3.278 \(\int (d+e x)^3 (a+b x+c x^2)^5 (d (6 b d+5 a e)+(12 c d^2+17 b d e+5 a e^2) x+e (29 c d+11 b e) x^2+17 c e^2 x^3) \, dx\)

Optimal. Leaf size=20 \[ (d+e x)^5 \left (a+b x+c x^2\right )^6 \]

[Out]

(d + e*x)^5*(a + b*x + c*x^2)^6

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Rubi [A]  time = 0.420043, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 75, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027, Rules used = {1624, 1590} \[ (d+e x)^5 \left (a+b x+c x^2\right )^6 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + b*x + c*x^2)^5*(d*(6*b*d + 5*a*e) + (12*c*d^2 + 17*b*d*e + 5*a*e^2)*x + e*(29*c*d + 11*b*
e)*x^2 + 17*c*e^2*x^3),x]

[Out]

(d + e*x)^5*(a + b*x + c*x^2)^6

Rule 1624

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m
+ 1)*PolynomialQuotient[Pq, d + e*x, x]*(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq
, x] && EqQ[PolynomialRemainder[Pq, d + e*x, x], 0]

Rule 1590

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x
, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps

\begin{align*} \int (d+e x)^3 \left (a+b x+c x^2\right )^5 \left (d (6 b d+5 a e)+\left (12 c d^2+17 b d e+5 a e^2\right ) x+e (29 c d+11 b e) x^2+17 c e^2 x^3\right ) \, dx &=\int (d+e x)^4 \left (a+b x+c x^2\right )^5 \left (6 b d+5 a e+(12 c d+11 b e) x+17 c e x^2\right ) \, dx\\ &=(d+e x)^5 \left (a+b x+c x^2\right )^6\\ \end{align*}

Mathematica [B]  time = 0.439293, size = 167, normalized size = 8.35 \[ x \left (20 a^3 x^2 (b+c x)^3 (d+e x)^5+15 a^2 x^3 (b+c x)^4 (d+e x)^5+6 a^5 (b+c x) (d+e x)^5+15 a^4 x (b+c x)^2 (d+e x)^5+a^6 e \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+6 a x^4 (b+c x)^5 (d+e x)^5+x^5 (b+c x)^6 (d+e x)^5\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a + b*x + c*x^2)^5*(d*(6*b*d + 5*a*e) + (12*c*d^2 + 17*b*d*e + 5*a*e^2)*x + e*(29*c*d +
11*b*e)*x^2 + 17*c*e^2*x^3),x]

[Out]

x*(6*a^5*(b + c*x)*(d + e*x)^5 + 15*a^4*x*(b + c*x)^2*(d + e*x)^5 + 20*a^3*x^2*(b + c*x)^3*(d + e*x)^5 + 15*a^
2*x^3*(b + c*x)^4*(d + e*x)^5 + 6*a*x^4*(b + c*x)^5*(d + e*x)^5 + x^5*(b + c*x)^6*(d + e*x)^5 + a^6*e*(5*d^4 +
10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4))

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Maple [B]  time = 0.049, size = 8419, normalized size = 421. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(c*x^2+b*x+a)^5*(d*(5*a*e+6*b*d)+(5*a*e^2+17*b*d*e+12*c*d^2)*x+e*(11*b*e+29*c*d)*x^2+17*c*e^2*x^
3),x)

[Out]

result too large to display

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Maxima [B]  time = 1.05573, size = 2402, normalized size = 120.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x+a)^5*(d*(5*a*e+6*b*d)+(5*a*e^2+17*b*d*e+12*c*d^2)*x+e*(11*b*e+29*c*d)*x^2+17*c*
e^2*x^3),x, algorithm="maxima")

