3.403 \(\int \frac {(d+e x^r)^3 (a+b \log (c x^n))}{x^6} \, dx\)
Optimal. Leaf size=183 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d^2 e x^{r-5} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {3 d e^2 x^{2 r-5} \left (a+b \log \left (c x^n\right )\right )}{5-2 r}-\frac {e^3 x^{3 r-5} \left (a+b \log \left (c x^n\right )\right )}{5-3 r}-\frac {b d^3 n}{25 x^5}-\frac {3 b d^2 e n x^{r-5}}{(5-r)^2}-\frac {3 b d e^2 n x^{2 r-5}}{(5-2 r)^2}-\frac {b e^3 n x^{3 r-5}}{(5-3 r)^2} \]
[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 = 0.41, 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 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 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])
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 = 0.42, size = 187, normalized size = 1.02 \[ \frac {a \left (-5 d^3+\frac {75 d^2 e x^r}{r-5}+\frac {75 d e^2 x^{2 r}}{2 r-5}+\frac {25 e^3 x^{3 r}}{3 r-5}\right )+5 b \log \left (c x^n\right ) \left (-d^3+\frac {15 d^2 e x^r}{r-5}+\frac {15 d e^2 x^{2 r}}{2 r-5}+\frac {5 e^3 x^{3 r}}{3 r-5}\right )+b n \left (-d^3-\frac {75 d^2 e x^r}{(r-5)^2}-\frac {75 d e^2 x^{2 r}}{(5-2 r)^2}-\frac {25 e^3 x^{3 r}}{(5-3 r)^2}\right )}{25 x^5} \]
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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fricas [B] time = 0.44, size = 981, normalized size = 5.36 \[ \text {result too large to display} \]
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{6}}\,{d x} \]
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)
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maple [C] time = 0.52, size = 4031, normalized size = 22.03 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d+e*x^r)^3*(b*ln(c*x^n)+a)/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)/
(2*r-5)/(r-5)*ln(x^n)-1/50*(440625*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r-234375*I*Pi*b*d*e^2*csgn(I*x
^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+36000*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^2*csgn(I*c)*x^r-96875*I*Pi*b*e^3*r^2*
csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+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)^2-6600*a*d^3*r^5+48250*a*d^3*r^4+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*x^n)*csgn(I*c*x^n)^2*x^r+140625*I*Pi*b*e^3*r*csgn(
I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3+72*b*d^3*n*r^6-1320*b*d^3*n*r^5+9650*b*d^3*n*r^4+156250*a*e^3*(x^r)^3+1
56250*a*d^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+360*a*d^3*r^6+31250*b*d^3*n+180*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2+181
250*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)-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+193750*a*e^3*r^2*(x^r)^3-281250*a*e^3*r*(x^r)^3+468750*a
*d*e^2*(x^r)^2+468750*a*d^2*e*x^r-187500*I*Pi*b*d^3*r*csgn(I*c*x^n)^2*csgn(I*c)+156250*b*d^3*ln(c)-36000*b*d^3
*n*r^3+72500*b*d^3*n*r^2-75000*b*d^3*n*r+2700*I*Pi*b*d^2*e*r^5*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+1350*I*Pi*b*d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)*cs
gn(I*c)*(x^r)^2-180000*a*d^3*r^3+362500*a*d^3*r^2-375000*a*d^3*r-937500*ln(c)*b*d*e^2*r*(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+71250
0*ln(c)*b*d*e^2*r^2*(x^r)^2-187500*b*d^2*e*n*r*x^r+82500*b*d*e^2*n*r^2*(x^r)^2+138750*b*d^2*e*n*r^2*x^r+187500
*I*Pi*b*d^3*r*csgn(I*c*x^n)^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)^3-90000*I
*Pi*b*d^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)-468750*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-1350*I*Pi*b*
d*e^2*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-180*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3-78125*I*Pi*b*d^3*csgn(I*c*x^n
)^3+440625*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+93750*b*d^2*e*n*x^r-255000*a*d*e^2*r^3*(x^r)^2+712
500*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+5000*I*Pi*b*e^3
*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-24125*I*Pi*b*d^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-21375*I*Pi*b
*d*e^2*r^4*csgn(I*c*x^n)^3*(x^r)^2-78125*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+36000*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-234375*I*Pi*b
*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+78125*I*Pi*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+181250*I*Pi*
b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-78125*I*Pi*b*d^3*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)*csgn(I*c)*(x^r)^3+21375*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-181875
*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^2*csgn(I*c)*x^r+21375*I*Pi*b*d*e^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-23437
5*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+300*I*Pi*b*e^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-468750*I*Pi*
b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+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)^2*x^r-515625*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r-1350
*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-5000*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^3*(x^r)^3+24125*I*Pi*b*d
^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+24125*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)-140625*I*Pi*b*e^3*r*csgn(I*x
^n)*csgn(I*c*x^n)^2*(x^r)^3-140625*I*Pi*b*e^3*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+234375*I*Pi*b*d^2*e*csgn(I*x
^n)*csgn(I*c*x^n)^2*x^r+234375*I*Pi*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+96875*I*Pi*b*e^3*r^2*csgn(I*x^n)*csg
n(I*c*x^n)^2*(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+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+200*b*
e^3*n*r^4*(x^r)^3-1031250*a*d^2*e*r*x^r+468750*ln(c)*b*d^2*e*x^r+468750*ln(c)*b*d*e^2*(x^r)^2-600*ln(c)*b*e^3*
r^5*(x^r)^3+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)^3-36000*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*x^r+234375*I*Pi*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)
*(x^r)^2-180*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-440625*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r+14
0625*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x^r)^3-3300*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2+181875*I*Pi*b*d^2*e*
r^3*csgn(I*c*x^n)^3*x^r+90000*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+31875*I*Pi*b*e^3*r^3*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)^3-3300*I*Pi*b*d^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)-15000
0*b*d*e^2*n*r*(x^r)^2-2700*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+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-45
000*b*d^2*e*n*r^3*x^r-2700*I*Pi*b*d^2*e*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-2700*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n
)^2*csgn(I*c)*x^r+1350*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2+2700*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r-2412
5*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^3-181250*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3-78125*I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^
3+78125*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+78125*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)+181875*I*Pi*b*d^2*e*
r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-356250*I*Pi*b*d*e^2*r^2*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*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-440625*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn
(I*c*x^n)*csgn(I*c)*x^r+515625*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+31875*I*Pi*b*e^3*r^3*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-127500*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-127500*I*
Pi*b*d*e^2*r^3*csgn(I*c*x^n)^2*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)^3-3187
5*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+180*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)-234375*I*Pi*b*
d*e^2*csgn(I*c*x^n)^3*(x^r)^2+78125*I*Pi*b*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-96875*I*Pi*b*e^3*r^2*csgn(I
*c*x^n)^3*(x^r)^3-90000*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-187500*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^
n)^2-36000*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+468750*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(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*x^n)*csgn(I*c*x^n)^2*(x^r)^3-300
*I*Pi*b*e^3*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+5000*I*Pi*b*e^3*r^4*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-181250
*I*Pi*b*d^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+96875*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-356
250*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2+515625*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r+187500*I*Pi*b*d^3*r*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/(-5+3*r)^2/x^5/(2*r-5)^2/(r-5)^2
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 r-6 equal to -1?
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^6} \,d x \]
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)
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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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