3.5.5 \(\int \frac {(d+e x^r)^3 (a+b \log (c x^n))}{x^{10}} \, dx\) [405]

Optimal. Leaf size=191 \[ -\frac {b d^3 n}{81 x^9}-\frac {b e^3 n x^{-3 (3-r)}}{9 (3-r)^2}-\frac {3 b d^2 e n x^{-9+r}}{(9-r)^2}-\frac {3 b d e^2 n x^{-9+2 r}}{(9-2 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {e^3 x^{-3 (3-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (3-r)}-\frac {3 d^2 e x^{-9+r} \left (a+b \log \left (c x^n\right )\right )}{9-r}-\frac {3 d e^2 x^{-9+2 r} \left (a+b \log \left (c x^n\right )\right )}{9-2 r} \]

[Out]

-1/81*b*d^3*n/x^9-1/9*b*e^3*n/(3-r)^2/(x^(9-3*r))-3*b*d^2*e*n*x^(-9+r)/(9-r)^2-3*b*d*e^2*n*x^(-9+2*r)/(9-2*r)^
2-1/9*d^3*(a+b*ln(c*x^n))/x^9-1/3*e^3*(a+b*ln(c*x^n))/(3-r)/(x^(9-3*r))-3*d^2*e*x^(-9+r)*(a+b*ln(c*x^n))/(9-r)
-3*d*e^2*x^(-9+2*r)*(a+b*ln(c*x^n))/(9-2*r)

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Rubi [A]
time = 0.28, antiderivative size = 191, 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 )}{9 x^9}-\frac {3 d^2 e x^{r-9} \left (a+b \log \left (c x^n\right )\right )}{9-r}-\frac {3 d e^2 x^{2 r-9} \left (a+b \log \left (c x^n\right )\right )}{9-2 r}-\frac {e^3 x^{-3 (3-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (3-r)}-\frac {b d^3 n}{81 x^9}-\frac {3 b d^2 e n x^{r-9}}{(9-r)^2}-\frac {3 b d e^2 n x^{2 r-9}}{(9-2 r)^2}-\frac {b e^3 n x^{-3 (3-r)}}{9 (3-r)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^10,x]

[Out]

-1/81*(b*d^3*n)/x^9 - (b*e^3*n)/(9*(3 - r)^2*x^(3*(3 - r))) - (3*b*d^2*e*n*x^(-9 + r))/(9 - r)^2 - (3*b*d*e^2*
n*x^(-9 + 2*r))/(9 - 2*r)^2 - (d^3*(a + b*Log[c*x^n]))/(9*x^9) - (e^3*(a + b*Log[c*x^n]))/(3*(3 - r)*x^(3*(3 -
r))) - (3*d^2*e*x^(-9 + r)*(a + b*Log[c*x^n]))/(9 - r) - (3*d*e^2*x^(-9 + 2*r)*(a + b*Log[c*x^n]))/(9 - 2*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^{10}} \, dx &=-\frac {1}{9} \left (\frac {d^3}{x^9}+\frac {3 e^3 x^{-3 (3-r)}}{3-r}+\frac {27 d^2 e x^{-9+r}}{9-r}+\frac {27 d e^2 x^{-9+2 r}}{9-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^3+\frac {27 d^2 e x^r}{-9+r}+\frac {27 d e^2 x^{2 r}}{-9+2 r}+\frac {3 e^3 x^{3 r}}{-3+r}}{9 x^{10}} \, dx\\ &=-\frac {1}{9} \left (\frac {d^3}{x^9}+\frac {3 e^3 x^{-3 (3-r)}}{3-r}+\frac {27 d^2 e x^{-9+r}}{9-r}+\frac {27 d e^2 x^{-9+2 r}}{9-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} (b n) \int \frac {-d^3+\frac {27 d^2 e x^r}{-9+r}+\frac {27 d e^2 x^{2 r}}{-9+2 r}+\frac {3 e^3 x^{3 r}}{-3+r}}{x^{10}} \, dx\\ &=-\frac {1}{9} \left (\frac {d^3}{x^9}+\frac {3 e^3 x^{-3 (3-r)}}{3-r}+\frac {27 d^2 e x^{-9+r}}{9-r}+\frac {27 d e^2 x^{-9+2 r}}{9-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} (b n) \int \left (-\frac {d^3}{x^{10}}+\frac {27 d^2 e x^{-10+r}}{-9+r}+\frac {27 d e^2 x^{2 (-5+r)}}{-9+2 r}+\frac {3 e^3 x^{-10+3 r}}{-3+r}\right ) \, dx\\ &=-\frac {b d^3 n}{81 x^9}-\frac {b e^3 n x^{-3 (3-r)}}{9 (3-r)^2}-\frac {3 b d^2 e n x^{-9+r}}{(9-r)^2}-\frac {3 b d e^2 n x^{-9+2 r}}{(9-2 r)^2}-\frac {1}{9} \left (\frac {d^3}{x^9}+\frac {3 e^3 x^{-3 (3-r)}}{3-r}+\frac {27 d^2 e x^{-9+r}}{9-r}+\frac {27 d e^2 x^{-9+2 r}}{9-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 160, normalized size = 0.84 \begin {gather*} \frac {-9 b d^3 n \log (x)-d^3 \left (9 a+b n-9 b n \log (x)+9 b \log \left (c x^n\right )\right )+\frac {243 d^2 e x^r \left (-b n+a (-9+r)+b (-9+r) \log \left (c x^n\right )\right )}{(-9+r)^2}+\frac {9 e^3 x^{3 r} \left (-b n+3 a (-3+r)+3 b (-3+r) \log \left (c x^n\right )\right )}{(-3+r)^2}+\frac {243 d e^2 x^{2 r} \left (-b n+a (-9+2 r)+b (-9+2 r) \log \left (c x^n\right )\right )}{(9-2 r)^2}}{81 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^10,x]

