3.22.47 \(\int (d+e x)^3 (a+b x+c x^2)^4 \, dx\) [2147]

Optimal. Leaf size=443 \[ \frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}{4 e^9}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^9}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^6}{3 e^9}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^7}{7 e^9}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^8}{8 e^9}-\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^9}{9 e^9}+\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{10}}{5 e^9}-\frac {4 c^3 (2 c d-b e) (d+e x)^{11}}{11 e^9}+\frac {c^4 (d+e x)^{12}}{12 e^9} \]

[Out]

1/4*(a*e^2-b*d*e+c*d^2)^4*(e*x+d)^4/e^9-4/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^5/e^9+1/3*(a*e^2-b*d*e+
c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^6/e^9-4/7*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d
^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^7/e^9+1/8*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20*c^3*d^2*e*(
-3*a*e+7*b*d)+6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d)^8/e^9-4/9*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c
*e*(-3*a*e+7*b*d))*(e*x+d)^9/e^9+1/5*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^10/e^9-4/11*c^3*(-b
*e+2*c*d)*(e*x+d)^11/e^9+1/12*c^4*(e*x+d)^12/e^9

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Rubi [A]
time = 0.44, antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \begin {gather*} \frac {(d+e x)^8 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{8 e^9}+\frac {c^2 (d+e x)^{10} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac {4 c (d+e x)^9 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{9 e^9}-\frac {4 (d+e x)^7 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^9}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9}-\frac {4 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^4}{4 e^9}-\frac {4 c^3 (d+e x)^{11} (2 c d-b e)}{11 e^9}+\frac {c^4 (d+e x)^{12}}{12 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + b*x + c*x^2)^4,x]

[Out]

((c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^4)/(4*e^9) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5)/(5*e
^9) + ((c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^6)/(3*e^9) - (4*(2*c
*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^7)/(7*e^9) + ((70*c^4*
d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*
e + a^2*e^2))*(d + e*x)^8)/(8*e^9) - (4*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^
9)/(9*e^9) + (c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^10)/(5*e^9) - (4*c^3*(2*c*d - b*e)*
(d + e*x)^11)/(11*e^9) + (c^4*(d + e*x)^12)/(12*e^9)

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (d+e x)^3 \left (a+b x+c x^2\right )^4 \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^3}{e^8}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^5}{e^8}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^6}{e^8}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^7}{e^8}+\frac {4 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^8}{e^8}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^9}{e^8}-\frac {4 c^3 (2 c d-b e) (d+e x)^{10}}{e^8}+\frac {c^4 (d+e x)^{11}}{e^8}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}{4 e^9}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^9}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^6}{3 e^9}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^7}{7 e^9}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^8}{8 e^9}-\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^9}{9 e^9}+\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{10}}{5 e^9}-\frac {4 c^3 (2 c d-b e) (d+e x)^{11}}{11 e^9}+\frac {c^4 (d+e x)^{12}}{12 e^9}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 611, normalized size = 1.38 \begin {gather*} a^4 d^3 x+\frac {1}{2} a^3 d^2 (4 b d+3 a e) x^2+\frac {1}{3} a^2 d \left (6 b^2 d^2+12 a b d e+a \left (4 c d^2+3 a e^2\right )\right ) x^3+\frac {1}{4} a \left (4 b^3 d^3+18 a b^2 d^2 e+12 a b d \left (c d^2+a e^2\right )+a^2 e \left (12 c d^2+a e^2\right )\right ) x^4+\frac {1}{5} \left (b^4 d^3+12 a b^3 d^2 e+4 a^2 b e \left (9 c d^2+a e^2\right )+6 a^2 c d \left (c d^2+2 a e^2\right )+6 a b^2 d \left (2 c d^2+3 a e^2\right )\right ) x^5+\frac {1}{6} \left (3 b^4 d^2 e+6 a b^2 e \left (6 c d^2+a e^2\right )+2 a^2 c e \left (9 c d^2+2 a e^2\right )+12 a b c d \left (c d^2+3 a e^2\right )+4 b^3 \left (c d^3+3 a d e^2\right )\right ) x^6+\frac {1}{7} \left (3 b^4 d e^2+12 a b c e \left (3 c d^2+a e^2\right )+6 b^2 c d \left (c d^2+6 a e^2\right )+2 a c^2 d \left (2 c d^2+9 a e^2\right )+4 b^3 \left (3 c d^2 e+a e^3\right )\right ) x^7+\frac {1}{8} \left (12 b^3 c d e^2+b^4 e^3+6 a c^2 e \left (2 c d^2+a e^2\right )+6 b^2 c e \left (3 c d^2+2 a e^2\right )+4 b c^2 d \left (c d^2+9 a e^2\right )\right ) x^8+\frac {1}{9} c \left (c^3 d^3+4 b^3 e^3+12 c^2 d e (b d+a e)+6 b c e^2 (3 b d+2 a e)\right ) x^9+\frac {1}{10} c^2 e \left (3 c^2 d^2+6 b^2 e^2+4 c e (3 b d+a e)\right ) x^{10}+\frac {1}{11} c^3 e^2 (3 c d+4 b e) x^{11}+\frac {1}{12} c^4 e^3 x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a + b*x + c*x^2)^4,x]

