))/(7-3*r)
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Rubi [A]
time = 0.28, antiderivative size = 183, normalized size of antiderivative = 1.00, 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 = {276, 2372, 12,
14} \begin {gather*} -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{r-7} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\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 {b d^3 n}{49 x^7}-\frac {3 b d^2 e n x^{r-7}}{(7-r)^2}-\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} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^8,x]
[Out]
-1/49*(b*d^3*n)/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)
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]]
Rule 276
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 2372
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]}, Dist[a + b*Log[c*x^n], u, 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])
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*}
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Mathematica [A]
time = 0.16, size = 164, normalized size = 0.90 \begin {gather*} \frac {-7 b d^3 n \log (x)-d^3 \left (7 a+b n-7 b n \log (x)+7 b \log \left (c x^n\right )\right )+\frac {147 d^2 e x^r \left (-b n+a (-7+r)+b (-7+r) \log \left (c x^n\right )\right )}{(-7+r)^2}+\frac {147 d e^2 x^{2 r} \left (-b n+a (-7+2 r)+b (-7+2 r) \log \left (c x^n\right )\right )}{(7-2 r)^2}+\frac {49 e^3 x^{3 r} \left (-b n+a (-7+3 r)+b (-7+3 r) \log \left (c x^n\right )\right )}{(7-3 r)^2}}{49 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^8,x]
[Out]
(-7*b*d^3*n*Log[x] - d^3*(7*a + b*n - 7*b*n*Log[x] + 7*b*Log[c*x^n]) + (147*d^2*e*x^r*(-(b*n) + a*(-7 + r) + b
*(-7 + r)*Log[c*x^n]))/(-7 + r)^2 + (147*d*e^2*x^(2*r)*(-(b*n) + a*(-7 + 2*r) + b*(-7 + 2*r)*Log[c*x^n]))/(7 -
2*r)^2 + (49*e^3*x^(3*r)*(-(b*n) + a*(-7 + 3*r) + b*(-7 + 3*r)*Log[c*x^n]))/(7 - 3*r)^2)/(49*x^7)
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.32, size = 4031, normalized size = 22.03
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| method |
result |
size |
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| risch |
\(\text {Expression too large to display}\) |
\(4031\) |
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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,method=_RETURNVERBOSE)
[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*e^3*(x^r)^3*a-974806*I*Pi*b*d^3*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x
^n)+2470629*I*Pi*b*d^2*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r+4941258*d^2*e*x^r*a+4941258*d*e^2*(x^r)^2*a-2369787*I*P
i*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r-588*I*Pi*b*e^3*r^5*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3+2369787*I*Pi*b*d^2*e*r^
2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+122451*I*Pi*b*e^3*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-123480*b*d
^2*e*n*r^3*x^r+3831996*ln(c)*b*d*e^2*r^2*(x^r)^2-7058940*ln(c)*b*d*e^2*r*(x^r)^2+504*a*d^3*r^6-12936*a*d^3*r^5
+132398*a*d^3*r^4-3529470*I*Pi*b*d*e^2*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-5292*I*Pi*b*d^2*e*r^5*csgn(I*c)*csg
n(I*c*x^n)^2*x^r-1915998*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-2646*I*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*c*x^n
)^2*(x^r)^2+1647086*a*d^3-979608*a*d*e^2*r^3*(x^r)^2+3831996*a*d*e^2*r^2*(x^r)^2-7058940*a*d*e^2*r*(x^r)^2-139
7382*a*d^2*e*r^3*x^r+4739574*a*d^2*e*r^2*x^r-7764834*a*d^2*e*r*x^r-252*I*Pi*b*d^3*r^6*csgn(I*c)*csgn(I*x^n)*cs
gn(I*c*x^n)+2470629*I*Pi*b*d*e^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+72*b*d^3*n*r^6-1848*b*d^3*n*r^5+18914*b*d^3
*n*r^4-6468*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2+66199*I*Pi*b*d^3*r^4*csgn(I*c)*csgn(I*c*x^n)^2+66199*I*
Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+698691*I*Pi*b*d^2*e*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+2646*
I*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+533022*b*d^2*e*n*r^2*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-6914