[Out]

(-9*b*d^3*n*Log[x] - d^3*(9*a + b*n - 9*b*n*Log[x] + 9*b*Log[c*x^n]) + (243*d^2*e*x^r*(-(b*n) + a*(-9 + r) + b
*(-9 + r)*Log[c*x^n]))/(-9 + r)^2 + (9*e^3*x^(3*r)*(-(b*n) + 3*a*(-3 + r) + 3*b*(-3 + r)*Log[c*x^n]))/(-3 + r)
^2 + (243*d*e^2*x^(2*r)*(-(b*n) + a*(-9 + 2*r) + b*(-9 + 2*r)*Log[c*x^n]))/(9 - 2*r)^2)/(81*x^9)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.33, size = 4027, normalized size = 21.08

method	result	size
risch	\(\text {Expression too large to display}\)	\(4027\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*x^r)^3*(a+b*ln(c*x^n))/x^10,x,method=_RETURNVERBOSE)

[Out]

-1/9*b*(-6*e^3*r^2*(x^r)^3-27*d*e^2*r^2*(x^r)^2+81*e^3*r*(x^r)^3+2*d^3*r^3-54*d^2*e*r^2*x^r+324*d*e^2*r*(x^r)^
2-243*e^3*(x^r)^3-33*d^3*r^2+405*d^2*e*r*x^r-729*d*e^2*(x^r)^2+162*d^3*r-729*d^2*e*x^r-243*d^3)/x^9/(-3+r)/(-9
+2*r)/(-9+r)*ln(x^n)-1/162*(1062882*e^3*(x^r)^3*a+3188646*d^2*e*x^r*a+3188646*d*e^2*(x^r)^2*a-104976*I*Pi*b*d^
3*r^3*csgn(I*c)*csgn(I*c*x^n)^2-104976*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2+3240*I*Pi*b*e^3*r^4*csgn(I*x
^n)*csgn(I*c*x^n)^2*(x^r)^3-925101*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r-531441*I*Pi*b*e^3*csgn(I*c)*csgn(I*x^n
)*csgn(I*c*x^n)*(x^r)^3+708588*I*Pi*b*d^3*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-29160*b*d^2*e*n*r^3*x^r+149590
8*ln(c)*b*d*e^2*r^2*(x^r)^2-3542940*ln(c)*b*d*e^2*r*(x^r)^2-1594323*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+72*a*d^3*
r^6-2376*a*d^3*r^5+31266*a*d^3*r^4-23328*I*Pi*b*d^2*e*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+972*I*Pi*b*d
^2*e*r^5*csgn(I*c*x^n)^3*x^r+1062882*a*d^3-297432*a*d*e^2*r^3*(x^r)^2+1495908*a*d*e^2*r^2*(x^r)^2-3542940*a*d*
e^2*r*(x^r)^2-424278*a*d^2*e*r^3*x^r+1850202*a*d^2*e*r^2*x^r-3897234*a*d^2*e*r*x^r+8*b*d^3*n*r^6-264*b*d^3*n*r
^5+3474*b*d^3*n*r^4+161838*b*d^2*e*n*r^2*x^r-314928*b*d*e^2*n*r*(x^r)^2-393660*b*d^2*e*n*r*x^r+486*b*d*e^2*n*r
^4*(x^r)^2-11664*b*d*e^2*n*r^3*(x^r)^2+1944*b*d^2*e*n*r^4*x^r-209952*a*d^3*r^3+761076*a*d^3*r^2-1417176*a*d^3*
r-708588*I*Pi*b*d^3*r*csgn(I*c)*csgn(I*c*x^n)^2+72*ln(c)*b*d^3*r^6-2376*ln(c)*b*d^3*r^5+31266*ln(c)*b*d^3*r^4-
209952*ln(c)*b*d^3*r^3+761076*ln(c)*b*d^3*r^2-1417176*ln(c)*b*d^3*r+118098*b*d^3*n-216*a*e^3*r^5*(x^r)^3+6480*
a*e^3*r^4*(x^r)^3+1062882*ln(c)*b*e^3*(x^r)^3+118098*b*e^3*n*(x^r)^3-74358*a*e^3*r^3*(x^r)^3+1062882*d^3*b*ln(
c)-23328*b*d^3*n*r^3+84564*b*d^3*n*r^2-157464*b*d^3*n*r+18954*b*e^3*n*r^2*(x^r)^3-78732*b*e^3*n*r*(x^r)^3+3542
94*b*d*e^2*n*(x^r)^2+354294*b*d^2*e*n*x^r+3188646*ln(c)*b*d^2*e*x^r-1948617*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*c*
x^n)^2*x^r-1594323*I*Pi*b*d*e^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+406782*a*e^3*r^2*(x^r)^3-1062882*a
*e^3*r*(x^r)^3+104976*I*Pi*b*d^3*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-380538*I*Pi*b*d^3*r^2*csgn(I*c)*csgn(
I*x^n)*csgn(I*c*x^n)+1594323*I*Pi*b*d^2*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r+212139*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^
3*x^r+3188646*ln(c)*b*d*e^2*(x^r)^2-74358*ln(c)*b*e^3*r^3*(x^r)^3+406782*ln(c)*b*e^3*r^2*(x^r)^3-1062882*ln(c)
