3.15.33 \(\int (d+e x)^m (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=206 \[ -\frac {6 b^5 (b d-a e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{m+5}}{e^7 (m+5)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{m+4}}{e^7 (m+4)}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{m+3}}{e^7 (m+3)}+\frac {(b d-a e)^6 (d+e x)^{m+1}}{e^7 (m+1)}-\frac {6 b (b d-a e)^5 (d+e x)^{m+2}}{e^7 (m+2)}+\frac {b^6 (d+e x)^{m+7}}{e^7 (m+7)} \]

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Rubi [A]  time = 0.11, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} \frac {15 b^2 (b d-a e)^4 (d+e x)^{m+3}}{e^7 (m+3)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{m+4}}{e^7 (m+4)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{m+5}}{e^7 (m+5)}-\frac {6 b^5 (b d-a e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac {(b d-a e)^6 (d+e x)^{m+1}}{e^7 (m+1)}-\frac {6 b (b d-a e)^5 (d+e x)^{m+2}}{e^7 (m+2)}+\frac {b^6 (d+e x)^{m+7}}{e^7 (m+7)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((b*d - a*e)^6*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (6*b*(b*d - a*e)^5*(d + e*x)^(2 + m))/(e^7*(2 + m)) + (15*b^
2*(b*d - a*e)^4*(d + e*x)^(3 + m))/(e^7*(3 + m)) - (20*b^3*(b*d - a*e)^3*(d + e*x)^(4 + m))/(e^7*(4 + m)) + (1
5*b^4*(b*d - a*e)^2*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (6*b^5*(b*d - a*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (
b^6*(d + e*x)^(7 + m))/(e^7*(7 + m))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^m \, dx\\ &=\int \left (\frac {(-b d+a e)^6 (d+e x)^m}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{1+m}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{2+m}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{3+m}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{4+m}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{5+m}}{e^6}+\frac {b^6 (d+e x)^{6+m}}{e^6}\right ) \, dx\\ &=\frac {(b d-a e)^6 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {6 b (b d-a e)^5 (d+e x)^{2+m}}{e^7 (2+m)}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{3+m}}{e^7 (3+m)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{4+m}}{e^7 (4+m)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{5+m}}{e^7 (5+m)}-\frac {6 b^5 (b d-a e) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {b^6 (d+e x)^{7+m}}{e^7 (7+m)}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 175, normalized size = 0.85 \begin {gather*} \frac {(d+e x)^{m+1} \left (-\frac {6 b^5 (d+e x)^5 (b d-a e)}{m+6}+\frac {15 b^4 (d+e x)^4 (b d-a e)^2}{m+5}-\frac {20 b^3 (d+e x)^3 (b d-a e)^3}{m+4}+\frac {15 b^2 (d+e x)^2 (b d-a e)^4}{m+3}-\frac {6 b (d+e x) (b d-a e)^5}{m+2}+\frac {(b d-a e)^6}{m+1}+\frac {b^6 (d+e x)^6}{m+7}\right )}{e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((d + e*x)^(1 + m)*((b*d - a*e)^6/(1 + m) - (6*b*(b*d - a*e)^5*(d + e*x))/(2 + m) + (15*b^2*(b*d - a*e)^4*(d +
e*x)^2)/(3 + m) - (20*b^3*(b*d - a*e)^3*(d + e*x)^3)/(4 + m) + (15*b^4*(b*d - a*e)^2*(d + e*x)^4)/(5 + m) - (
6*b^5*(b*d - a*e)*(d + e*x)^5)/(6 + m) + (b^6*(d + e*x)^6)/(7 + m)))/e^7

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IntegrateAlgebraic [F]  time = 0.74, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

Defer[IntegrateAlgebraic][(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3, x]

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fricas [B]  time = 0.45, size = 2230, normalized size = 10.83

