3.398 \(\int x^4 (d+e x^r)^3 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=151 \[ \frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{r+5}}{r+5}+\frac {15 d e^2 x^{2 r+5}}{2 r+5}+\frac {5 e^3 x^{3 r+5}}{3 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d^3 n x^5-\frac {3 b d^2 e n x^{r+5}}{(r+5)^2}-\frac {3 b d e^2 n x^{2 r+5}}{(2 r+5)^2}-\frac {b e^3 n x^{3 r+5}}{(3 r+5)^2} \]

[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*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*ln(c*x^n))

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Rubi [A]  time = 0.38, antiderivative size = 151, 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 = {270, 2334, 12, 14} \[ \frac {1}{5} \left (\frac {15 d^2 e x^{r+5}}{r+5}+d^3 x^5+\frac {15 d e^2 x^{2 r+5}}{2 r+5}+\frac {5 e^3 x^{3 r+5}}{3 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b d^2 e n x^{r+5}}{(r+5)^2}-\frac {1}{25} b d^3 n x^5-\frac {3 b d e^2 n x^{2 r+5}}{(2 r+5)^2}-\frac {b e^3 n x^{3 r+5}}{(3 r+5)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

-(b*d^3*n*x^5)/25 - (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 x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{5} \left (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 {1}{5} x^4 \left (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}\right ) \, dx\\ &=\frac {1}{5} \left (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 x^4 \left (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}\right ) \, dx\\ &=\frac {1}{5} \left (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 (d^3 x^4+\frac {15 d e^2 x^{2 (2+r)}}{5+2 r}+\frac {15 d^2 e x^{4+r}}{5+r}+\frac {5 e^3 x^{4+3 r}}{5+3 r}\right ) \, dx\\ &=-\frac {1}{25} b d^3 n 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 (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 = 0.37, size = 184, normalized size = 1.22 \[ \frac {1}{25} x^5 \left (5 a \left (d^3+\frac {15 d^2 e x^r}{r+5}+\frac {15 d e^2 x^{2 r}}{2 r+5}+\frac {5 e^3 x^{3 r}}{3 r+5}\right )+5 b \log \left (c x^n\right ) \left (d^3+\frac {15 d^2 e x^r}{r+5}+\frac {15 d e^2 x^{2 r}}{2 r+5}+\frac {5 e^3 x^{3 r}}{3 r+5}\right )+b n \left (-d^3-\frac {75 d^2 e x^r}{(r+5)^2}-\frac {75 d e^2 x^{2 r}}{(2 r+5)^2}-\frac {25 e^3 x^{3 r}}{(3 r+5)^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

(x^5*(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) +
5*a*(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)) + 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

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fricas [B]  time = 0.50, size = 1023, normalized size = 6.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/25*(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 + 1
5625*b*d^3)*x^5*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)*x^5*log(x) - (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)*x^5 + 25*((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)*x^5*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)*x^5*log(x) + (12*a*e^3*r^5 - 625*b*e^3*n - 4*(b*e^3*n - 5
0*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)*x^5)*x^(3*r) + 75*((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 + 625
0*b*d*e^2*r + 3125*b*d*e^2)*x^5*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 + 4750*b*d
*e^2*n*r^2 + 6250*b*d*e^2*n*r + 3125*b*d*e^2*n)*x^5*log(x) + (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 - 25
0*(4*b*d*e^2*n - 25*a*d*e^2)*r)*x^5)*x^(2*r) + 75*((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)*x^5*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)*x^5*log(x) + (36*a*d^2*e*r^5 - 625*b*d^2*e*n
- 12*(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)*x^5)*x^r)/(36*r^6 + 660*r^5 + 4825*r^4 + 18000*r^3 + 3625
0*r^2 + 37500*r + 15625)