[Out]

a^4*d^3*x + (a^3*d^2*(4*b*d + 3*a*e)*x^2)/2 + (a^2*d*(6*b^2*d^2 + 12*a*b*d*e + a*(4*c*d^2 + 3*a*e^2))*x^3)/3 +
(a*(4*b^3*d^3 + 18*a*b^2*d^2*e + 12*a*b*d*(c*d^2 + a*e^2) + a^2*e*(12*c*d^2 + a*e^2))*x^4)/4 + ((b^4*d^3 + 12
*a*b^3*d^2*e + 4*a^2*b*e*(9*c*d^2 + a*e^2) + 6*a^2*c*d*(c*d^2 + 2*a*e^2) + 6*a*b^2*d*(2*c*d^2 + 3*a*e^2))*x^5)
/5 + ((3*b^4*d^2*e + 6*a*b^2*e*(6*c*d^2 + a*e^2) + 2*a^2*c*e*(9*c*d^2 + 2*a*e^2) + 12*a*b*c*d*(c*d^2 + 3*a*e^2
) + 4*b^3*(c*d^3 + 3*a*d*e^2))*x^6)/6 + ((3*b^4*d*e^2 + 12*a*b*c*e*(3*c*d^2 + a*e^2) + 6*b^2*c*d*(c*d^2 + 6*a*
e^2) + 2*a*c^2*d*(2*c*d^2 + 9*a*e^2) + 4*b^3*(3*c*d^2*e + a*e^3))*x^7)/7 + ((12*b^3*c*d*e^2 + b^4*e^3 + 6*a*c^
2*e*(2*c*d^2 + a*e^2) + 6*b^2*c*e*(3*c*d^2 + 2*a*e^2) + 4*b*c^2*d*(c*d^2 + 9*a*e^2))*x^8)/8 + (c*(c^3*d^3 + 4*
b^3*e^3 + 12*c^2*d*e*(b*d + a*e) + 6*b*c*e^2*(3*b*d + 2*a*e))*x^9)/9 + (c^2*e*(3*c^2*d^2 + 6*b^2*e^2 + 4*c*e*(
3*b*d + a*e))*x^10)/10 + (c^3*e^2*(3*c*d + 4*b*e)*x^11)/11 + (c^4*e^3*x^12)/12

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Maple [A]
time = 0.73, size = 747, normalized size = 1.69 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(c*x^2+b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