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

(a^6*d*e^6*m^6 + 720*b^6*d^7 - 5040*a*b^5*d^6*e + 15120*a^2*b^4*d^5*e^2 - 25200*a^3*b^3*d^4*e^3 + 25200*a^4*b^
2*d^3*e^4 - 15120*a^5*b*d^2*e^5 + 5040*a^6*d*e^6 + (b^6*e^7*m^6 + 21*b^6*e^7*m^5 + 175*b^6*e^7*m^4 + 735*b^6*e
^7*m^3 + 1624*b^6*e^7*m^2 + 1764*b^6*e^7*m + 720*b^6*e^7)*x^7 + (5040*a*b^5*e^7 + (b^6*d*e^6 + 6*a*b^5*e^7)*m^
6 + 3*(5*b^6*d*e^6 + 44*a*b^5*e^7)*m^5 + 5*(17*b^6*d*e^6 + 228*a*b^5*e^7)*m^4 + 15*(15*b^6*d*e^6 + 328*a*b^5*e
^7)*m^3 + 2*(137*b^6*d*e^6 + 5547*a*b^5*e^7)*m^2 + 12*(10*b^6*d*e^6 + 1019*a*b^5*e^7)*m)*x^6 - 3*(2*a^5*b*d^2*
e^5 - 9*a^6*d*e^6)*m^5 + 3*(5040*a^2*b^4*e^7 + (2*a*b^5*d*e^6 + 5*a^2*b^4*e^7)*m^6 - (2*b^6*d^2*e^5 - 34*a*b^5
*d*e^6 - 115*a^2*b^4*e^7)*m^5 - 5*(4*b^6*d^2*e^5 - 42*a*b^5*d*e^6 - 207*a^2*b^4*e^7)*m^4 - 5*(14*b^6*d^2*e^5 -
118*a*b^5*d*e^6 - 925*a^2*b^4*e^7)*m^3 - 4*(25*b^6*d^2*e^5 - 187*a*b^5*d*e^6 - 2680*a^2*b^4*e^7)*m^2 - 12*(4*
b^6*d^2*e^5 - 28*a*b^5*d*e^6 - 1005*a^2*b^4*e^7)*m)*x^5 + 5*(6*a^4*b^2*d^3*e^4 - 30*a^5*b*d^2*e^5 + 59*a^6*d*e
^6)*m^4 + 5*(5040*a^3*b^3*e^7 + (3*a^2*b^4*d*e^6 + 4*a^3*b^3*e^7)*m^6 - 3*(2*a*b^5*d^2*e^5 - 19*a^2*b^4*d*e^6
- 32*a^3*b^3*e^7)*m^5 + (6*b^6*d^3*e^4 - 78*a*b^5*d^2*e^5 + 393*a^2*b^4*d*e^6 + 904*a^3*b^3*e^7)*m^4 + 3*(12*b
^6*d^3*e^4 - 106*a*b^5*d^2*e^5 + 401*a^2*b^4*d*e^6 + 1408*a^3*b^3*e^7)*m^3 + 2*(33*b^6*d^3*e^4 - 249*a*b^5*d^2
*e^5 + 810*a^2*b^4*d*e^6 + 5090*a^3*b^3*e^7)*m^2 + 36*(b^6*d^3*e^4 - 7*a*b^5*d^2*e^5 + 21*a^2*b^4*d*e^6 + 328*
a^3*b^3*e^7)*m)*x^4 - 15*(8*a^3*b^3*d^4*e^3 - 44*a^4*b^2*d^3*e^4 + 98*a^5*b*d^2*e^5 - 111*a^6*d*e^6)*m^3 + 5*(
5040*a^4*b^2*e^7 + (4*a^3*b^3*d*e^6 + 3*a^4*b^2*e^7)*m^6 - 3*(4*a^2*b^4*d^2*e^5 - 28*a^3*b^3*d*e^6 - 25*a^4*b^
2*e^7)*m^5 + (24*a*b^5*d^3*e^4 - 192*a^2*b^4*d^2*e^5 + 652*a^3*b^3*d*e^6 + 741*a^4*b^2*e^7)*m^4 - 3*(8*b^6*d^4
*e^3 - 80*a*b^5*d^3*e^4 + 332*a^2*b^4*d^2*e^5 - 756*a^3*b^3*d*e^6 - 1219*a^4*b^2*e^7)*m^3 - 8*(9*b^6*d^4*e^3 -
69*a*b^5*d^3*e^4 + 228*a^2*b^4*d^2*e^5 - 422*a^3*b^3*d*e^6 - 1167*a^4*b^2*e^7)*m^2 - 12*(4*b^6*d^4*e^3 - 28*a
*b^5*d^3*e^4 + 84*a^2*b^4*d^2*e^5 - 140*a^3*b^3*d*e^6 - 949*a^4*b^2*e^7)*m)*x^3 + 2*(180*a^2*b^4*d^5*e^2 - 108
0*a^3*b^3*d^4*e^3 + 2685*a^4*b^2*d^3*e^4 - 3525*a^5*b*d^2*e^5 + 2552*a^6*d*e^6)*m^2 + 3*(5040*a^5*b*e^7 + (5*a
^4*b^2*d*e^6 + 2*a^5*b*e^7)*m^6 - (20*a^3*b^3*d^2*e^5 - 115*a^4*b^2*d*e^6 - 52*a^5*b*e^7)*m^5 + 5*(12*a^2*b^4*
d^3*e^4 - 76*a^3*b^3*d^2*e^5 + 201*a^4*b^2*d*e^6 + 108*a^5*b*e^7)*m^4 - 5*(24*a*b^5*d^4*e^3 - 168*a^2*b^4*d^3*
e^4 + 500*a^3*b^3*d^2*e^5 - 817*a^4*b^2*d*e^6 - 568*a^5*b*e^7)*m^3 + 2*(60*b^6*d^5*e^2 - 480*a*b^5*d^4*e^3 + 1
650*a^2*b^4*d^3*e^4 - 3170*a^3*b^3*d^2*e^5 + 3695*a^4*b^2*d*e^6 + 3929*a^5*b*e^7)*m^2 + 12*(10*b^6*d^5*e^2 - 7
0*a*b^5*d^4*e^3 + 210*a^2*b^4*d^3*e^4 - 350*a^3*b^3*d^2*e^5 + 350*a^4*b^2*d*e^6 + 879*a^5*b*e^7)*m)*x^2 - 12*(
60*a*b^5*d^6*e - 390*a^2*b^4*d^5*e^2 + 1070*a^3*b^3*d^4*e^3 - 1595*a^4*b^2*d^3*e^4 + 1377*a^5*b*d^2*e^5 - 669*
a^6*d*e^6)*m + (5040*a^6*e^7 + (6*a^5*b*d*e^6 + a^6*e^7)*m^6 - 3*(10*a^4*b^2*d^2*e^5 - 50*a^5*b*d*e^6 - 9*a^6*
e^7)*m^5 + 5*(24*a^3*b^3*d^3*e^4 - 132*a^4*b^2*d^2*e^5 + 294*a^5*b*d*e^6 + 59*a^6*e^7)*m^4 - 15*(24*a^2*b^4*d^
4*e^3 - 144*a^3*b^3*d^3*e^4 + 358*a^4*b^2*d^2*e^5 - 470*a^5*b*d*e^6 - 111*a^6*e^7)*m^3 + 4*(180*a*b^5*d^5*e^2
- 1170*a^2*b^4*d^4*e^3 + 3210*a^3*b^3*d^3*e^4 - 4785*a^4*b^2*d^2*e^5 + 4131*a^5*b*d*e^6 + 1276*a^6*e^7)*m^2 -
36*(20*b^6*d^6*e - 140*a*b^5*d^5*e^2 + 420*a^2*b^4*d^4*e^3 - 700*a^3*b^3*d^3*e^4 + 700*a^4*b^2*d^2*e^5 - 420*a
^5*b*d*e^6 - 223*a^6*e^7)*m)*x)*(e*x + d)^m/(e^7*m^7 + 28*e^7*m^6 + 322*e^7*m^5 + 1960*e^7*m^4 + 6769*e^7*m^3
+ 13132*e^7*m^2 + 13068*e^7*m + 5040*e^7)