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giac [B]  time = 0.60, size = 1588, normalized size = 10.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/25*(180*b*d^3*n*r^6*x^5*log(x) + 2700*b*d^2*n*r^5*x^5*x^r*e*log(x) - 36*b*d^3*n*r^6*x^5 + 180*b*d^3*r^6*x^5*
log(c) + 2700*b*d^2*r^5*x^5*x^r*e*log(c) + 3300*b*d^3*n*r^5*x^5*log(x) + 1350*b*d*n*r^5*x^5*x^(2*r)*e^2*log(x)
+ 36000*b*d^2*n*r^4*x^5*x^r*e*log(x) - 660*b*d^3*n*r^5*x^5 + 180*a*d^3*r^6*x^5 - 2700*b*d^2*n*r^4*x^5*x^r*e +
2700*a*d^2*r^5*x^5*x^r*e + 3300*b*d^3*r^5*x^5*log(c) + 1350*b*d*r^5*x^5*x^(2*r)*e^2*log(c) + 36000*b*d^2*r^4*
x^5*x^r*e*log(c) + 24125*b*d^3*n*r^4*x^5*log(x) + 300*b*n*r^5*x^5*x^(3*r)*e^3*log(x) + 21375*b*d*n*r^4*x^5*x^(
2*r)*e^2*log(x) + 181875*b*d^2*n*r^3*x^5*x^r*e*log(x) - 4825*b*d^3*n*r^4*x^5 + 3300*a*d^3*r^5*x^5 - 675*b*d*n*
r^4*x^5*x^(2*r)*e^2 + 1350*a*d*r^5*x^5*x^(2*r)*e^2 - 22500*b*d^2*n*r^3*x^5*x^r*e + 36000*a*d^2*r^4*x^5*x^r*e +
24125*b*d^3*r^4*x^5*log(c) + 300*b*r^5*x^5*x^(3*r)*e^3*log(c) + 21375*b*d*r^4*x^5*x^(2*r)*e^2*log(c) + 181875
*b*d^2*r^3*x^5*x^r*e*log(c) + 90000*b*d^3*n*r^3*x^5*log(x) + 5000*b*n*r^4*x^5*x^(3*r)*e^3*log(x) + 127500*b*d*
n*r^3*x^5*x^(2*r)*e^2*log(x) + 440625*b*d^2*n*r^2*x^5*x^r*e*log(x) - 18000*b*d^3*n*r^3*x^5 + 24125*a*d^3*r^4*x
^5 - 100*b*n*r^4*x^5*x^(3*r)*e^3 + 300*a*r^5*x^5*x^(3*r)*e^3 - 9000*b*d*n*r^3*x^5*x^(2*r)*e^2 + 21375*a*d*r^4*
x^5*x^(2*r)*e^2 - 69375*b*d^2*n*r^2*x^5*x^r*e + 181875*a*d^2*r^3*x^5*x^r*e + 90000*b*d^3*r^3*x^5*log(c) + 5000
*b*r^4*x^5*x^(3*r)*e^3*log(c) + 127500*b*d*r^3*x^5*x^(2*r)*e^2*log(c) + 440625*b*d^2*r^2*x^5*x^r*e*log(c) + 18