[Out]

c^6*e^5*x^17 + (5*c^6*d*e^4 + 6*b*c^5*e^5)*x^16 + (10*c^6*d^2*e^3 + 30*b*c^5*d*e^4 + 3*(5*b^2*c^4 + 2*a*c^5)*e
^5)*x^15 + 5*(2*c^6*d^3*e^2 + 12*b*c^5*d^2*e^3 + 3*(5*b^2*c^4 + 2*a*c^5)*d*e^4 + 2*(2*b^3*c^3 + 3*a*b*c^4)*e^5
)*x^14 + 5*(c^6*d^4*e + 12*b*c^5*d^3*e^2 + 6*(5*b^2*c^4 + 2*a*c^5)*d^2*e^3 + 10*(2*b^3*c^3 + 3*a*b*c^4)*d*e^4
+ 3*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*e^5)*x^13 + (c^6*d^5 + 30*b*c^5*d^4*e + 30*(5*b^2*c^4 + 2*a*c^5)*d^3*e^2
+ 100*(2*b^3*c^3 + 3*a*b*c^4)*d^2*e^3 + 75*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d*e^4 + 6*(b^5*c + 10*a*b^3*c^2
+ 10*a^2*b*c^3)*e^5)*x^12 + (6*b*c^5*d^5 + 15*(5*b^2*c^4 + 2*a*c^5)*d^4*e + 100*(2*b^3*c^3 + 3*a*b*c^4)*d^3*e^
2 + 150*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^2*e^3 + 30*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d*e^4 + (b^6 + 30
*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*e^5)*x^11 + (3*(5*b^2*c^4 + 2*a*c^5)*d^5 + 50*(2*b^3*c^3 + 3*a*b*c^4)*
d^4*e + 150*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^3*e^2 + 60*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^2*e^3 + 5*(
b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d*e^4 + 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*e^5)*x^10 + 5*
(2*(2*b^3*c^3 + 3*a*b*c^4)*d^5 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^4*e + 12*(b^5*c + 10*a*b^3*c^2 + 10*a^
2*b*c^3)*d^3*e^2 + 2*(b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^2*e^3 + 6*(a*b^5 + 10*a^2*b^3*c + 10*a
^3*b*c^2)*d*e^4 + 3*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*e^5)*x^9 + 5*(3*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^5 +
6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^4*e + 2*(b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^3*e^2 + 1
2*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^2*e^3 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d*e^4 + 2*(2*a^3*b^3 +
3*a^4*b*c)*e^5)*x^8 + (6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^5 + 5*(b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20
*a^3*c^3)*d^4*e + 60*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^3*e^2 + 150*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^2
*e^3 + 50*(2*a^3*b^3 + 3*a^4*b*c)*d*e^4 + 3*(5*a^4*b^2 + 2*a^5*c)*e^5)*x^7 + (6*a^5*b*e^5 + (b^6 + 30*a*b^4*c
+ 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^5 + 30*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^4*e + 150*(a^2*b^4 + 4*a^3*b^2
*c + a^4*c^2)*d^3*e^2 + 100*(2*a^3*b^3 + 3*a^4*b*c)*d^2*e^3 + 15*(5*a^4*b^2 + 2*a^5*c)*d*e^4)*x^6 + (30*a^5*b*
d*e^4 + a^6*e^5 + 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5 + 75*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e + 1
00*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^2 + 30*(5*a^4*b^2 + 2*a^5*c)*d^2*e^3)*x^5 + 5*(12*a^5*b*d^2*e^3 + a^6*d*e^4 +
3*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^5 + 10*(2*a^3*b^3 + 3*a^4*b*c)*d^4*e + 6*(5*a^4*b^2 + 2*a^5*c)*d^3*e^2)
*x^4 + 5*(12*a^5*b*d^3*e^2 + 2*a^6*d^2*e^3 + 2*(2*a^3*b^3 + 3*a^4*b*c)*d^5 + 3*(5*a^4*b^2 + 2*a^5*c)*d^4*e)*x^
3 + (30*a^5*b*d^4*e + 10*a^6*d^3*e^2 + 3*(5*a^4*b^2 + 2*a^5*c)*d^5)*x^2 + (6*a^5*b*d^5 + 5*a^6*d^4*e)*x

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Fricas [B]  time = 1.12122, size = 5253, normalized size = 262.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x+a)^5*(d*(5*a*e+6*b*d)+(5*a*e^2+17*b*d*e+12*c*d^2)*x+e*(11*b*e+29*c*d)*x^2+17*c*
e^2*x^3),x, algorithm="fricas")