*b*e^3*r*(x^r)^3-216*ln(c)*b*e^3*r^5*(x^r)^3+6480*ln(c)*b*e^3*r^4*(x^r)^3+72*b*e^3*n*r^4*(x^r)^3-1944*b*e^3*n*
r^3*(x^r)^3-972*a*d*e^2*r^5*(x^r)^2+27702*a*d*e^2*r^4*(x^r)^2-1944*a*d^2*e*r^5*x^r+46656*a*d^2*e*r^4*x^r+15943
23*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+15633*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+37179*I*Pi*b*
e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3-203391*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3-747954*I*Pi*b*d*e^2*r^2*csgn(I*c
*x^n)^3*(x^r)^2-531441*I*Pi*b*e^3*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3-531441*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c
*x^n)^2*(x^r)^3+96228*b*d*e^2*n*r^2*(x^r)^2+925101*I*Pi*b*d^2*e*r^2*csgn(I*c)*csgn(I*c*x^n)^2*x^r+925101*I*Pi*
b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+37179*I*Pi*b*e^3*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-1
944*ln(c)*b*d^2*e*r^5*x^r+36*I*Pi*b*d^3*r^6*csgn(I*c)*csgn(I*c*x^n)^2-13851*I*Pi*b*d*e^2*r^4*csgn(I*c)*csgn(I*
x^n)*csgn(I*c*x^n)*(x^r)^2+108*I*Pi*b*e^3*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-148716*I*Pi*b*d*e^2*
r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+531441*I*Pi*b*e^3*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3+177147
0*I*Pi*b*d*e^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+1948617*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*x^n)*csgn
(I*c*x^n)*x^r+104976*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3-380538*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3-1594323*I*Pi*b*d^2*e
*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+212139*I*Pi*b*d^2*e*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+486*I
*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2-203391*I*Pi*b*e^3*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I
*c*x^n)*(x^r)^3+747954*I*Pi*b*d*e^2*r^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2+747954*I*Pi*b*d*e^2*r^2*csgn(I*x^n)*
csgn(I*c*x^n)^2*(x^r)^2-486*I*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-3240*I*Pi*b*e^3*r^4*csgn(I*c)*c
sgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3+531441*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x^r)^3-1594323*I*Pi*b*d*e^2*csgn(I*c*x^n
)^3*(x^r)^2-108*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+203391*I*Pi*b*e^3*r^2*csgn(I*c)*csgn(I*c*x^
n)^2*(x^r)^3+108*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3-3240*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3-148716*I*P
i*b*d*e^2*r^3*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-972*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+27702*ln(
c)*b*d*e^2*r^4*(x^r)^2-424278*ln(c)*b*d^2*e*r^3*x^r+1850202*ln(c)*b*d^2*e*r^2*x^r-3897234*ln(c)*b*d^2*e*r*x^r-
297432*ln(c)*b*d*e^2*r^3*(x^r)^2+148716*I*Pi*b*d*e^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+972*I*Pi*
b*d^2*e*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+531441*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+3805
38*I*Pi*b*d^3*r^2*csgn(I*c)*csgn(I*c*x^n)^2+380538*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-212139*I*Pi*b*d^
2*e*r^3*csgn(I*c)*csgn(I*c*x^n)^2*x^r-486*I*Pi*...