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giac [B]  time = 0.30, size = 3877, normalized size = 18.82

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

((x*e + d)^m*b^6*m^6*x^7*e^7 + (x*e + d)^m*b^6*d*m^6*x^6*e^6 + 6*(x*e + d)^m*a*b^5*m^6*x^6*e^7 + 21*(x*e + d)^
m*b^6*m^5*x^7*e^7 + 6*(x*e + d)^m*a*b^5*d*m^6*x^5*e^6 + 15*(x*e + d)^m*b^6*d*m^5*x^6*e^6 - 6*(x*e + d)^m*b^6*d
^2*m^5*x^5*e^5 + 15*(x*e + d)^m*a^2*b^4*m^6*x^5*e^7 + 132*(x*e + d)^m*a*b^5*m^5*x^6*e^7 + 175*(x*e + d)^m*b^6*
m^4*x^7*e^7 + 15*(x*e + d)^m*a^2*b^4*d*m^6*x^4*e^6 + 102*(x*e + d)^m*a*b^5*d*m^5*x^5*e^6 + 85*(x*e + d)^m*b^6*
d*m^4*x^6*e^6 - 30*(x*e + d)^m*a*b^5*d^2*m^5*x^4*e^5 - 60*(x*e + d)^m*b^6*d^2*m^4*x^5*e^5 + 30*(x*e + d)^m*b^6
*d^3*m^4*x^4*e^4 + 20*(x*e + d)^m*a^3*b^3*m^6*x^4*e^7 + 345*(x*e + d)^m*a^2*b^4*m^5*x^5*e^7 + 1140*(x*e + d)^m
*a*b^5*m^4*x^6*e^7 + 735*(x*e + d)^m*b^6*m^3*x^7*e^7 + 20*(x*e + d)^m*a^3*b^3*d*m^6*x^3*e^6 + 285*(x*e + d)^m*
a^2*b^4*d*m^5*x^4*e^6 + 630*(x*e + d)^m*a*b^5*d*m^4*x^5*e^6 + 225*(x*e + d)^m*b^6*d*m^3*x^6*e^6 - 60*(x*e + d)
^m*a^2*b^4*d^2*m^5*x^3*e^5 - 390*(x*e + d)^m*a*b^5*d^2*m^4*x^4*e^5 - 210*(x*e + d)^m*b^6*d^2*m^3*x^5*e^5 + 120
*(x*e + d)^m*a*b^5*d^3*m^4*x^3*e^4 + 180*(x*e + d)^m*b^6*d^3*m^3*x^4*e^4 - 120*(x*e + d)^m*b^6*d^4*m^3*x^3*e^3
+ 15*(x*e + d)^m*a^4*b^2*m^6*x^3*e^7 + 480*(x*e + d)^m*a^3*b^3*m^5*x^4*e^7 + 3105*(x*e + d)^m*a^2*b^4*m^4*x^5
*e^7 + 4920*(x*e + d)^m*a*b^5*m^3*x^6*e^7 + 1624*(x*e + d)^m*b^6*m^2*x^7*e^7 + 15*(x*e + d)^m*a^4*b^2*d*m^6*x^
2*e^6 + 420*(x*e + d)^m*a^3*b^3*d*m^5*x^3*e^6 + 1965*(x*e + d)^m*a^2*b^4*d*m^4*x^4*e^6 + 1770*(x*e + d)^m*a*b^
5*d*m^3*x^5*e^6 + 274*(x*e + d)^m*b^6*d*m^2*x^6*e^6 - 60*(x*e + d)^m*a^3*b^3*d^2*m^5*x^2*e^5 - 960*(x*e + d)^m
*a^2*b^4*d^2*m^4*x^3*e^5 - 1590*(x*e + d)^m*a*b^5*d^2*m^3*x^4*e^5 - 300*(x*e + d)^m*b^6*d^2*m^2*x^5*e^5 + 180*
(x*e + d)^m*a^2*b^4*d^3*m^4*x^2*e^4 + 1200*(x*e + d)^m*a*b^5*d^3*m^3*x^3*e^4 + 330*(x*e + d)^m*b^6*d^3*m^2*x^4
*e^4 - 360*(x*e + d)^m*a*b^5*d^4*m^3*x^2*e^3 - 360*(x*e + d)^m*b^6*d^4*m^2*x^3*e^3 + 360*(x*e + d)^m*b^6*d^5*m
^2*x^2*e^2 + 6*(x*e + d)^m*a^5*b*m^6*x^2*e^7 + 375*(x*e + d)^m*a^4*b^2*m^5*x^3*e^7 + 4520*(x*e + d)^m*a^3*b^3*