[Out]

x^17*e^5*c^6 + 5*x^16*e^4*d*c^6 + 6*x^16*e^5*c^5*b + 10*x^15*e^3*d^2*c^6 + 30*x^15*e^4*d*c^5*b + 15*x^15*e^5*c
^4*b^2 + 6*x^15*e^5*c^5*a + 10*x^14*e^2*d^3*c^6 + 60*x^14*e^3*d^2*c^5*b + 75*x^14*e^4*d*c^4*b^2 + 20*x^14*e^5*
c^3*b^3 + 30*x^14*e^4*d*c^5*a + 30*x^14*e^5*c^4*b*a + 5*x^13*e*d^4*c^6 + 60*x^13*e^2*d^3*c^5*b + 150*x^13*e^3*
d^2*c^4*b^2 + 100*x^13*e^4*d*c^3*b^3 + 15*x^13*e^5*c^2*b^4 + 60*x^13*e^3*d^2*c^5*a + 150*x^13*e^4*d*c^4*b*a +
60*x^13*e^5*c^3*b^2*a + 15*x^13*e^5*c^4*a^2 + x^12*d^5*c^6 + 30*x^12*e*d^4*c^5*b + 150*x^12*e^2*d^3*c^4*b^2 +
200*x^12*e^3*d^2*c^3*b^3 + 75*x^12*e^4*d*c^2*b^4 + 6*x^12*e^5*c*b^5 + 60*x^12*e^2*d^3*c^5*a + 300*x^12*e^3*d^2
*c^4*b*a + 300*x^12*e^4*d*c^3*b^2*a + 60*x^12*e^5*c^2*b^3*a + 75*x^12*e^4*d*c^4*a^2 + 60*x^12*e^5*c^3*b*a^2 +
6*x^11*d^5*c^5*b + 75*x^11*e*d^4*c^4*b^2 + 200*x^11*e^2*d^3*c^3*b^3 + 150*x^11*e^3*d^2*c^2*b^4 + 30*x^11*e^4*d
*c*b^5 + x^11*e^5*b^6 + 30*x^11*e*d^4*c^5*a + 300*x^11*e^2*d^3*c^4*b*a + 600*x^11*e^3*d^2*c^3*b^2*a + 300*x^11
*e^4*d*c^2*b^3*a + 30*x^11*e^5*c*b^4*a + 150*x^11*e^3*d^2*c^4*a^2 + 300*x^11*e^4*d*c^3*b*a^2 + 90*x^11*e^5*c^2
*b^2*a^2 + 20*x^11*e^5*c^3*a^3 + 15*x^10*d^5*c^4*b^2 + 100*x^10*e*d^4*c^3*b^3 + 150*x^10*e^2*d^3*c^2*b^4 + 60*
x^10*e^3*d^2*c*b^5 + 5*x^10*e^4*d*b^6 + 6*x^10*d^5*c^5*a + 150*x^10*e*d^4*c^4*b*a + 600*x^10*e^2*d^3*c^3*b^2*a
+ 600*x^10*e^3*d^2*c^2*b^3*a + 150*x^10*e^4*d*c*b^4*a + 6*x^10*e^5*b^5*a + 150*x^10*e^2*d^3*c^4*a^2 + 600*x^1
0*e^3*d^2*c^3*b*a^2 + 450*x^10*e^4*d*c^2*b^2*a^2 + 60*x^10*e^5*c*b^3*a^2 + 100*x^10*e^4*d*c^3*a^3 + 60*x^10*e^
5*c^2*b*a^3 + 20*x^9*d^5*c^3*b^3 + 75*x^9*e*d^4*c^2*b^4 + 60*x^9*e^2*d^3*c*b^5 + 10*x^9*e^3*d^2*b^6 + 30*x^9*d
^5*c^4*b*a + 300*x^9*e*d^4*c^3*b^2*a + 600*x^9*e^2*d^3*c^2*b^3*a + 300*x^9*e^3*d^2*c*b^4*a + 30*x^9*e^4*d*b^5*
a + 75*x^9*e*d^4*c^4*a^2 + 600*x^9*e^2*d^3*c^3*b*a^2 + 900*x^9*e^3*d^2*c^2*b^2*a^2 + 300*x^9*e^4*d*c*b^3*a^2 +
15*x^9*e^5*b^4*a^2 + 200*x^9*e^3*d^2*c^3*a^3 + 300*x^9*e^4*d*c^2*b*a^3 + 60*x^9*e^5*c*b^2*a^3 + 15*x^9*e^5*c^