88*a*d^3*r^3+1949612*a*d^3*r^2-2823576*a*d^3*r-345744*I*Pi*b*d^3*r^3*csgn(I*c)*csgn(I*c*x^n)^2-345744*I*Pi*b*d
^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2+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+235298*b*d^3*n-1176*a*e^3*r^5*(x^r)^3+27440*a*e^3*r
^4*(x^r)^3+1647086*ln(c)*b*e^3*(x^r)^3+235298*b*e^3*n*(x^r)^3-244902*a*e^3*r^3*(x^r)^3+1647086*d^3*b*ln(c)-122
451*I*Pi*b*e^3*r^3*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3-98784*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r-122451*I*Pi*b*
e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-98784*b*d^3*n*r^3+278516*b*d^3*n*r^2-403368*b*d^3*n*r+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+4941258*ln(c)*b*d^2*e*x^r
-3882417*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r-1411788*I*Pi*b*d^3*r*csgn(I*c)*csgn(I*c*x^n)^2-1411788*I
*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2-823543*I*Pi*b*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1042034*a*e^3*r^
2*(x^r)^3-2117682*a*e^3*r*(x^r)^3+6468*I*Pi*b*d^3*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+4941258*ln(c)*b*d*e^
2*(x^r)^2-244902*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)^3-1176*ln
(c)*b*e^3*r^5*(x^r)^3+27440*ln(c)*b*e^3*r^4*(x^r)^3+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+98784*I*Pi*b*d^2*e*r^4
*csgn(I*c)*csgn(I*c*x^n)^2*x^r+98784*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-5292*I*Pi*b*d^2*e*r^5*cs
gn(I*x^n)*csgn(I*c*x^n)^2*x^r-698691*I*Pi*b*d^2*e*r^3*csgn(I*c)*csgn(I*c*x^n)^2*x^r-521017*I*Pi*b*e^3*r^2*csgn
(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-489804*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+316932*b*d
*e^2*n*r^2*(x^r)^2-10584*ln(c)*b*d^2*e*r^5*x^r+122451*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3-521017*I*Pi*b*e^3
*r^2*csgn(I*c*x^n)^3*(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)^2-66199*I*Pi*b*d^3*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+823543*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^
2+6468*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3-66199*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^3+698691*I*Pi*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)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-2369787*I*Pi*b*d^2*e*r^2*csgn(I*
c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+3882417*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r-58653*I*Pi*b*d*e^2*r^4*csgn(I*c*x^
n)^3*(x^r)^2+1915998*I*Pi*b*d*e^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-823543*I*Pi*b*d^3*csgn(I*c*x^n)^3+588*
I*Pi*b*e^3*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3+1058841*I*Pi*b*e^3*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c
*x^n)*(x^r)^3-2646*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+117306*ln(c)*b*d*e^2*r^4*(x^r)^2-13973
82*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+3529470*I*Pi*b*d*e^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+3882417*I*Pi*b*d^2*e*r*csgn(I*c)*csgn
(I*x^n)*csgn(I*c*x^n)*x^r-58653*I*Pi*b*d*e^2*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-98784*I*Pi*b*d^2*
e*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-13720*I*Pi*b*e^3*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3
+2369787*I*Pi*b*d^2*e*r^2*csgn(I*c)*csgn(I*c*x^...
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(r-8>0)', see `assume?` for mor
e details)Is
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 844 vs.
\(2 (172) = 344\).