1/12*c^4*e^3*x^12+1/11*(4*b*c^3*e^3+3*c^4*d*e^2)*x^11+1/10*(3*d^2*e*c^4+12*d*e^2*b*c^3+e^3*(2*(2*a*c+b^2)*c^2+
4*b^2*c^2))*x^10+1/9*(d^3*c^4+12*b*c^3*d^2*e+3*d*e^2*(2*(2*a*c+b^2)*c^2+4*b^2*c^2)+e^3*(4*a*b*c^2+4*(2*a*c+b^2
)*b*c))*x^9+1/8*(4*c^3*b*d^3+3*d^2*e*(2*(2*a*c+b^2)*c^2+4*b^2*c^2)+3*d*e^2*(4*a*b*c^2+4*(2*a*c+b^2)*b*c)+e^3*(
2*a^2*c^2+8*a*c*b^2+(2*a*c+b^2)^2))*x^8+1/7*(d^3*(2*(2*a*c+b^2)*c^2+4*b^2*c^2)+3*d^2*e*(4*a*b*c^2+4*(2*a*c+b^2
)*b*c)+3*d*e^2*(2*a^2*c^2+8*a*c*b^2+(2*a*c+b^2)^2)+e^3*(4*c*a^2*b+4*a*b*(2*a*c+b^2)))*x^7+1/6*(d^3*(4*a*b*c^2+
4*(2*a*c+b^2)*b*c)+3*d^2*e*(2*a^2*c^2+8*a*c*b^2+(2*a*c+b^2)^2)+3*d*e^2*(4*c*a^2*b+4*a*b*(2*a*c+b^2))+e^3*(2*a^
2*(2*a*c+b^2)+4*a^2*b^2))*x^6+1/5*(d^3*(2*a^2*c^2+8*a*c*b^2+(2*a*c+b^2)^2)+3*d^2*e*(4*c*a^2*b+4*a*b*(2*a*c+b^2
))+3*d*e^2*(2*a^2*(2*a*c+b^2)+4*a^2*b^2)+4*a^3*b*e^3)*x^5+1/4*(d^3*(4*c*a^2*b+4*a*b*(2*a*c+b^2))+3*d^2*e*(2*a^
2*(2*a*c+b^2)+4*a^2*b^2)+12*d*e^2*a^3*b+e^3*a^4)*x^4+1/3*(d^3*(2*a^2*(2*a*c+b^2)+4*a^2*b^2)+12*d^2*e*a^3*b+3*d
*e^2*a^4)*x^3+1/2*(3*a^4*d^2*e+4*a^3*b*d^3)*x^2+a^4*d^3*x

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Maxima [A]
time = 0.27, size = 632, normalized size = 1.43 \begin {gather*} \frac {1}{12} \, c^{4} x^{12} e^{3} + \frac {1}{11} \, {\left (3 \, c^{4} d e^{2} + 4 \, b c^{3} e^{3}\right )} x^{11} + \frac {1}{10} \, {\left (3 \, c^{4} d^{2} e + 12 \, b c^{3} d e^{2} + 6 \, b^{2} c^{2} e^{3} + 4 \, a c^{3} e^{3}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{3} + 12 \, b c^{3} d^{2} e + 4 \, b^{3} c e^{3} + 12 \, a b c^{2} e^{3} + 6 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d\right )} x^{9} + \frac {1}{8} \, {\left (4 \, b c^{3} d^{3} + b^{4} e^{3} + 12 \, a b^{2} c e^{3} + 6 \, a^{2} c^{2} e^{3} + 6 \, {\left (3 \, b^{2} c^{2} e + 2 \, a c^{3} e\right )} d^{2} + 12 \, {\left (b^{3} c e^{2} + 3 \, a b c^{2} e^{2}\right )} d\right )} x^{8} + a^{4} d^{3} x + \frac {1}{7} \, {\left (4 \, a b^{3} e^{3} + 12 \, a^{2} b c e^{3} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} + 12 \, {\left (b^{3} c e + 3 \, a b c^{2} e\right )} d^{2} + 3 \, {\left (b^{4} e^{2} + 12 \, a b^{2} c e^{2} + 6 \, a^{2} c^{2} e^{2}\right )} d\right )} x^{7} + \frac {1}{6} \, {\left (6 \, a^{2} b^{2} e^{3} + 4 \, a^{3} c e^{3} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} + 3 \, {\left (b^{4} e + 12 \, a b^{2} c e + 6 \, a^{2} c^{2} e\right )} d^{2} + 12 \, {\left (a b^{3} e^{2} + 3 \, a^{2} b c e^{2}\right )} d\right )} x^{6} + \frac {1}{5} \, {\left (4 \, a^{3} b e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} + 12 \, {\left (a b^{3} e + 3 \, a^{2} b c e\right )} d^{2} + 6 \, {\left (3 \, a^{2} b^{2} e^{2} + 2 \, a^{3} c e^{2}\right )} d\right )} x^{5} + \frac {1}{4} \, {\left (12 \, a^{3} b d e^{2} + a^{4} e^{3} + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} + 6 \, {\left (3 \, a^{2} b^{2} e + 2 \, a^{3} c e\right )} d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (12 \, a^{3} b d^{2} e + 3 \, a^{4} d e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x+a)^4,x, algorithm="maxima")