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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^10,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-10>0)', see `assume?` for mo
re details)I

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 844 vs. \(2 (173) = 346\).
time = 0.37, size = 844, normalized size = 4.42 \begin {gather*} -\frac {4 \, {\left (b d^{3} n + 9 \, a d^{3}\right )} r^{6} - 132 \, {\left (b d^{3} n + 9 \, a d^{3}\right )} r^{5} + 59049 \, b d^{3} n + 1737 \, {\left (b d^{3} n + 9 \, a d^{3}\right )} r^{4} + 531441 \, a d^{3} - 11664 \, {\left (b d^{3} n + 9 \, a d^{3}\right )} r^{3} + 42282 \, {\left (b d^{3} n + 9 \, a d^{3}\right )} r^{2} - 78732 \, {\left (b d^{3} n + 9 \, a d^{3}\right )} r - 9 \, {\left (3 \, {\left (4 \, b r^{5} - 120 \, b r^{4} + 1377 \, b r^{3} - 7533 \, b r^{2} + 19683 \, b r - 19683 \, b\right )} e^{3} \log \left (c\right ) + 3 \, {\left (4 \, b n r^{5} - 120 \, b n r^{4} + 1377 \, b n r^{3} - 7533 \, b n r^{2} + 19683 \, b n r - 19683 \, b n\right )} e^{3} \log \left (x\right ) + {\left (12 \, a r^{5} - 4 \, {\left (b n + 90 \, a\right )} r^{4} + 27 \, {\left (4 \, b n + 153 \, a\right )} r^{3} - 81 \, {\left (13 \, b n + 279 \, a\right )} r^{2} - 6561 \, b n + 2187 \, {\left (2 \, b n + 27 \, a\right )} r - 59049 \, a\right )} e^{3}\right )} x^{3 \, r} - 243 \, {\left ({\left (2 \, b d r^{5} - 57 \, b d r^{4} + 612 \, b d r^{3} - 3078 \, b d r^{2} + 7290 \, b d r - 6561 \, b d\right )} e^{2} \log \left (c\right ) + {\left (2 \, b d n r^{5} - 57 \, b d n r^{4} + 612 \, b d n r^{3} - 3078 \, b d n r^{2} + 7290 \, b d n r - 6561 \, b d n\right )} e^{2} \log \left (x\right ) + {\left (2 \, a d r^{5} - {\left (b d n + 57 \, a d\right )} r^{4} + 12 \, {\left (2 \, b d n + 51 \, a d\right )} r^{3} - 729 \, b d n - 18 \, {\left (11 \, b d n + 171 \, a d\right )} r^{2} - 6561 \, a d + 162 \, {\left (4 \, b d n + 45 \, a d\right )} r\right )} e^{2}\right )} x^{2 \, r} - 243 \, {\left ({\left (4 \, b d^{2} r^{5} - 96 \, b d^{2} r^{4} + 873 \, b d^{2} r^{3} - 3807 \, b d^{2} r^{2} + 8019 \, b d^{2} r - 6561 \, b d^{2}\right )} e \log \left (c\right ) + {\left (4 \, b d^{2} n r^{5} - 96 \, b d^{2} n r^{4} + 873 \, b d^{2} n r^{3} - 3807 \, b d^{2} n r^{2} + 8019 \, b d^{2} n r - 6561 \, b d^{2} n\right )} e \log \left (x\right ) + {\left (4 \, a d^{2} r^{5} - 4 \, {\left (b d^{2} n + 24 \, a d^{2}\right )} r^{4} - 729 \, b d^{2} n + 3 \, {\left (20 \, b d^{2} n + 291 \, a d^{2}\right )} r^{3} - 6561 \, a d^{2} - 9 \, {\left (37 \, b d^{2} n + 423 \, a d^{2}\right )} r^{2} + 81 \, {\left (10 \, b d^{2} n + 99 \, a d^{2}\right )} r\right )} e\right )} x^{r} + 9 \, {\left (4 \, b d^{3} r^{6} - 132 \, b d^{3} r^{5} + 1737 \, b d^{3} r^{4} - 11664 \, b d^{3} r^{3} + 42282 \, b d^{3} r^{2} - 78732 \, b d^{3} r + 59049 \, b d^{3}\right )} \log \left (c\right ) + 9 \, {\left (4 \, b d^{3} n r^{6} - 132 \, b d^{3} n r^{5} + 1737 \, b d^{3} n r^{4} - 11664 \, b d^{3} n r^{3} + 42282 \, b d^{3} n r^{2} - 78732 \, b d^{3} n r + 59049 \, b d^{3} n\right )} \log \left (x\right )}{81 \, {\left (4 \, r^{6} - 132 \, r^{5} + 1737 \, r^{4} - 11664 \, r^{3} + 42282 \, r^{2} - 78732 \, r + 59049\right )} x^{9}} \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^10,x, algorithm="fricas")