time = 0.37, size = 844, normalized size = 4.61 \begin {gather*} -\frac {36 \, {\left (b d^{3} n + 7 \, a d^{3}\right )} r^{6} - 924 \, {\left (b d^{3} n + 7 \, a d^{3}\right )} r^{5} + 117649 \, b d^{3} n + 9457 \, {\left (b d^{3} n + 7 \, a d^{3}\right )} r^{4} + 823543 \, a d^{3} - 49392 \, {\left (b d^{3} n + 7 \, a d^{3}\right )} r^{3} + 139258 \, {\left (b d^{3} n + 7 \, a d^{3}\right )} r^{2} - 201684 \, {\left (b d^{3} n + 7 \, a d^{3}\right )} r - 49 \, {\left ({\left (12 \, b r^{5} - 280 \, b r^{4} + 2499 \, b r^{3} - 10633 \, b r^{2} + 21609 \, b r - 16807 \, b\right )} e^{3} \log \left (c\right ) + {\left (12 \, b n r^{5} - 280 \, b n r^{4} + 2499 \, b n r^{3} - 10633 \, b n r^{2} + 21609 \, b n r - 16807 \, b n\right )} e^{3} \log \left (x\right ) + {\left (12 \, a r^{5} - 4 \, {\left (b n + 70 \, a\right )} r^{4} + 21 \, {\left (4 \, b n + 119 \, a\right )} r^{3} - 49 \, {\left (13 \, b n + 217 \, a\right )} r^{2} - 2401 \, b n + 1029 \, {\left (2 \, b n + 21 \, a\right )} r - 16807 \, a\right )} e^{3}\right )} x^{3 \, r} - 147 \, {\left ({\left (18 \, b d r^{5} - 399 \, b d r^{4} + 3332 \, b d r^{3} - 13034 \, b d r^{2} + 24010 \, b d r - 16807 \, b d\right )} e^{2} \log \left (c\right ) + {\left (18 \, b d n r^{5} - 399 \, b d n r^{4} + 3332 \, b d n r^{3} - 13034 \, b d n r^{2} + 24010 \, b d n r - 16807 \, b d n\right )} e^{2} \log \left (x\right ) + {\left (18 \, a d r^{5} - 3 \, {\left (3 \, b d n + 133 \, a d\right )} r^{4} + 28 \, {\left (6 \, b d n + 119 \, a d\right )} r^{3} - 2401 \, b d n - 98 \, {\left (11 \, b d n + 133 \, a d\right )} r^{2} - 16807 \, a d + 686 \, {\left (4 \, b d n + 35 \, a d\right )} r\right )} e^{2}\right )} x^{2 \, r} - 147 \, {\left ({\left (36 \, b d^{2} r^{5} - 672 \, b d^{2} r^{4} + 4753 \, b d^{2} r^{3} - 16121 \, b d^{2} r^{2} + 26411 \, b d^{2} r - 16807 \, b d^{2}\right )} e \log \left (c\right ) + {\left (36 \, b d^{2} n r^{5} - 672 \, b d^{2} n r^{4} + 4753 \, b d^{2} n r^{3} - 16121 \, b d^{2} n r^{2} + 26411 \, b d^{2} n r - 16807 \, b d^{2} n\right )} e \log \left (x\right ) + {\left (36 \, a d^{2} r^{5} - 12 \, {\left (3 \, b d^{2} n + 56 \, a d^{2}\right )} r^{4} - 2401 \, b d^{2} n + 7 \, {\left (60 \, b d^{2} n + 679 \, a d^{2}\right )} r^{3} - 16807 \, a d^{2} - 49 \, {\left (37 \, b d^{2} n + 329 \, a d^{2}\right )} r^{2} + 343 \, {\left (10 \, b d^{2} n + 77 \, a d^{2}\right )} r\right )} e\right )} x^{r} + 7 \, {\left (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 + 117649 \, b d^{3}\right )} \log \left (c\right ) + 7 \, {\left (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\right )} \log \left (x\right )}{49 \, {\left (36 \, r^{6} - 924 \, r^{5} + 9457 \, r^{4} - 49392 \, r^{3} + 139258 \, r^{2} - 201684 \, r + 117649\right )} x^{7}} \end {gather*}
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*b*r^5 - 280*b*r^4 + 2499*b*r^3 - 10633*b*r^2 + 21609*b*r - 16807*b)*e^3*log(c) + (12*b*n*r^5 - 28
0*b*n*r^4 + 2499*b*n*r^3 - 10633*b*n*r^2 + 21609*b*n*r - 16807*b*n)*e^3*log(x) + (12*a*r^5 - 4*(b*n + 70*a)*r^
4 + 21*(4*b*n + 119*a)*r^3 - 49*(13*b*n + 217*a)*r^2 - 2401*b*n + 1029*(2*b*n + 21*a)*r - 16807*a)*e^3)*x^(3*r
) - 147*((18*b*d*r^5 - 399*b*d*r^4 + 3332*b*d*r^3 - 13034*b*d*r^2 + 24010*b*d*r - 16807*b*d)*e^2*log(c) + (18*
b*d*n*r^5 - 399*b*d*n*r^4 + 3332*b*d*n*r^3 - 13034*b*d*n*r^2 + 24010*b*d*n*r - 16807*b*d*n)*e^2*log(x) + (18*a
*d*r^5 - 3*(3*b*d*n + 133*a*d)*r^4 + 28*(6*b*d*n + 119*a*d)*r^3 - 2401*b*d*n - 98*(11*b*d*n + 133*a*d)*r^2 - 1
6807*a*d + 686*(4*b*d*n + 35*a*d)*r)*e^2)*x^(2*r) - 147*((36*b*d^2*r^5 - 672*b*d^2*r^4 + 4753*b*d^2*r^3 - 1612
1*b*d^2*r^2 + 26411*b*d^2*r - 16807*b*d^2)*e*log(c) + (36*b*d^2*n*r^5 - 672*b*d^2*n*r^4 + 4753*b*d^2*n*r^3 - 1
6121*b*d^2*n*r^2 + 26411*b*d^2*n*r - 16807*b*d^2*n)*e*log(x) + (36*a*d^2*r^5 - 12*(3*b*d^2*n + 56*a*d^2)*r^4 -
2401*b*d^2*n + 7*(60*b*d^2*n + 679*a*d^2)*r^3 - 16807*a*d^2 - 49*(37*b*d^2*n + 329*a*d^2)*r^2 + 343*(10*b*d^2
*n + 77*a*d^2)*r)*e)*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 + 117649*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 - 49
392*r^3 + 139258*r^2 - 201684*r + 117649)*x^7)
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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((x^r*e + d)^3*(b*log(c*x^n) + a)/x^8, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^8,x)
[Out]
int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^8, x)
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