[Out]

1/12*c^4*x^12*e^3 + 1/11*(3*c^4*d*e^2 + 4*b*c^3*e^3)*x^11 + 1/10*(3*c^4*d^2*e + 12*b*c^3*d*e^2 + 6*b^2*c^2*e^3
+ 4*a*c^3*e^3)*x^10 + 1/9*(c^4*d^3 + 12*b*c^3*d^2*e + 4*b^3*c*e^3 + 12*a*b*c^2*e^3 + 6*(3*b^2*c^2*e^2 + 2*a*c
^3*e^2)*d)*x^9 + 1/8*(4*b*c^3*d^3 + b^4*e^3 + 12*a*b^2*c*e^3 + 6*a^2*c^2*e^3 + 6*(3*b^2*c^2*e + 2*a*c^3*e)*d^2
+ 12*(b^3*c*e^2 + 3*a*b*c^2*e^2)*d)*x^8 + a^4*d^3*x + 1/7*(4*a*b^3*e^3 + 12*a^2*b*c*e^3 + 2*(3*b^2*c^2 + 2*a*
c^3)*d^3 + 12*(b^3*c*e + 3*a*b*c^2*e)*d^2 + 3*(b^4*e^2 + 12*a*b^2*c*e^2 + 6*a^2*c^2*e^2)*d)*x^7 + 1/6*(6*a^2*b
^2*e^3 + 4*a^3*c*e^3 + 4*(b^3*c + 3*a*b*c^2)*d^3 + 3*(b^4*e + 12*a*b^2*c*e + 6*a^2*c^2*e)*d^2 + 12*(a*b^3*e^2
+ 3*a^2*b*c*e^2)*d)*x^6 + 1/5*(4*a^3*b*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3 + 12*(a*b^3*e + 3*a^2*b*c*e)*d
^2 + 6*(3*a^2*b^2*e^2 + 2*a^3*c*e^2)*d)*x^5 + 1/4*(12*a^3*b*d*e^2 + a^4*e^3 + 4*(a*b^3 + 3*a^2*b*c)*d^3 + 6*(3
*a^2*b^2*e + 2*a^3*c*e)*d^2)*x^4 + 1/3*(12*a^3*b*d^2*e + 3*a^4*d*e^2 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3)*x^3 + 1/2*
(4*a^3*b*d^3 + 3*a^4*d^2*e)*x^2

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Fricas [A]
time = 4.72, size = 611, normalized size = 1.38 \begin {gather*} \frac {1}{9} \, c^{4} d^{3} x^{9} + \frac {1}{2} \, b c^{3} d^{3} x^{8} + \frac {2}{7} \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{7} + \frac {2}{3} \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x^{6} + 2 \, a^{3} b d^{3} x^{2} + \frac {1}{5} \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} x^{5} + a^{4} d^{3} x + {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3} x^{3} + \frac {1}{27720} \, {\left (2310 \, c^{4} x^{12} + 10080 \, b c^{3} x^{11} + 5544 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{10} + 12320 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{9} + 22176 \, a^{3} b x^{5} + 3465 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{8} + 6930 \, a^{4} x^{4} + 15840 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{7} + 9240 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x^{6}\right )} e^{3} + \frac {1}{2310} \, {\left (630 \, c^{4} d x^{11} + 2772 \, b c^{3} d x^{10} + 1540 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{9} + 3465 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{8} + 6930 \, a^{3} b d x^{4} + 990 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{7} + 2310 \, a^{4} d x^{3} + 4620 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x^{6} + 2772 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d x^{5}\right )} e^{2} + \frac {1}{420} \, {\left (126 \, c^{4} d^{2} x^{10} + 560 \, b c^{3} d^{2} x^{9} + 315 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{8} + 720 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{7} + 1680 \, a^{3} b d^{2} x^{3} + 210 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x^{6} + 630 \, a^{4} d^{2} x^{2} + 1008 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} x^{5} + 630 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} x^{4}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x+a)^4,x, algorithm="fricas")

[Out]