[Out]

-1/81*(4*(b*d^3*n + 9*a*d^3)*r^6 - 132*(b*d^3*n + 9*a*d^3)*r^5 + 59049*b*d^3*n + 1737*(b*d^3*n + 9*a*d^3)*r^4
+ 531441*a*d^3 - 11664*(b*d^3*n + 9*a*d^3)*r^3 + 42282*(b*d^3*n + 9*a*d^3)*r^2 - 78732*(b*d^3*n + 9*a*d^3)*r -
9*(3*(4*b*r^5 - 120*b*r^4 + 1377*b*r^3 - 7533*b*r^2 + 19683*b*r - 19683*b)*e^3*log(c) + 3*(4*b*n*r^5 - 120*b*
n*r^4 + 1377*b*n*r^3 - 7533*b*n*r^2 + 19683*b*n*r - 19683*b*n)*e^3*log(x) + (12*a*r^5 - 4*(b*n + 90*a)*r^4 + 2
7*(4*b*n + 153*a)*r^3 - 81*(13*b*n + 279*a)*r^2 - 6561*b*n + 2187*(2*b*n + 27*a)*r - 59049*a)*e^3)*x^(3*r) - 2
43*((2*b*d*r^5 - 57*b*d*r^4 + 612*b*d*r^3 - 3078*b*d*r^2 + 7290*b*d*r - 6561*b*d)*e^2*log(c) + (2*b*d*n*r^5 -
57*b*d*n*r^4 + 612*b*d*n*r^3 - 3078*b*d*n*r^2 + 7290*b*d*n*r - 6561*b*d*n)*e^2*log(x) + (2*a*d*r^5 - (b*d*n +
57*a*d)*r^4 + 12*(2*b*d*n + 51*a*d)*r^3 - 729*b*d*n - 18*(11*b*d*n + 171*a*d)*r^2 - 6561*a*d + 162*(4*b*d*n +
45*a*d)*r)*e^2)*x^(2*r) - 243*((4*b*d^2*r^5 - 96*b*d^2*r^4 + 873*b*d^2*r^3 - 3807*b*d^2*r^2 + 8019*b*d^2*r - 6
561*b*d^2)*e*log(c) + (4*b*d^2*n*r^5 - 96*b*d^2*n*r^4 + 873*b*d^2*n*r^3 - 3807*b*d^2*n*r^2 + 8019*b*d^2*n*r -
6561*b*d^2*n)*e*log(x) + (4*a*d^2*r^5 - 4*(b*d^2*n + 24*a*d^2)*r^4 - 729*b*d^2*n + 3*(20*b*d^2*n + 291*a*d^2)*
r^3 - 6561*a*d^2 - 9*(37*b*d^2*n + 423*a*d^2)*r^2 + 81*(10*b*d^2*n + 99*a*d^2)*r)*e)*x^r + 9*(4*b*d^3*r^6 - 13
2*b*d^3*r^5 + 1737*b*d^3*r^4 - 11664*b*d^3*r^3 + 42282*b*d^3*r^2 - 78732*b*d^3*r + 59049*b*d^3)*log(c) + 9*(4*
b*d^3*n*r^6 - 132*b*d^3*n*r^5 + 1737*b*d^3*n*r^4 - 11664*b*d^3*n*r^3 + 42282*b*d^3*n*r^2 - 78732*b*d^3*n*r + 5
9049*b*d^3*n)*log(x))/((4*r^6 - 132*r^5 + 1737*r^4 - 11664*r^3 + 42282*r^2 - 78732*r + 59049)*x^9)

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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**10,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^10,x, algorithm="giac")

[Out]

integrate((x^r*e + d)^3*(b*log(c*x^n) + a)/x^10, 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^{10}} \,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^10,x)

[Out]

int(((d + e*x^r)^3*(a + b*log(c*x^n)))/x^10, x)

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