2*a^4 + 15*x^8*d^5*c^2*b^4 + 30*x^8*e*d^4*c*b^5 + 10*x^8*e^2*d^3*b^6 + 60*x^8*d^5*c^3*b^2*a + 300*x^8*e*d^4*c^
2*b^3*a + 300*x^8*e^2*d^3*c*b^4*a + 60*x^8*e^3*d^2*b^5*a + 15*x^8*d^5*c^4*a^2 + 300*x^8*e*d^4*c^3*b*a^2 + 900*
x^8*e^2*d^3*c^2*b^2*a^2 + 600*x^8*e^3*d^2*c*b^3*a^2 + 75*x^8*e^4*d*b^4*a^2 + 200*x^8*e^2*d^3*c^3*a^3 + 600*x^8
*e^3*d^2*c^2*b*a^3 + 300*x^8*e^4*d*c*b^2*a^3 + 20*x^8*e^5*b^3*a^3 + 75*x^8*e^4*d*c^2*a^4 + 30*x^8*e^5*c*b*a^4
+ 6*x^7*d^5*c*b^5 + 5*x^7*e*d^4*b^6 + 60*x^7*d^5*c^2*b^3*a + 150*x^7*e*d^4*c*b^4*a + 60*x^7*e^2*d^3*b^5*a + 60
*x^7*d^5*c^3*b*a^2 + 450*x^7*e*d^4*c^2*b^2*a^2 + 600*x^7*e^2*d^3*c*b^3*a^2 + 150*x^7*e^3*d^2*b^4*a^2 + 100*x^7
*e*d^4*c^3*a^3 + 600*x^7*e^2*d^3*c^2*b*a^3 + 600*x^7*e^3*d^2*c*b^2*a^3 + 100*x^7*e^4*d*b^3*a^3 + 150*x^7*e^3*d
^2*c^2*a^4 + 150*x^7*e^4*d*c*b*a^4 + 15*x^7*e^5*b^2*a^4 + 6*x^7*e^5*c*a^5 + x^6*d^5*b^6 + 30*x^6*d^5*c*b^4*a +
30*x^6*e*d^4*b^5*a + 90*x^6*d^5*c^2*b^2*a^2 + 300*x^6*e*d^4*c*b^3*a^2 + 150*x^6*e^2*d^3*b^4*a^2 + 20*x^6*d^5*
c^3*a^3 + 300*x^6*e*d^4*c^2*b*a^3 + 600*x^6*e^2*d^3*c*b^2*a^3 + 200*x^6*e^3*d^2*b^3*a^3 + 150*x^6*e^2*d^3*c^2*
a^4 + 300*x^6*e^3*d^2*c*b*a^4 + 75*x^6*e^4*d*b^2*a^4 + 30*x^6*e^4*d*c*a^5 + 6*x^6*e^5*b*a^5 + 6*x^5*d^5*b^5*a
+ 60*x^5*d^5*c*b^3*a^2 + 75*x^5*e*d^4*b^4*a^2 + 60*x^5*d^5*c^2*b*a^3 + 300*x^5*e*d^4*c*b^2*a^3 + 200*x^5*e^2*d
^3*b^3*a^3 + 75*x^5*e*d^4*c^2*a^4 + 300*x^5*e^2*d^3*c*b*a^4 + 150*x^5*e^3*d^2*b^2*a^4 + 60*x^5*e^3*d^2*c*a^5 +
30*x^5*e^4*d*b*a^5 + x^5*e^5*a^6 + 15*x^4*d^5*b^4*a^2 + 60*x^4*d^5*c*b^2*a^3 + 100*x^4*e*d^4*b^3*a^3 + 15*x^4
*d^5*c^2*a^4 + 150*x^4*e*d^4*c*b*a^4 + 150*x^4*e^2*d^3*b^2*a^4 + 60*x^4*e^2*d^3*c*a^5 + 60*x^4*e^3*d^2*b*a^5 +
5*x^4*e^4*d*a^6 + 20*x^3*d^5*b^3*a^3 + 30*x^3*d^5*c*b*a^4 + 75*x^3*e*d^4*b^2*a^4 + 30*x^3*e*d^4*c*a^5 + 60*x^
3*e^2*d^3*b*a^5 + 10*x^3*e^3*d^2*a^6 + 15*x^2*d^5*b^2*a^4 + 6*x^2*d^5*c*a^5 + 30*x^2*e*d^4*b*a^5 + 10*x^2*e^2*
d^3*a^6 + 6*x*d^5*b*a^5 + 5*x*e*d^4*a^6