1/9*c^4*d^3*x^9 + 1/2*b*c^3*d^3*x^8 + 2/7*(3*b^2*c^2 + 2*a*c^3)*d^3*x^7 + 2/3*(b^3*c + 3*a*b*c^2)*d^3*x^6 + 2*
a^3*b*d^3*x^2 + 1/5*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*x^5 + a^4*d^3*x + (a*b^3 + 3*a^2*b*c)*d^3*x^4 + 2/3*(3*
a^2*b^2 + 2*a^3*c)*d^3*x^3 + 1/27720*(2310*c^4*x^12 + 10080*b*c^3*x^11 + 5544*(3*b^2*c^2 + 2*a*c^3)*x^10 + 123
20*(b^3*c + 3*a*b*c^2)*x^9 + 22176*a^3*b*x^5 + 3465*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*x^8 + 6930*a^4*x^4 + 15840*
(a*b^3 + 3*a^2*b*c)*x^7 + 9240*(3*a^2*b^2 + 2*a^3*c)*x^6)*e^3 + 1/2310*(630*c^4*d*x^11 + 2772*b*c^3*d*x^10 + 1
540*(3*b^2*c^2 + 2*a*c^3)*d*x^9 + 3465*(b^3*c + 3*a*b*c^2)*d*x^8 + 6930*a^3*b*d*x^4 + 990*(b^4 + 12*a*b^2*c +
6*a^2*c^2)*d*x^7 + 2310*a^4*d*x^3 + 4620*(a*b^3 + 3*a^2*b*c)*d*x^6 + 2772*(3*a^2*b^2 + 2*a^3*c)*d*x^5)*e^2 + 1
/420*(126*c^4*d^2*x^10 + 560*b*c^3*d^2*x^9 + 315*(3*b^2*c^2 + 2*a*c^3)*d^2*x^8 + 720*(b^3*c + 3*a*b*c^2)*d^2*x
^7 + 1680*a^3*b*d^2*x^3 + 210*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*x^6 + 630*a^4*d^2*x^2 + 1008*(a*b^3 + 3*a^2*b
*c)*d^2*x^5 + 630*(3*a^2*b^2 + 2*a^3*c)*d^2*x^4)*e

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Sympy [A]
time = 0.06, size = 777, normalized size = 1.75 \begin {gather*} a^{4} d^{3} x + \frac {c^{4} e^{3} x^{12}}{12} + x^{11} \cdot \left (\frac {4 b c^{3} e^{3}}{11} + \frac {3 c^{4} d e^{2}}{11}\right ) + x^{10} \cdot \left (\frac {2 a c^{3} e^{3}}{5} + \frac {3 b^{2} c^{2} e^{3}}{5} + \frac {6 b c^{3} d e^{2}}{5} + \frac {3 c^{4} d^{2} e}{10}\right ) + x^{9} \cdot \left (\frac {4 a b c^{2} e^{3}}{3} + \frac {4 a c^{3} d e^{2}}{3} + \frac {4 b^{3} c e^{3}}{9} + 2 b^{2} c^{2} d e^{2} + \frac {4 b c^{3} d^{2} e}{3} + \frac {c^{4} d^{3}}{9}\right ) + x^{8} \cdot \left (\frac {3 a^{2} c^{2} e^{3}}{4} + \frac {3 a b^{2} c e^{3}}{2} + \frac {9 a b c^{2} d e^{2}}{2} + \frac {3 a c^{3} d^{2} e}{2} + \frac {b^{4} e^{3}}{8} + \frac {3 b^{3} c d e^{2}}{2} + \frac {9 b^{2} c^{2} d^{2} e}{4} + \frac {b c^{3} d^{3}}{2}\right ) + x^{7} \cdot \left (\frac {12 a^{2} b c e^{3}}{7} + \frac {18 a^{2} c^{2} d e^{2}}{7} + \frac {4 a b^{3} e^{3}}{7} + \frac {36 a b^{2} c d e^{2}}{7} + \frac {36 a b c^{2} d^{2} e}{7} + \frac {4 a c^{3} d^{3}}{7} + \frac {3 b^{4} d e^{2}}{7} + \frac {12 b^{3} c d^{2} e}{7} + \frac {6 b^{2} c^{2} d^{3}}{7}\right ) + x^{6} \cdot \left (\frac {2 a^{3} c e^{3}}{3} + a^{2} b^{2} e^{3} + 6 a^{2} b c d e^{2} + 3 a^{2} c^{2} d^{2} e + 2 a b^{3} d e^{2} + 6 a b^{2} c d^{2} e + 2 a b c^{2} d^{3} + \frac {b^{4} d^{2} e}{2} + \frac {2 b^{3} c d^{3}}{3}\right ) + x^{5} \cdot \left (\frac {4 a^{3} b e^{3}}{5} + \frac {12 a^{3} c d e^{2}}{5} + \frac {18 a^{2} b^{2} d e^{2}}{5} + \frac {36 a^{2} b c d^{2} e}{5} + \frac {6 a^{2} c^{2} d^{3}}{5} + \frac {12 a b^{3} d^{2} e}{5} + \frac {12 a b^{2} c d^{3}}{5} + \frac {b^{4} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{4} e^{3}}{4} + 3 a^{3} b d e^{2} + 3 a^{3} c d^{2} e + \frac {9 a^{2} b^{2} d^{2} e}{2} + 3 a^{2} b c d^{3} + a b^{3} d^{3}\right ) + x^{3} \left (a^{4} d e^{2} + 4 a^{3} b d^{2} e + \frac {4 a^{3} c d^{3}}{3} + 2 a^{2} b^{2} d^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{4} d^{2} e}{2} + 2 a^{3} b d^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(c*x**2+b*x+a)**4,x)