m^4*x^4*e^7 + 13875*(x*e + d)^m*a^2*b^4*m^3*x^5*e^7 + 11094*(x*e + d)^m*a*b^5*m^2*x^6*e^7 + 1764*(x*e + d)^m*b
^6*m*x^7*e^7 + 6*(x*e + d)^m*a^5*b*d*m^6*x*e^6 + 345*(x*e + d)^m*a^4*b^2*d*m^5*x^2*e^6 + 3260*(x*e + d)^m*a^3*
b^3*d*m^4*x^3*e^6 + 6015*(x*e + d)^m*a^2*b^4*d*m^3*x^4*e^6 + 2244*(x*e + d)^m*a*b^5*d*m^2*x^5*e^6 + 120*(x*e +
d)^m*b^6*d*m*x^6*e^6 - 30*(x*e + d)^m*a^4*b^2*d^2*m^5*x*e^5 - 1140*(x*e + d)^m*a^3*b^3*d^2*m^4*x^2*e^5 - 4980
*(x*e + d)^m*a^2*b^4*d^2*m^3*x^3*e^5 - 2490*(x*e + d)^m*a*b^5*d^2*m^2*x^4*e^5 - 144*(x*e + d)^m*b^6*d^2*m*x^5*
e^5 + 120*(x*e + d)^m*a^3*b^3*d^3*m^4*x*e^4 + 2520*(x*e + d)^m*a^2*b^4*d^3*m^3*x^2*e^4 + 2760*(x*e + d)^m*a*b^
5*d^3*m^2*x^3*e^4 + 180*(x*e + d)^m*b^6*d^3*m*x^4*e^4 - 360*(x*e + d)^m*a^2*b^4*d^4*m^3*x*e^3 - 2880*(x*e + d)
^m*a*b^5*d^4*m^2*x^2*e^3 - 240*(x*e + d)^m*b^6*d^4*m*x^3*e^3 + 720*(x*e + d)^m*a*b^5*d^5*m^2*x*e^2 + 360*(x*e
+ d)^m*b^6*d^5*m*x^2*e^2 - 720*(x*e + d)^m*b^6*d^6*m*x*e + (x*e + d)^m*a^6*m^6*x*e^7 + 156*(x*e + d)^m*a^5*b*m
^5*x^2*e^7 + 3705*(x*e + d)^m*a^4*b^2*m^4*x^3*e^7 + 21120*(x*e + d)^m*a^3*b^3*m^3*x^4*e^7 + 32160*(x*e + d)^m*
a^2*b^4*m^2*x^5*e^7 + 12228*(x*e + d)^m*a*b^5*m*x^6*e^7 + 720*(x*e + d)^m*b^6*x^7*e^7 + (x*e + d)^m*a^6*d*m^6*
e^6 + 150*(x*e + d)^m*a^5*b*d*m^5*x*e^6 + 3015*(x*e + d)^m*a^4*b^2*d*m^4*x^2*e^6 + 11340*(x*e + d)^m*a^3*b^3*d
*m^3*x^3*e^6 + 8100*(x*e + d)^m*a^2*b^4*d*m^2*x^4*e^6 + 1008*(x*e + d)^m*a*b^5*d*m*x^5*e^6 - 6*(x*e + d)^m*a^5
*b*d^2*m^5*e^5 - 660*(x*e + d)^m*a^4*b^2*d^2*m^4*x*e^5 - 7500*(x*e + d)^m*a^3*b^3*d^2*m^3*x^2*e^5 - 9120*(x*e
+ d)^m*a^2*b^4*d^2*m^2*x^3*e^5 - 1260*(x*e + d)^m*a*b^5*d^2*m*x^4*e^5 + 30*(x*e + d)^m*a^4*b^2*d^3*m^4*e^4 + 2
160*(x*e + d)^m*a^3*b^3*d^3*m^3*x*e^4 + 9900*(x*e + d)^m*a^2*b^4*d^3*m^2*x^2*e^4 + 1680*(x*e + d)^m*a*b^5*d^3*
m*x^3*e^4 - 120*(x*e + d)^m*a^3*b^3*d^4*m^3*e^3 - 4680*(x*e + d)^m*a^2*b^4*d^4*m^2*x*e^3 - 2520*(x*e + d)^m*a*
b^5*d^4*m*x^2*e^3 + 360*(x*e + d)^m*a^2*b^4*d^5*m^2*e^2 + 5040*(x*e + d)^m*a*b^5*d^5*m*x*e^2 - 720*(x*e + d)^m
*a*b^5*d^6*m*e + 720*(x*e + d)^m*b^6*d^7 + 27*(x*e + d)^m*a^6*m^5*x*e^7 + 1620*(x*e + d)^m*a^5*b*m^4*x^2*e^7 +
18285*(x*e + d)^m*a^4*b^2*m^3*x^3*e^7 + 50900*(x*e + d)^m*a^3*b^3*m^2*x^4*e^7 + 36180*(x*e + d)^m*a^2*b^4*m*x