________________________________________________________________________________________

Sympy [B]  time = 0.393559, size = 2281, normalized size = 114.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(c*x**2+b*x+a)**5*(d*(5*a*e+6*b*d)+(5*a*e**2+17*b*d*e+12*c*d**2)*x+e*(11*b*e+29*c*d)*x**2
+17*c*e**2*x**3),x)

[Out]

c**6*e**5*x**17 + x**16*(6*b*c**5*e**5 + 5*c**6*d*e**4) + x**15*(6*a*c**5*e**5 + 15*b**2*c**4*e**5 + 30*b*c**5
*d*e**4 + 10*c**6*d**2*e**3) + x**14*(30*a*b*c**4*e**5 + 30*a*c**5*d*e**4 + 20*b**3*c**3*e**5 + 75*b**2*c**4*d
*e**4 + 60*b*c**5*d**2*e**3 + 10*c**6*d**3*e**2) + x**13*(15*a**2*c**4*e**5 + 60*a*b**2*c**3*e**5 + 150*a*b*c*
*4*d*e**4 + 60*a*c**5*d**2*e**3 + 15*b**4*c**2*e**5 + 100*b**3*c**3*d*e**4 + 150*b**2*c**4*d**2*e**3 + 60*b*c*
*5*d**3*e**2 + 5*c**6*d**4*e) + x**12*(60*a**2*b*c**3*e**5 + 75*a**2*c**4*d*e**4 + 60*a*b**3*c**2*e**5 + 300*a
*b**2*c**3*d*e**4 + 300*a*b*c**4*d**2*e**3 + 60*a*c**5*d**3*e**2 + 6*b**5*c*e**5 + 75*b**4*c**2*d*e**4 + 200*b
**3*c**3*d**2*e**3 + 150*b**2*c**4*d**3*e**2 + 30*b*c**5*d**4*e + c**6*d**5) + x**11*(20*a**3*c**3*e**5 + 90*a
**2*b**2*c**2*e**5 + 300*a**2*b*c**3*d*e**4 + 150*a**2*c**4*d**2*e**3 + 30*a*b**4*c*e**5 + 300*a*b**3*c**2*d*e
**4 + 600*a*b**2*c**3*d**2*e**3 + 300*a*b*c**4*d**3*e**2 + 30*a*c**5*d**4*e + b**6*e**5 + 30*b**5*c*d*e**4 + 1
50*b**4*c**2*d**2*e**3 + 200*b**3*c**3*d**3*e**2 + 75*b**2*c**4*d**4*e + 6*b*c**5*d**5) + x**10*(60*a**3*b*c**
2*e**5 + 100*a**3*c**3*d*e**4 + 60*a**2*b**3*c*e**5 + 450*a**2*b**2*c**2*d*e**4 + 600*a**2*b*c**3*d**2*e**3 +
150*a**2*c**4*d**3*e**2 + 6*a*b**5*e**5 + 150*a*b**4*c*d*e**4 + 600*a*b**3*c**2*d**2*e**3 + 600*a*b**2*c**3*d*
*3*e**2 + 150*a*b*c**4*d**4*e + 6*a*c**5*d**5 + 5*b**6*d*e**4 + 60*b**5*c*d**2*e**3 + 150*b**4*c**2*d**3*e**2
+ 100*b**3*c**3*d**4*e + 15*b**2*c**4*d**5) + x**9*(15*a**4*c**2*e**5 + 60*a**3*b**2*c*e**5 + 300*a**3*b*c**2*
d*e**4 + 200*a**3*c**3*d**2*e**3 + 15*a**2*b**4*e**5 + 300*a**2*b**3*c*d*e**4 + 900*a**2*b**2*c**2*d**2*e**3 +
600*a**2*b*c**3*d**3*e**2 + 75*a**2*c**4*d**4*e + 30*a*b**5*d*e**4 + 300*a*b**4*c*d**2*e**3 + 600*a*b**3*c**2