[Out]

a**4*d**3*x + c**4*e**3*x**12/12 + x**11*(4*b*c**3*e**3/11 + 3*c**4*d*e**2/11) + x**10*(2*a*c**3*e**3/5 + 3*b*
*2*c**2*e**3/5 + 6*b*c**3*d*e**2/5 + 3*c**4*d**2*e/10) + x**9*(4*a*b*c**2*e**3/3 + 4*a*c**3*d*e**2/3 + 4*b**3*
c*e**3/9 + 2*b**2*c**2*d*e**2 + 4*b*c**3*d**2*e/3 + c**4*d**3/9) + x**8*(3*a**2*c**2*e**3/4 + 3*a*b**2*c*e**3/
2 + 9*a*b*c**2*d*e**2/2 + 3*a*c**3*d**2*e/2 + b**4*e**3/8 + 3*b**3*c*d*e**2/2 + 9*b**2*c**2*d**2*e/4 + b*c**3*
d**3/2) + x**7*(12*a**2*b*c*e**3/7 + 18*a**2*c**2*d*e**2/7 + 4*a*b**3*e**3/7 + 36*a*b**2*c*d*e**2/7 + 36*a*b*c
**2*d**2*e/7 + 4*a*c**3*d**3/7 + 3*b**4*d*e**2/7 + 12*b**3*c*d**2*e/7 + 6*b**2*c**2*d**3/7) + x**6*(2*a**3*c*e
**3/3 + a**2*b**2*e**3 + 6*a**2*b*c*d*e**2 + 3*a**2*c**2*d**2*e + 2*a*b**3*d*e**2 + 6*a*b**2*c*d**2*e + 2*a*b*
c**2*d**3 + b**4*d**2*e/2 + 2*b**3*c*d**3/3) + x**5*(4*a**3*b*e**3/5 + 12*a**3*c*d*e**2/5 + 18*a**2*b**2*d*e**
2/5 + 36*a**2*b*c*d**2*e/5 + 6*a**2*c**2*d**3/5 + 12*a*b**3*d**2*e/5 + 12*a*b**2*c*d**3/5 + b**4*d**3/5) + x**
4*(a**4*e**3/4 + 3*a**3*b*d*e**2 + 3*a**3*c*d**2*e + 9*a**2*b**2*d**2*e/2 + 3*a**2*b*c*d**3 + a*b**3*d**3) + x
**3*(a**4*d*e**2 + 4*a**3*b*d**2*e + 4*a**3*c*d**3/3 + 2*a**2*b**2*d**3) + x**2*(3*a**4*d**2*e/2 + 2*a**3*b*d*
*3)