^5*e^7 + 5040*(x*e + d)^m*a*b^5*x^6*e^7 + 27*(x*e + d)^m*a^6*d*m^5*e^6 + 1470*(x*e + d)^m*a^5*b*d*m^4*x*e^6 +
12255*(x*e + d)^m*a^4*b^2*d*m^3*x^2*e^6 + 16880*(x*e + d)^m*a^3*b^3*d*m^2*x^3*e^6 + 3780*(x*e + d)^m*a^2*b^4*d
*m*x^4*e^6 - 150*(x*e + d)^m*a^5*b*d^2*m^4*e^5 - 5370*(x*e + d)^m*a^4*b^2*d^2*m^3*x*e^5 - 19020*(x*e + d)^m*a^
3*b^3*d^2*m^2*x^2*e^5 - 5040*(x*e + d)^m*a^2*b^4*d^2*m*x^3*e^5 + 660*(x*e + d)^m*a^4*b^2*d^3*m^3*e^4 + 12840*(
x*e + d)^m*a^3*b^3*d^3*m^2*x*e^4 + 7560*(x*e + d)^m*a^2*b^4*d^3*m*x^2*e^4 - 2160*(x*e + d)^m*a^3*b^3*d^4*m^2*e
^3 - 15120*(x*e + d)^m*a^2*b^4*d^4*m*x*e^3 + 4680*(x*e + d)^m*a^2*b^4*d^5*m*e^2 - 5040*(x*e + d)^m*a*b^5*d^6*e
+ 295*(x*e + d)^m*a^6*m^4*x*e^7 + 8520*(x*e + d)^m*a^5*b*m^3*x^2*e^7 + 46680*(x*e + d)^m*a^4*b^2*m^2*x^3*e^7
+ 59040*(x*e + d)^m*a^3*b^3*m*x^4*e^7 + 15120*(x*e + d)^m*a^2*b^4*x^5*e^7 + 295*(x*e + d)^m*a^6*d*m^4*e^6 + 70
50*(x*e + d)^m*a^5*b*d*m^3*x*e^6 + 22170*(x*e + d)^m*a^4*b^2*d*m^2*x^2*e^6 + 8400*(x*e + d)^m*a^3*b^3*d*m*x^3*
e^6 - 1470*(x*e + d)^m*a^5*b*d^2*m^3*e^5 - 19140*(x*e + d)^m*a^4*b^2*d^2*m^2*x*e^5 - 12600*(x*e + d)^m*a^3*b^3
*d^2*m*x^2*e^5 + 5370*(x*e + d)^m*a^4*b^2*d^3*m^2*e^4 + 25200*(x*e + d)^m*a^3*b^3*d^3*m*x*e^4 - 12840*(x*e + d
)^m*a^3*b^3*d^4*m*e^3 + 15120*(x*e + d)^m*a^2*b^4*d^5*e^2 + 1665*(x*e + d)^m*a^6*m^3*x*e^7 + 23574*(x*e + d)^m
*a^5*b*m^2*x^2*e^7 + 56940*(x*e + d)^m*a^4*b^2*m*x^3*e^7 + 25200*(x*e + d)^m*a^3*b^3*x^4*e^7 + 1665*(x*e + d)^
m*a^6*d*m^3*e^6 + 16524*(x*e + d)^m*a^5*b*d*m^2*x*e^6 + 12600*(x*e + d)^m*a^4*b^2*d*m*x^2*e^6 - 7050*(x*e + d)
^m*a^5*b*d^2*m^2*e^5 - 25200*(x*e + d)^m*a^4*b^2*d^2*m*x*e^5 + 19140*(x*e + d)^m*a^4*b^2*d^3*m*e^4 - 25200*(x*
e + d)^m*a^3*b^3*d^4*e^3 + 5104*(x*e + d)^m*a^6*m^2*x*e^7 + 31644*(x*e + d)^m*a^5*b*m*x^2*e^7 + 25200*(x*e + d
)^m*a^4*b^2*x^3*e^7 + 5104*(x*e + d)^m*a^6*d*m^2*e^6 + 15120*(x*e + d)^m*a^5*b*d*m*x*e^6 - 16524*(x*e + d)^m*a
^5*b*d^2*m*e^5 + 25200*(x*e + d)^m*a^4*b^2*d^3*e^4 + 8028*(x*e + d)^m*a^6*m*x*e^7 + 15120*(x*e + d)^m*a^5*b*x^
2*e^7 + 8028*(x*e + d)^m*a^6*d*m*e^6 - 15120*(x*e + d)^m*a^5*b*d^2*e^5 + 5040*(x*e + d)^m*a^6*x*e^7 + 5040*(x*
e + d)^m*a^6*d*e^6)/(m^7*e^7 + 28*m^6*e^7 + 322*m^5*e^7 + 1960*m^4*e^7 + 6769*m^3*e^7 + 13132*m^2*e^7 + 13068*
m*e^7 + 5040*e^7)