1250*b*d^3*n*r^2*x^5*log(x) + 31875*b*n*r^3*x^5*x^(3*r)*e^3*log(x) + 356250*b*d*n*r^2*x^5*x^(2*r)*e^2*log(x) +
515625*b*d^2*n*r*x^5*x^r*e*log(x) - 36250*b*d^3*n*r^2*x^5 + 90000*a*d^3*r^3*x^5 - 1500*b*n*r^3*x^5*x^(3*r)*e^
3 + 5000*a*r^4*x^5*x^(3*r)*e^3 - 41250*b*d*n*r^2*x^5*x^(2*r)*e^2 + 127500*a*d*r^3*x^5*x^(2*r)*e^2 - 93750*b*d^
2*n*r*x^5*x^r*e + 440625*a*d^2*r^2*x^5*x^r*e + 181250*b*d^3*r^2*x^5*log(c) + 31875*b*r^3*x^5*x^(3*r)*e^3*log(c
) + 356250*b*d*r^2*x^5*x^(2*r)*e^2*log(c) + 515625*b*d^2*r*x^5*x^r*e*log(c) + 187500*b*d^3*n*r*x^5*log(x) + 96
875*b*n*r^2*x^5*x^(3*r)*e^3*log(x) + 468750*b*d*n*r*x^5*x^(2*r)*e^2*log(x) + 234375*b*d^2*n*x^5*x^r*e*log(x) -
37500*b*d^3*n*r*x^5 + 181250*a*d^3*r^2*x^5 - 8125*b*n*r^2*x^5*x^(3*r)*e^3 + 31875*a*r^3*x^5*x^(3*r)*e^3 - 750
00*b*d*n*r*x^5*x^(2*r)*e^2 + 356250*a*d*r^2*x^5*x^(2*r)*e^2 - 46875*b*d^2*n*x^5*x^r*e + 515625*a*d^2*r*x^5*x^r
*e + 187500*b*d^3*r*x^5*log(c) + 96875*b*r^2*x^5*x^(3*r)*e^3*log(c) + 468750*b*d*r*x^5*x^(2*r)*e^2*log(c) + 23
4375*b*d^2*x^5*x^r*e*log(c) + 78125*b*d^3*n*x^5*log(x) + 140625*b*n*r*x^5*x^(3*r)*e^3*log(x) + 234375*b*d*n*x^
5*x^(2*r)*e^2*log(x) - 15625*b*d^3*n*x^5 + 187500*a*d^3*r*x^5 - 18750*b*n*r*x^5*x^(3*r)*e^3 + 96875*a*r^2*x^5*
x^(3*r)*e^3 - 46875*b*d*n*x^5*x^(2*r)*e^2 + 468750*a*d*r*x^5*x^(2*r)*e^2 + 234375*a*d^2*x^5*x^r*e + 78125*b*d^
3*x^5*log(c) + 140625*b*r*x^5*x^(3*r)*e^3*log(c) + 234375*b*d*x^5*x^(2*r)*e^2*log(c) + 78125*b*n*x^5*x^(3*r)*e
^3*log(x) + 78125*a*d^3*x^5 - 15625*b*n*x^5*x^(3*r)*e^3 + 140625*a*r*x^5*x^(3*r)*e^3 + 234375*a*d*x^5*x^(2*r)*
e^2 + 78125*b*x^5*x^(3*r)*e^3*log(c) + 78125*a*x^5*x^(3*r)*e^3)/(36*r^6 + 660*r^5 + 4825*r^4 + 18000*r^3 + 362
50*r^2 + 37500*r + 15625)