*d**3*e**2 + 300*a*b**2*c**3*d**4*e + 30*a*b*c**4*d**5 + 10*b**6*d**2*e**3 + 60*b**5*c*d**3*e**2 + 75*b**4*c**
2*d**4*e + 20*b**3*c**3*d**5) + x**8*(30*a**4*b*c*e**5 + 75*a**4*c**2*d*e**4 + 20*a**3*b**3*e**5 + 300*a**3*b*
*2*c*d*e**4 + 600*a**3*b*c**2*d**2*e**3 + 200*a**3*c**3*d**3*e**2 + 75*a**2*b**4*d*e**4 + 600*a**2*b**3*c*d**2
*e**3 + 900*a**2*b**2*c**2*d**3*e**2 + 300*a**2*b*c**3*d**4*e + 15*a**2*c**4*d**5 + 60*a*b**5*d**2*e**3 + 300*
a*b**4*c*d**3*e**2 + 300*a*b**3*c**2*d**4*e + 60*a*b**2*c**3*d**5 + 10*b**6*d**3*e**2 + 30*b**5*c*d**4*e + 15*
b**4*c**2*d**5) + x**7*(6*a**5*c*e**5 + 15*a**4*b**2*e**5 + 150*a**4*b*c*d*e**4 + 150*a**4*c**2*d**2*e**3 + 10
0*a**3*b**3*d*e**4 + 600*a**3*b**2*c*d**2*e**3 + 600*a**3*b*c**2*d**3*e**2 + 100*a**3*c**3*d**4*e + 150*a**2*b
**4*d**2*e**3 + 600*a**2*b**3*c*d**3*e**2 + 450*a**2*b**2*c**2*d**4*e + 60*a**2*b*c**3*d**5 + 60*a*b**5*d**3*e
**2 + 150*a*b**4*c*d**4*e + 60*a*b**3*c**2*d**5 + 5*b**6*d**4*e + 6*b**5*c*d**5) + x**6*(6*a**5*b*e**5 + 30*a*
*5*c*d*e**4 + 75*a**4*b**2*d*e**4 + 300*a**4*b*c*d**2*e**3 + 150*a**4*c**2*d**3*e**2 + 200*a**3*b**3*d**2*e**3
+ 600*a**3*b**2*c*d**3*e**2 + 300*a**3*b*c**2*d**4*e + 20*a**3*c**3*d**5 + 150*a**2*b**4*d**3*e**2 + 300*a**2
*b**3*c*d**4*e + 90*a**2*b**2*c**2*d**5 + 30*a*b**5*d**4*e + 30*a*b**4*c*d**5 + b**6*d**5) + x**5*(a**6*e**5 +
30*a**5*b*d*e**4 + 60*a**5*c*d**2*e**3 + 150*a**4*b**2*d**2*e**3 + 300*a**4*b*c*d**3*e**2 + 75*a**4*c**2*d**4
*e + 200*a**3*b**3*d**3*e**2 + 300*a**3*b**2*c*d**4*e + 60*a**3*b*c**2*d**5 + 75*a**2*b**4*d**4*e + 60*a**2*b*
*3*c*d**5 + 6*a*b**5*d**5) + x**4*(5*a**6*d*e**4 + 60*a**5*b*d**2*e**3 + 60*a**5*c*d**3*e**2 + 150*a**4*b**2*d
**3*e**2 + 150*a**4*b*c*d**4*e + 15*a**4*c**2*d**5 + 100*a**3*b**3*d**4*e + 60*a**3*b**2*c*d**5 + 15*a**2*b**4
*d**5) + x**3*(10*a**6*d**2*e**3 + 60*a**5*b*d**3*e**2 + 30*a**5*c*d**4*e + 75*a**4*b**2*d**4*e + 30*a**4*b*c*
d**5 + 20*a**3*b**3*d**5) + x**2*(10*a**6*d**3*e**2 + 30*a**5*b*d**4*e + 6*a**5*c*d**5 + 15*a**4*b**2*d**5) +
x*(5*a**6*d**4*e + 6*a**5*b*d**5)