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maple [B]  time = 0.08, size = 2157, normalized size = 10.47

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

(e*x+d)^(m+1)*(b^6*e^6*m^6*x^6+6*a*b^5*e^6*m^6*x^5+21*b^6*e^6*m^5*x^6+15*a^2*b^4*e^6*m^6*x^4+132*a*b^5*e^6*m^5
*x^5-6*b^6*d*e^5*m^5*x^5+175*b^6*e^6*m^4*x^6+20*a^3*b^3*e^6*m^6*x^3+345*a^2*b^4*e^6*m^5*x^4-30*a*b^5*d*e^5*m^5
*x^4+1140*a*b^5*e^6*m^4*x^5-90*b^6*d*e^5*m^4*x^5+735*b^6*e^6*m^3*x^6+15*a^4*b^2*e^6*m^6*x^2+480*a^3*b^3*e^6*m^
5*x^3-60*a^2*b^4*d*e^5*m^5*x^3+3105*a^2*b^4*e^6*m^4*x^4-510*a*b^5*d*e^5*m^4*x^4+4920*a*b^5*e^6*m^3*x^5+30*b^6*
d^2*e^4*m^4*x^4-510*b^6*d*e^5*m^3*x^5+1624*b^6*e^6*m^2*x^6+6*a^5*b*e^6*m^6*x+375*a^4*b^2*e^6*m^5*x^2-60*a^3*b^
3*d*e^5*m^5*x^2+4520*a^3*b^3*e^6*m^4*x^3-1140*a^2*b^4*d*e^5*m^4*x^3+13875*a^2*b^4*e^6*m^3*x^4+120*a*b^5*d^2*e^
4*m^4*x^3-3150*a*b^5*d*e^5*m^3*x^4+11094*a*b^5*e^6*m^2*x^5+300*b^6*d^2*e^4*m^3*x^4-1350*b^6*d*e^5*m^2*x^5+1764
*b^6*e^6*m*x^6+a^6*e^6*m^6+156*a^5*b*e^6*m^5*x-30*a^4*b^2*d*e^5*m^5*x+3705*a^4*b^2*e^6*m^4*x^2-1260*a^3*b^3*d*
e^5*m^4*x^2+21120*a^3*b^3*e^6*m^3*x^3+180*a^2*b^4*d^2*e^4*m^4*x^2-7860*a^2*b^4*d*e^5*m^3*x^3+32160*a^2*b^4*e^6
*m^2*x^4+1560*a*b^5*d^2*e^4*m^3*x^3-8850*a*b^5*d*e^5*m^2*x^4+12228*a*b^5*e^6*m*x^5-120*b^6*d^3*e^3*m^3*x^3+105
0*b^6*d^2*e^4*m^2*x^4-1644*b^6*d*e^5*m*x^5+720*b^6*e^6*x^6+27*a^6*e^6*m^5-6*a^5*b*d*e^5*m^5+1620*a^5*b*e^6*m^4
*x-690*a^4*b^2*d*e^5*m^4*x+18285*a^4*b^2*e^6*m^3*x^2+120*a^3*b^3*d^2*e^4*m^4*x-9780*a^3*b^3*d*e^5*m^3*x^2+5090
0*a^3*b^3*e^6*m^2*x^3+2880*a^2*b^4*d^2*e^4*m^3*x^2-24060*a^2*b^4*d*e^5*m^2*x^3+36180*a^2*b^4*e^6*m*x^4-360*a*b
^5*d^3*e^3*m^3*x^2+6360*a*b^5*d^2*e^4*m^2*x^3-11220*a*b^5*d*e^5*m*x^4+5040*a*b^5*e^6*x^5-720*b^6*d^3*e^3*m^2*x
^3+1500*b^6*d^2*e^4*m*x^4-720*b^6*d*e^5*x^5+295*a^6*e^6*m^4-150*a^5*b*d*e^5*m^4+8520*a^5*b*e^6*m^3*x+30*a^4*b^
2*d^2*e^4*m^4-6030*a^4*b^2*d*e^5*m^3*x+46680*a^4*b^2*e^6*m^2*x^2+2280*a^3*b^3*d^2*e^4*m^3*x-34020*a^3*b^3*d*e^
5*m^2*x^2+59040*a^3*b^3*e^6*m*x^3-360*a^2*b^4*d^3*e^3*m^3*x+14940*a^2*b^4*d^2*e^4*m^2*x^2-32400*a^2*b^4*d*e^5*
m*x^3+15120*a^2*b^4*e^6*x^4-3600*a*b^5*d^3*e^3*m^2*x^2+9960*a*b^5*d^2*e^4*m*x^3-5040*a*b^5*d*e^5*x^4+360*b^6*d
^4*e^2*m^2*x^2-1320*b^6*d^3*e^3*m*x^3+720*b^6*d^2*e^4*x^4+1665*a^6*e^6*m^3-1470*a^5*b*d*e^5*m^3+23574*a^5*b*e^
6*m^2*x+660*a^4*b^2*d^2*e^4*m^3-24510*a^4*b^2*d*e^5*m^2*x+56940*a^4*b^2*e^6*m*x^2-120*a^3*b^3*d^3*e^3*m^3+1500
0*a^3*b^3*d^2*e^4*m^2*x-50640*a^3*b^3*d*e^5*m*x^2+25200*a^3*b^3*e^6*x^3-5040*a^2*b^4*d^3*e^3*m^2*x+27360*a^2*b
^4*d^2*e^4*m*x^2-15120*a^2*b^4*d*e^5*x^3+720*a*b^5*d^4*e^2*m^2*x-8280*a*b^5*d^3*e^3*m*x^2+5040*a*b^5*d^2*e^4*x
^3+1080*b^6*d^4*e^2*m*x^2-720*b^6*d^3*e^3*x^3+5104*a^6*e^6*m^2-7050*a^5*b*d*e^5*m^2+31644*a^5*b*e^6*m*x+5370*a
^4*b^2*d^2*e^4*m^2-44340*a^4*b^2*d*e^5*m*x+25200*a^4*b^2*e^6*x^2-2160*a^3*b^3*d^3*e^3*m^2+38040*a^3*b^3*d^2*e^
4*m*x-25200*a^3*b^3*d*e^5*x^2+360*a^2*b^4*d^4*e^2*m^2-19800*a^2*b^4*d^3*e^3*m*x+15120*a^2*b^4*d^2*e^4*x^2+5760
*a*b^5*d^4*e^2*m*x-5040*a*b^5*d^3*e^3*x^2-720*b^6*d^5*e*m*x+720*b^6*d^4*e^2*x^2+8028*a^6*e^6*m-16524*a^5*b*d*e
^5*m+15120*a^5*b*e^6*x+19140*a^4*b^2*d^2*e^4*m-25200*a^4*b^2*d*e^5*x-12840*a^3*b^3*d^3*e^3*m+25200*a^3*b^3*d^2
*e^4*x+4680*a^2*b^4*d^4*e^2*m-15120*a^2*b^4*d^3*e^3*x-720*a*b^5*d^5*e*m+5040*a*b^5*d^4*e^2*x-720*b^6*d^5*e*x+5
040*a^6*e^6-15120*a^5*b*d*e^5+25200*a^4*b^2*d^2*e^4-25200*a^3*b^3*d^3*e^3+15120*a^2*b^4*d^4*e^2-5040*a*b^5*d^5
*e+720*b^6*d^6)/e^7/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)

