3.5.3 \(\int \frac {(d+e x^r)^3 (a+b \log (c x^n))}{x^6} \, dx\) [403]

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

[Out]

-1/25*b*d^3*n/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)^2-1
/5*d^3*(a+b*ln(c*x^n))/x^5-3*d^2*e*x^(-5+r)*(a+b*ln(c*x^n))/(5-r)-3*d*e^2*x^(-5+2*r)*(a+b*ln(c*x^n))/(5-2*r)-e
^3*x^(-5+3*r)*(a+b*ln(c*x^n))/(5-3*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 )}{5 x^5}-\frac {3 d^2 e x^{r-5} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\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 {b d^3 n}{25 x^5}-\frac {3 b d^2 e n x^{r-5}}{(5-r)^2}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/25*(b*d^3*n)/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)

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^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*}

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Mathematica [A]
time = 0.16, size = 164, normalized size = 0.90 \begin {gather*} \frac {-5 b d^3 n \log (x)-d^3 \left (5 a+b n-5 b n \log (x)+5 b \log \left (c x^n\right )\right )+\frac {75 d^2 e x^r \left (-b n+a (-5+r)+b (-5+r) \log \left (c x^n\right )\right )}{(-5+r)^2}+\frac {75 d e^2 x^{2 r} \left (-b n+a (-5+2 r)+b (-5+2 r) \log \left (c x^n\right )\right )}{(5-2 r)^2}+\frac {25 e^3 x^{3 r} \left (-b n+a (-5+3 r)+b (-5+3 r) \log \left (c x^n\right )\right )}{(5-3 r)^2}}{25 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.34, size = 4031, normalized size = 22.03

method	result	size
risch	\(\text {Expression too large to display}\)	\(4031\)

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,method=_RETURNVERBOSE)

[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*(156250*e^3*(x^r)^3*a+468750*d^2*e*x^r*a+468750*d*e^2*(x^r)^2*a+181250*I*Pi*b*d^3
*r^2*csgn(I*c)*csgn(I*c*x^n)^2+181250*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+180*I*Pi*b*d^3*r^6*csgn(I*x^n
)*csgn(I*c*x^n)^2-45000*b*d^2*e*n*r^3*x^r+712500*ln(c)*b*d*e^2*r^2*(x^r)^2-937500*ln(c)*b*d*e^2*r*(x^r)^2+3187
5*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+360*a*d^3*r^6-6600*a*d^3*r^5+48250*a*d^3*r^4+5000*I*Pi*b*e^3*r^4*csgn
(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+468750*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(x^r)^2+156250*a*d^3-255000*a*d*e^2*r^3*
(x^r)^2+712500*a*d*e^2*r^2*(x^r)^2-937500*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-1031
250*a*d^2*e*r*x^r+72*b*d^3*n*r^6-1320*b*d^3*n*r^5+9650*b*d^3*n*r^4+78125*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^
2-356250*I*Pi*b*d*e^2*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+138750*b*d^2*e*n*r^2*x^r-150000*b*d*e^2*
n*r*(x^r)^2-187500*b*d^2*e*n*r*x^r+1350*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-180000*a*d^3*r^3+362500*a*d^3*r^2-375000*a*d^3*r+3300*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^3+24125*I*Pi*b*d^3*r^4*c
sgn(I*c)*csgn(I*c*x^n)^2+24125*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2-96875*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3
*(x^r)^3+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+31250*b*d^3*n-600*a*e^3*r^5*(x^r)^3+10000*a*e^3*r^4*(x^r)^3+156250*ln(c)*b*e^3*(x
^r)^3+31250*b*e^3*n*(x^r)^3-63750*a*e^3*r^3*(x^r)^3+156250*d^3*b*ln(c)-78125*I*Pi*b*d^3*csgn(I*c*x^n)^3-36000*
b*d^3*n*r^3+72500*b*d^3*n*r^2-75000*b*d^3*n*r+16250*b*e^3*n*r^2*(x^r)^3-37500*b*e^3*n*r*(x^r)^3+93750*b*d*e^2*