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maple [C]  time = 0.51, size = 4031, normalized size = 26.70 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d+e*x^r)^3*(b*ln(c*x^n)+a),x)

[Out]

1/5*x^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)/(5+3*r)/(2*
r+5)/(r+5)*ln(x^n)-1/50*x^5*(-6600*a*d^3*r^5-48250*a*d^3*r^4+36000*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*csgn(I*c*x^n)*
csgn(I*c)*x^r+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)^3-440625*I*Pi*b*d^2*e*r^2*csgn(I
*x^n)*csgn(I*c*x^n)^2*x^r+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)^3-156250*a*d^3-3
60*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^2-37
5000*ln(c)*b*d^3*r+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)^2-360*a*d^3*r^6+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)^3-193750*a*e^3*r^2*(x^r)^3-281250*a*e^3*r*(x^r)^3-46875
0*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+2137
5*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)
*csgn(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)^2-71
2500*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+187
500*I*Pi*b*d^3*r*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)+180*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3+3300*I*Pi*b*d^3*r^5*csg
n(I*c*x^n)^3+24125*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^3+90000*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^3-180*I*Pi*b*d^3*r^6*csgn
(I*x^n)*csgn(I*c*x^n)^2-90000*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^2*csgn(I*c)-181250*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^2*c
sgn(I*c)-440625*I*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r-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+234375*I*Pi*b*d*e^2*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)*(x^r)^2+78125*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+440625*I*Pi*b*d^2*e*r^2*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)*x^r+93750*b*d^2*e*n*x^r-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-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)^3-181875*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)^2-36000*I*Pi*b*d^2*e*r^4*csgn(I*x^n)*c
sgn(I*c*x^n)^2*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+356250*I*Pi*b*d*
e^2*r^2*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-5156
25*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+234375*I*Pi*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-13
50*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+356250*I*Pi*b*d*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-140625*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+440625*I
*Pi*b*d^2*e*r^2*csgn(I*c*x^n)^3*x^r+78125*I*Pi*b*d^3*csgn(I*c*x^n)^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)^3-78125*I*Pi*b*e^3*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)^3*(x^r)^3-180*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^2*csgn(I*c)-3300*I*P
i*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)^2+234375*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2+181875*I*Pi*b*d^2*e*r^3*cs
gn(I*c*x^n)^3*x^r-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)^2-5000*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)+90000*I*Pi*b*d^3*r^3*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+181250*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3+96875*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)
^3+31875*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3+234375*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+300*I*Pi*b*e^3*r^5*csg
n(I*c*x^n)^3*(x^r)^3+5000*I*Pi*b*e^3*r^4*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)
+150000*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+45000*b*d^2*e*n*r^3*x^r+5000*I*Pi*b*e^3*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-2700*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-21375*I*Pi*b*d*e^2*r^4
*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+1350*I*Pi*b*d*e^2*r^5*csgn(I*c*x^n)^3*(x^r)^2-5000*I*Pi*b*e^3*r^4*csgn(I*
x^n)*csgn(I*c*x^n)^2*(x^r)^3+2700*I*Pi*b*d^2*e*r^5*csgn(I*c*x^n)^3*x^r+36000*I*Pi*b*d^2*e*r^4*csgn(I*c*x^n)^3*
x^r-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)-78125*I*Pi
*b*e^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+181875*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+12750
0*I*Pi*b*d*e^2*r^3*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)
*csgn(I*c)*x^r+31875*I*Pi*b*e^3*r^3*csgn(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)^2-234375*I*Pi*b*d^2*e*cs
gn(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)*csgn(I*c*x^n)^2*(x^r)^3-31875*I*Pi*b*e^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-31875*I*Pi*b*e^3*r^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)^2-90000*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+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+515625*I*Pi*b*d^2*e*r*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+187500*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+180*I*Pi*b*d^3*r^6*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/(5+3*r)^2/(2*r+5)^2/(r+5)^2

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maxima [A]  time = 1.38, size = 228, normalized size = 1.51 \[ -\frac {1}{25} \, b d^{3} n x^{5} + \frac {1}{5} \, b d^{3} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d^{3} x^{5} + \frac {b e^{3} x^{3 \, r + 5} \log \left (c x^{n}\right )}{3 \, r + 5} + \frac {3 \, b d e^{2} x^{2 \, r + 5} \log \left (c x^{n}\right )}{2 \, r + 5} + \frac {3 \, b d^{2} e x^{r + 5} \log \left (c x^{n}\right )}{r + 5} - \frac {b e^{3} n x^{3 \, r + 5}}{{\left (3 \, r + 5\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 5}}{3 \, r + 5} - \frac {3 \, b d e^{2} n x^{2 \, r + 5}}{{\left (2 \, r + 5\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 5}}{2 \, r + 5} - \frac {3 \, b d^{2} e n x^{r + 5}}{{\left (r + 5\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 5}}{r + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/25*b*d^3*n*x^5 + 1/5*b*d^3*x^5*log(c*x^n) + 1/5*a*d^3*x^5 + b*e^3*x^(3*r + 5)*log(c*x^n)/(3*r + 5) + 3*b*d*
e^2*x^(2*r + 5)*log(c*x^n)/(2*r + 5) + 3*b*d^2*e*x^(r + 5)*log(c*x^n)/(r + 5) - b*e^3*n*x^(3*r + 5)/(3*r + 5)^
2 + a*e^3*x^(3*r + 5)/(3*r + 5) - 3*b*d*e^2*n*x^(2*r + 5)/(2*r + 5)^2 + 3*a*d*e^2*x^(2*r + 5)/(2*r + 5) - 3*b*
d^2*e*n*x^(r + 5)/(r + 5)^2 + 3*a*d^2*e*x^(r + 5)/(r + 5)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d + e*x^r)^3*(a + b*log(c*x^n)),x)

[Out]

int(x^4*(d + e*x^r)^3*(a + b*log(c*x^n)), x)

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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(x**4*(d+e*x**r)**3*(a+b*ln(c*x**n)),x)

[Out]

Timed out

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