________________________________________________________________________________________

Giac [B]  time = 1.20794, size = 3217, normalized size = 160.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x+a)^5*(d*(5*a*e+6*b*d)+(5*a*e^2+17*b*d*e+12*c*d^2)*x+e*(11*b*e+29*c*d)*x^2+17*c*
e^2*x^3),x, algorithm="giac")

[Out]

c^6*x^17*e^5 + 5*c^6*d*x^16*e^4 + 10*c^6*d^2*x^15*e^3 + 10*c^6*d^3*x^14*e^2 + 5*c^6*d^4*x^13*e + c^6*d^5*x^12
+ 6*b*c^5*x^16*e^5 + 30*b*c^5*d*x^15*e^4 + 60*b*c^5*d^2*x^14*e^3 + 60*b*c^5*d^3*x^13*e^2 + 30*b*c^5*d^4*x^12*e
+ 6*b*c^5*d^5*x^11 + 15*b^2*c^4*x^15*e^5 + 6*a*c^5*x^15*e^5 + 75*b^2*c^4*d*x^14*e^4 + 30*a*c^5*d*x^14*e^4 + 1
50*b^2*c^4*d^2*x^13*e^3 + 60*a*c^5*d^2*x^13*e^3 + 150*b^2*c^4*d^3*x^12*e^2 + 60*a*c^5*d^3*x^12*e^2 + 75*b^2*c^
4*d^4*x^11*e + 30*a*c^5*d^4*x^11*e + 15*b^2*c^4*d^5*x^10 + 6*a*c^5*d^5*x^10 + 20*b^3*c^3*x^14*e^5 + 30*a*b*c^4
*x^14*e^5 + 100*b^3*c^3*d*x^13*e^4 + 150*a*b*c^4*d*x^13*e^4 + 200*b^3*c^3*d^2*x^12*e^3 + 300*a*b*c^4*d^2*x^12*
e^3 + 200*b^3*c^3*d^3*x^11*e^2 + 300*a*b*c^4*d^3*x^11*e^2 + 100*b^3*c^3*d^4*x^10*e + 150*a*b*c^4*d^4*x^10*e +
20*b^3*c^3*d^5*x^9 + 30*a*b*c^4*d^5*x^9 + 15*b^4*c^2*x^13*e^5 + 60*a*b^2*c^3*x^13*e^5 + 15*a^2*c^4*x^13*e^5 +
75*b^4*c^2*d*x^12*e^4 + 300*a*b^2*c^3*d*x^12*e^4 + 75*a^2*c^4*d*x^12*e^4 + 150*b^4*c^2*d^2*x^11*e^3 + 600*a*b^
2*c^3*d^2*x^11*e^3 + 150*a^2*c^4*d^2*x^11*e^3 + 150*b^4*c^2*d^3*x^10*e^2 + 600*a*b^2*c^3*d^3*x^10*e^2 + 150*a^
2*c^4*d^3*x^10*e^2 + 75*b^4*c^2*d^4*x^9*e + 300*a*b^2*c^3*d^4*x^9*e + 75*a^2*c^4*d^4*x^9*e + 15*b^4*c^2*d^5*x^
8 + 60*a*b^2*c^3*d^5*x^8 + 15*a^2*c^4*d^5*x^8 + 6*b^5*c*x^12*e^5 + 60*a*b^3*c^2*x^12*e^5 + 60*a^2*b*c^3*x^12*e
^5 + 30*b^5*c*d*x^11*e^4 + 300*a*b^3*c^2*d*x^11*e^4 + 300*a^2*b*c^3*d*x^11*e^4 + 60*b^5*c*d^2*x^10*e^3 + 600*a
*b^3*c^2*d^2*x^10*e^3 + 600*a^2*b*c^3*d^2*x^10*e^3 + 60*b^5*c*d^3*x^9*e^2 + 600*a*b^3*c^2*d^3*x^9*e^2 + 600*a^
2*b*c^3*d^3*x^9*e^2 + 30*b^5*c*d^4*x^8*e + 300*a*b^3*c^2*d^4*x^8*e + 300*a^2*b*c^3*d^4*x^8*e + 6*b^5*c*d^5*x^7
+ 60*a*b^3*c^2*d^5*x^7 + 60*a^2*b*c^3*d^5*x^7 + b^6*x^11*e^5 + 30*a*b^4*c*x^11*e^5 + 90*a^2*b^2*c^2*x^11*e^5