n*(x^r)^2+93750*b*d^2*e*n*x^r+468750*ln(c)*b*d^2*e*x^r-234375*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2-234375*I*Pi
*b*d^2*e*csgn(I*c*x^n)^3*x^r-5000*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3+36000*I*Pi*b*d^2*e*r^4*csgn(I*c)*csgn
(I*c*x^n)^2*x^r+36000*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+356250*I*Pi*b*d*e^2*r^2*csgn(I*c)*csgn(
I*c*x^n)^2*(x^r)^2+193750*a*e^3*r^2*(x^r)^3-281250*a*e^3*r*(x^r)^3+181875*I*Pi*b*d^2*e*r^3*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)*x^r+1350*I*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+127500*I*Pi*b*d*e^2*r^3
*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+2700*I*Pi*b*d^2*e*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+468
750*ln(c)*b*d*e^2*(x^r)^2-63750*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)^3-600*ln(c)*b*e^3*r^5*(x^r)^3+10000*ln(c)*b*e^3*r^4*(x^r)^3+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+72000*a*d^2*e*r^4*x^r+356250*I*Pi*b
*d*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+440625*I*Pi*b*d^2*e*r^2*csgn(I*c)*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)^2*x^r-21375*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2+96875*I*Pi*b*e^3
*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-356250*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2+31875*I*Pi*b*e^3*r^3*
csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-468750*I*Pi*b*d*e^2*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-187500*I*P
i*b*d^3*r*csgn(I*c)*csgn(I*c*x^n)^2-187500*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2+234375*I*Pi*b*d*e^2*csgn(I
*c)*csgn(I*c*x^n)^2*(x^r)^2+234375*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+82500*b*d*e^2*n*r^2*(x^r)^
2-5400*ln(c)*b*d^2*e*r^5*x^r+187500*I*Pi*b*d^3*r*csgn(I*c*x^n)^3+468750*I*Pi*b*d*e^2*r*csgn(I*c)*csgn(I*x^n)*c
sgn(I*c*x^n)*(x^r)^2+515625*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-78125*I*Pi*b*e^3*csgn(I*c*x
^n)^3*(x^r)^3+78125*I*Pi*b*d^3*csgn(I*c)*csgn(I*c*x^n)^2+90000*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3-181875*I*Pi*b*d^
2*e*r^3*csgn(I*c)*csgn(I*c*x^n)^2*x^r-1350*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-2700*I*Pi*b*d^
2*e*r^5*csgn(I*c)*csgn(I*c*x^n)^2*x^r-96875*I*Pi*b*e^3*r^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-5000*I*
Pi*b*e^3*r^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-234375*I*Pi*b*d*e^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^
n)*(x^r)^2-1350*I*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-3300*I*Pi*b*d^3*r^5*csgn(I*c)*csgn(I*c*x^n)
^2-3300*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2-440625*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r+5000*I*Pi*b*e^3
*r^4*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3+234375*I*Pi*b*d^2*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r-300*I*Pi*b*e^3*r^5*cs
gn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+127500*I*Pi*b*d*e^2*r^3*csgn(I*c*x^n)^3*(x^r)^2+140625*I*Pi*b*e^3*r*csgn(I*c
*x^n)^3*(x^r)^3+300*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3+42750*ln(c)*b*d*e^2*r^4*(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)^2-468750*I*P
i*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-515625*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r-127500*I*P
i*b*d*e^2*r^3*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2...