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maxima [B]  time = 1.52, size = 818, normalized size = 3.97 \begin {gather*} \frac {6 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a^{5} b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a^{6}}{e {\left (m + 1\right )}} + \frac {15 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a^{4} b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {20 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} a^{3} b^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {15 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} a^{2} b^{4}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {6 \, {\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m} a b^{5}}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{6}} + \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{7} x^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d e^{6} x^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} e^{5} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} e^{4} x^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} e^{3} x^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} e^{2} x^{2} - 720 \, d^{6} e m x + 720 \, d^{7}\right )} {\left (e x + d\right )}^{m} b^{6}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

6*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^5*b/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*a^6/(e*(m + 1)
) + 15*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*a^4*b^2/((m^3 + 6*m^2
+ 11*m + 6)*e^3) + 20*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2
*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*a^3*b^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 15*((m^4 + 10*m^3
+ 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 1
2*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*a^2*b^4/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m
+ 120)*e^5) + 6*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 +
24*m)*d*e^5*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 +
m)*d^4*e^2*x^2 + 120*d^5*e*m*x - 120*d^6)*(e*x + d)^m*a*b^5/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 17
64*m + 720)*e^6) + ((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 + 15*m^5 + 85*
m^4 + 225*m^3 + 274*m^2 + 120*m)*d*e^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5 + 30*(m^4 +
6*m^3 + 11*m^2 + 6*m)*d^3*e^4*x^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*e^3*x^3 + 360*(m^2 + m)*d^5*e^2*x^2 - 720*d^6
*e*m*x + 720*d^7)*(e*x + d)^m*b^6/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)
*e^7)

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mupad [B]  time = 1.64, size = 1898, normalized size = 9.21