+ 20*a^3*c^3*x^11*e^5 + 5*b^6*d*x^10*e^4 + 150*a*b^4*c*d*x^10*e^4 + 450*a^2*b^2*c^2*d*x^10*e^4 + 100*a^3*c^3*d
*x^10*e^4 + 10*b^6*d^2*x^9*e^3 + 300*a*b^4*c*d^2*x^9*e^3 + 900*a^2*b^2*c^2*d^2*x^9*e^3 + 200*a^3*c^3*d^2*x^9*e
^3 + 10*b^6*d^3*x^8*e^2 + 300*a*b^4*c*d^3*x^8*e^2 + 900*a^2*b^2*c^2*d^3*x^8*e^2 + 200*a^3*c^3*d^3*x^8*e^2 + 5*
b^6*d^4*x^7*e + 150*a*b^4*c*d^4*x^7*e + 450*a^2*b^2*c^2*d^4*x^7*e + 100*a^3*c^3*d^4*x^7*e + b^6*d^5*x^6 + 30*a
*b^4*c*d^5*x^6 + 90*a^2*b^2*c^2*d^5*x^6 + 20*a^3*c^3*d^5*x^6 + 6*a*b^5*x^10*e^5 + 60*a^2*b^3*c*x^10*e^5 + 60*a
^3*b*c^2*x^10*e^5 + 30*a*b^5*d*x^9*e^4 + 300*a^2*b^3*c*d*x^9*e^4 + 300*a^3*b*c^2*d*x^9*e^4 + 60*a*b^5*d^2*x^8*
e^3 + 600*a^2*b^3*c*d^2*x^8*e^3 + 600*a^3*b*c^2*d^2*x^8*e^3 + 60*a*b^5*d^3*x^7*e^2 + 600*a^2*b^3*c*d^3*x^7*e^2
+ 600*a^3*b*c^2*d^3*x^7*e^2 + 30*a*b^5*d^4*x^6*e + 300*a^2*b^3*c*d^4*x^6*e + 300*a^3*b*c^2*d^4*x^6*e + 6*a*b^
5*d^5*x^5 + 60*a^2*b^3*c*d^5*x^5 + 60*a^3*b*c^2*d^5*x^5 + 15*a^2*b^4*x^9*e^5 + 60*a^3*b^2*c*x^9*e^5 + 15*a^4*c
^2*x^9*e^5 + 75*a^2*b^4*d*x^8*e^4 + 300*a^3*b^2*c*d*x^8*e^4 + 75*a^4*c^2*d*x^8*e^4 + 150*a^2*b^4*d^2*x^7*e^3 +
600*a^3*b^2*c*d^2*x^7*e^3 + 150*a^4*c^2*d^2*x^7*e^3 + 150*a^2*b^4*d^3*x^6*e^2 + 600*a^3*b^2*c*d^3*x^6*e^2 + 1
50*a^4*c^2*d^3*x^6*e^2 + 75*a^2*b^4*d^4*x^5*e + 300*a^3*b^2*c*d^4*x^5*e + 75*a^4*c^2*d^4*x^5*e + 15*a^2*b^4*d^
5*x^4 + 60*a^3*b^2*c*d^5*x^4 + 15*a^4*c^2*d^5*x^4 + 20*a^3*b^3*x^8*e^5 + 30*a^4*b*c*x^8*e^5 + 100*a^3*b^3*d*x^
7*e^4 + 150*a^4*b*c*d*x^7*e^4 + 200*a^3*b^3*d^2*x^6*e^3 + 300*a^4*b*c*d^2*x^6*e^3 + 200*a^3*b^3*d^3*x^5*e^2 +
300*a^4*b*c*d^3*x^5*e^2 + 100*a^3*b^3*d^4*x^4*e + 150*a^4*b*c*d^4*x^4*e + 20*a^3*b^3*d^5*x^3 + 30*a^4*b*c*d^5*
x^3 + 15*a^4*b^2*x^7*e^5 + 6*a^5*c*x^7*e^5 + 75*a^4*b^2*d*x^6*e^4 + 30*a^5*c*d*x^6*e^4 + 150*a^4*b^2*d^2*x^5*e
^3 + 60*a^5*c*d^2*x^5*e^3 + 150*a^4*b^2*d^3*x^4*e^2 + 60*a^5*c*d^3*x^4*e^2 + 75*a^4*b^2*d^4*x^3*e + 30*a^5*c*d
^4*x^3*e + 15*a^4*b^2*d^5*x^2 + 6*a^5*c*d^5*x^2 + 6*a^5*b*x^6*e^5 + 30*a^5*b*d*x^5*e^4 + 60*a^5*b*d^2*x^4*e^3
+ 60*a^5*b*d^3*x^3*e^2 + 30*a^5*b*d^4*x^2*e + 6*a^5*b*d^5*x + a^6*x^5*e^5 + 5*a^6*d*x^4*e^4 + 10*a^6*d^2*x^3*e
^3 + 10*a^6*d^3*x^2*e^2 + 5*a^6*d^4*x*e