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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^6,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-6>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 + 5 \, a d^{3}\right )} r^{6} - 660 \, {\left (b d^{3} n + 5 \, a d^{3}\right )} r^{5} + 15625 \, b d^{3} n + 4825 \, {\left (b d^{3} n + 5 \, a d^{3}\right )} r^{4} + 78125 \, a d^{3} - 18000 \, {\left (b d^{3} n + 5 \, a d^{3}\right )} r^{3} + 36250 \, {\left (b d^{3} n + 5 \, a d^{3}\right )} r^{2} - 37500 \, {\left (b d^{3} n + 5 \, a d^{3}\right )} r - 25 \, {\left ({\left (12 \, b r^{5} - 200 \, b r^{4} + 1275 \, b r^{3} - 3875 \, b r^{2} + 5625 \, b r - 3125 \, b\right )} e^{3} \log \left (c\right ) + {\left (12 \, b n r^{5} - 200 \, b n r^{4} + 1275 \, b n r^{3} - 3875 \, b n r^{2} + 5625 \, b n r - 3125 \, b n\right )} e^{3} \log \left (x\right ) + {\left (12 \, a r^{5} - 4 \, {\left (b n + 50 \, a\right )} r^{4} + 15 \, {\left (4 \, b n + 85 \, a\right )} r^{3} - 25 \, {\left (13 \, b n + 155 \, a\right )} r^{2} - 625 \, b n + 375 \, {\left (2 \, b n + 15 \, a\right )} r - 3125 \, a\right )} e^{3}\right )} x^{3 \, r} - 75 \, {\left ({\left (18 \, b d r^{5} - 285 \, b d r^{4} + 1700 \, b d r^{3} - 4750 \, b d r^{2} + 6250 \, b d r - 3125 \, b d\right )} e^{2} \log \left (c\right ) + {\left (18 \, b d n r^{5} - 285 \, b d n r^{4} + 1700 \, b d n r^{3} - 4750 \, b d n r^{2} + 6250 \, b d n r - 3125 \, b d n\right )} e^{2} \log \left (x\right ) + {\left (18 \, a d r^{5} - 3 \, {\left (3 \, b d n + 95 \, a d\right )} r^{4} + 20 \, {\left (6 \, b d n + 85 \, a d\right )} r^{3} - 625 \, b d n - 50 \, {\left (11 \, b d n + 95 \, a d\right )} r^{2} - 3125 \, a d + 250 \, {\left (4 \, b d n + 25 \, a d\right )} r\right )} e^{2}\right )} x^{2 \, r} - 75 \, {\left ({\left (36 \, b d^{2} r^{5} - 480 \, b d^{2} r^{4} + 2425 \, b d^{2} r^{3} - 5875 \, b d^{2} r^{2} + 6875 \, b d^{2} r - 3125 \, b d^{2}\right )} e \log \left (c\right ) + {\left (36 \, b d^{2} n r^{5} - 480 \, b d^{2} n r^{4} + 2425 \, b d^{2} n r^{3} - 5875 \, b d^{2} n r^{2} + 6875 \, b d^{2} n r - 3125 \, b d^{2} n\right )} e \log \left (x\right ) + {\left (36 \, a d^{2} r^{5} - 12 \, {\left (3 \, b d^{2} n + 40 \, a d^{2}\right )} r^{4} - 625 \, b d^{2} n + 25 \, {\left (12 \, b d^{2} n + 97 \, a d^{2}\right )} r^{3} - 3125 \, a d^{2} - 25 \, {\left (37 \, b d^{2} n + 235 \, a d^{2}\right )} r^{2} + 625 \, {\left (2 \, b d^{2} n + 11 \, a d^{2}\right )} r\right )} e\right )} x^{r} + 5 \, {\left (36 \, b d^{3} r^{6} - 660 \, b d^{3} r^{5} + 4825 \, 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}\right )} \log \left (c\right ) + 5 \, {\left (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\right )} \log \left (x\right )}{25 \, {\left (36 \, r^{6} - 660 \, r^{5} + 4825 \, r^{4} - 18000 \, r^{3} + 36250 \, r^{2} - 37500 \, r + 15625\right )} x^{5}} \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^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*b*r^5 - 200*b*r^4 + 1275*b*r^3 - 3875*b*r^2 + 5625*b*r - 3125*b)*e^3*log(c) + (12*b*n*r^5 - 200*b*n*r
^4 + 1275*b*n*r^3 - 3875*b*n*r^2 + 5625*b*n*r - 3125*b*n)*e^3*log(x) + (12*a*r^5 - 4*(b*n + 50*a)*r^4 + 15*(4*
b*n + 85*a)*r^3 - 25*(13*b*n + 155*a)*r^2 - 625*b*n + 375*(2*b*n + 15*a)*r - 3125*a)*e^3)*x^(3*r) - 75*((18*b*
d*r^5 - 285*b*d*r^4 + 1700*b*d*r^3 - 4750*b*d*r^2 + 6250*b*d*r - 3125*b*d)*e^2*log(c) + (18*b*d*n*r^5 - 285*b*
d*n*r^4 + 1700*b*d*n*r^3 - 4750*b*d*n*r^2 + 6250*b*d*n*r - 3125*b*d*n)*e^2*log(x) + (18*a*d*r^5 - 3*(3*b*d*n +
95*a*d)*r^4 + 20*(6*b*d*n + 85*a*d)*r^3 - 625*b*d*n - 50*(11*b*d*n + 95*a*d)*r^2 - 3125*a*d + 250*(4*b*d*n +
25*a*d)*r)*e^2)*x^(2*r) - 75*((36*b*d^2*r^5 - 480*b*d^2*r^4 + 2425*b*d^2*r^3 - 5875*b*d^2*r^2 + 6875*b*d^2*r -
3125*b*d^2)*e*log(c) + (36*b*d^2*n*r^5 - 480*b*d^2*n*r^4 + 2425*b*d^2*n*r^3 - 5875*b*d^2*n*r^2 + 6875*b*d^2*n
*r - 3125*b*d^2*n)*e*log(x) + (36*a*d^2*r^5 - 12*(3*b*d^2*n + 40*a*d^2)*r^4 - 625*b*d^2*n + 25*(12*b*d^2*n + 9
7*a*d^2)*r^3 - 3125*a*d^2 - 25*(37*b*d^2*n + 235*a*d^2)*r^2 + 625*(2*b*d^2*n + 11*a*d^2)*r)*e)*x^r + 5*(36*b*d
^3*r^6 - 660*b*d^3*r^5 + 4825*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)] 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**6,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^6,x, algorithm="giac")

[Out]

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

[Out]

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

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