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((d + e*x)^m*(720*b^6*d^7 + 5040*a^6*d*e^6 - 15120*a^5*b*d^2*e^5 + 5104*a^6*d*e^6*m^2 + 1665*a^6*d*e^6*m^3 + 2
95*a^6*d*e^6*m^4 + 27*a^6*d*e^6*m^5 + a^6*d*e^6*m^6 + 15120*a^2*b^4*d^5*e^2 - 25200*a^3*b^3*d^4*e^3 + 25200*a^
4*b^2*d^3*e^4 - 5040*a*b^5*d^6*e + 8028*a^6*d*e^6*m - 720*a*b^5*d^6*e*m + 360*a^2*b^4*d^5*e^2*m^2 - 2160*a^3*b
^3*d^4*e^3*m^2 + 5370*a^4*b^2*d^3*e^4*m^2 - 120*a^3*b^3*d^4*e^3*m^3 + 660*a^4*b^2*d^3*e^4*m^3 + 30*a^4*b^2*d^3
*e^4*m^4 - 16524*a^5*b*d^2*e^5*m + 4680*a^2*b^4*d^5*e^2*m - 12840*a^3*b^3*d^4*e^3*m + 19140*a^4*b^2*d^3*e^4*m
- 7050*a^5*b*d^2*e^5*m^2 - 1470*a^5*b*d^2*e^5*m^3 - 150*a^5*b*d^2*e^5*m^4 - 6*a^5*b*d^2*e^5*m^5))/(e^7*(13068*
m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (x*(d + e*x)^m*(5040*a^6*e^7 + 8028*a^
6*e^7*m + 5104*a^6*e^7*m^2 + 1665*a^6*e^7*m^3 + 295*a^6*e^7*m^4 + 27*a^6*e^7*m^5 + a^6*e^7*m^6 - 720*b^6*d^6*e
*m + 15120*a^5*b*d*e^6*m - 4680*a^2*b^4*d^4*e^3*m^2 + 12840*a^3*b^3*d^3*e^4*m^2 - 19140*a^4*b^2*d^2*e^5*m^2 -
360*a^2*b^4*d^4*e^3*m^3 + 2160*a^3*b^3*d^3*e^4*m^3 - 5370*a^4*b^2*d^2*e^5*m^3 + 120*a^3*b^3*d^3*e^4*m^4 - 660*
a^4*b^2*d^2*e^5*m^4 - 30*a^4*b^2*d^2*e^5*m^5 + 5040*a*b^5*d^5*e^2*m + 16524*a^5*b*d*e^6*m^2 + 7050*a^5*b*d*e^6
*m^3 + 1470*a^5*b*d*e^6*m^4 + 150*a^5*b*d*e^6*m^5 + 6*a^5*b*d*e^6*m^6 - 15120*a^2*b^4*d^4*e^3*m + 25200*a^3*b^
3*d^3*e^4*m - 25200*a^4*b^2*d^2*e^5*m + 720*a*b^5*d^5*e^2*m^2))/(e^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^
4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (b^6*x^7*(d + e*x)^m*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m
^6 + 720))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040) + (3*b*x^2*(m + 1)*(d +
e*x)^m*(5040*a^5*e^5 + 5508*a^5*e^5*m + 120*b^5*d^5*m + 2350*a^5*e^5*m^2 + 490*a^5*e^5*m^3 + 50*a^5*e^5*m^4 +
2*a^5*e^5*m^5 - 840*a*b^4*d^4*e*m + 4200*a^4*b*d*e^4*m + 780*a^2*b^3*d^3*e^2*m^2 - 2140*a^3*b^2*d^2*e^3*m^2 +
60*a^2*b^3*d^3*e^2*m^3 - 360*a^3*b^2*d^2*e^3*m^3 - 20*a^3*b^2*d^2*e^3*m^4 - 120*a*b^4*d^4*e*m^2 + 3190*a^4*b*
d*e^4*m^2 + 895*a^4*b*d*e^4*m^3 + 110*a^4*b*d*e^4*m^4 + 5*a^4*b*d*e^4*m^5 + 2520*a^2*b^3*d^3*e^2*m - 4200*a^3*
b^2*d^2*e^3*m))/(e^5*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (3*b^4*x^5
*(d + e*x)^m*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)*(210*a^2*e^2 + 65*a^2*e^2*m - 2*b^2*d^2*m + 5*a^2*e^2*m^2 + 1
4*a*b*d*e*m + 2*a*b*d*e*m^2))/(e^2*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)
) + (5*b^3*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(840*a^3*e^3 + 428*a^3*e^3*m + 6*b^3*d^3*m + 72*a^3*e^3*m^
2 + 4*a^3*e^3*m^3 - 42*a*b^2*d^2*e*m + 126*a^2*b*d*e^2*m - 6*a*b^2*d^2*e*m^2 + 39*a^2*b*d*e^2*m^2 + 3*a^2*b*d*
e^2*m^3))/(e^3*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (b^5*x^6*(d + e*
x)^m*(42*a*e + 6*a*e*m + b*d*m)*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(e*(13068*m + 13132*m^2 + 676
9*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (5*b^2*x^3*(d + e*x)^m*(3*m + m^2 + 2)*(2520*a^4*e^4 + 19
14*a^4*e^4*m - 24*b^4*d^4*m + 537*a^4*e^4*m^2 + 66*a^4*e^4*m^3 + 3*a^4*e^4*m^4 + 168*a*b^3*d^3*e*m + 840*a^3*b
*d*e^3*m - 156*a^2*b^2*d^2*e^2*m^2 - 12*a^2*b^2*d^2*e^2*m^3 + 24*a*b^3*d^3*e*m^2 + 428*a^3*b*d*e^3*m^2 + 72*a^
3*b*d*e^3*m^3 + 4*a^3*b*d*e^3*m^4 - 504*a^2*b^2*d^2*e^2*m))/(e^4*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 +
322*m^5 + 28*m^6 + m^7 + 5040))

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sympy [F(-1)]  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((e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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