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Giac [A]
time = 2.46, size = 754, normalized size = 1.70 \begin {gather*} \frac {1}{12} \, c^{4} x^{12} e^{3} + \frac {3}{11} \, c^{4} d x^{11} e^{2} + \frac {3}{10} \, c^{4} d^{2} x^{10} e + \frac {1}{9} \, c^{4} d^{3} x^{9} + \frac {4}{11} \, b c^{3} x^{11} e^{3} + \frac {6}{5} \, b c^{3} d x^{10} e^{2} + \frac {4}{3} \, b c^{3} d^{2} x^{9} e + \frac {1}{2} \, b c^{3} d^{3} x^{8} + \frac {3}{5} \, b^{2} c^{2} x^{10} e^{3} + \frac {2}{5} \, a c^{3} x^{10} e^{3} + 2 \, b^{2} c^{2} d x^{9} e^{2} + \frac {4}{3} \, a c^{3} d x^{9} e^{2} + \frac {9}{4} \, b^{2} c^{2} d^{2} x^{8} e + \frac {3}{2} \, a c^{3} d^{2} x^{8} e + \frac {6}{7} \, b^{2} c^{2} d^{3} x^{7} + \frac {4}{7} \, a c^{3} d^{3} x^{7} + \frac {4}{9} \, b^{3} c x^{9} e^{3} + \frac {4}{3} \, a b c^{2} x^{9} e^{3} + \frac {3}{2} \, b^{3} c d x^{8} e^{2} + \frac {9}{2} \, a b c^{2} d x^{8} e^{2} + \frac {12}{7} \, b^{3} c d^{2} x^{7} e + \frac {36}{7} \, a b c^{2} d^{2} x^{7} e + \frac {2}{3} \, b^{3} c d^{3} x^{6} + 2 \, a b c^{2} d^{3} x^{6} + \frac {1}{8} \, b^{4} x^{8} e^{3} + \frac {3}{2} \, a b^{2} c x^{8} e^{3} + \frac {3}{4} \, a^{2} c^{2} x^{8} e^{3} + \frac {3}{7} \, b^{4} d x^{7} e^{2} + \frac {36}{7} \, a b^{2} c d x^{7} e^{2} + \frac {18}{7} \, a^{2} c^{2} d x^{7} e^{2} + \frac {1}{2} \, b^{4} d^{2} x^{6} e + 6 \, a b^{2} c d^{2} x^{6} e + 3 \, a^{2} c^{2} d^{2} x^{6} e + \frac {1}{5} \, b^{4} d^{3} x^{5} + \frac {12}{5} \, a b^{2} c d^{3} x^{5} + \frac {6}{5} \, a^{2} c^{2} d^{3} x^{5} + \frac {4}{7} \, a b^{3} x^{7} e^{3} + \frac {12}{7} \, a^{2} b c x^{7} e^{3} + 2 \, a b^{3} d x^{6} e^{2} + 6 \, a^{2} b c d x^{6} e^{2} + \frac {12}{5} \, a b^{3} d^{2} x^{5} e + \frac {36}{5} \, a^{2} b c d^{2} x^{5} e + a b^{3} d^{3} x^{4} + 3 \, a^{2} b c d^{3} x^{4} + a^{2} b^{2} x^{6} e^{3} + \frac {2}{3} \, a^{3} c x^{6} e^{3} + \frac {18}{5} \, a^{2} b^{2} d x^{5} e^{2} + \frac {12}{5} \, a^{3} c d x^{5} e^{2} + \frac {9}{2} \, a^{2} b^{2} d^{2} x^{4} e + 3 \, a^{3} c d^{2} x^{4} e + 2 \, a^{2} b^{2} d^{3} x^{3} + \frac {4}{3} \, a^{3} c d^{3} x^{3} + \frac {4}{5} \, a^{3} b x^{5} e^{3} + 3 \, a^{3} b d x^{4} e^{2} + 4 \, a^{3} b d^{2} x^{3} e + 2 \, a^{3} b d^{3} x^{2} + \frac {1}{4} \, a^{4} x^{4} e^{3} + a^{4} d x^{3} e^{2} + \frac {3}{2} \, a^{4} d^{2} x^{2} e + a^{4} d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x+a)^4,x, algorithm="giac")

[Out]

1/12*c^4*x^12*e^3 + 3/11*c^4*d*x^11*e^2 + 3/10*c^4*d^2*x^10*e + 1/9*c^4*d^3*x^9 + 4/11*b*c^3*x^11*e^3 + 6/5*b*
c^3*d*x^10*e^2 + 4/3*b*c^3*d^2*x^9*e + 1/2*b*c^3*d^3*x^8 + 3/5*b^2*c^2*x^10*e^3 + 2/5*a*c^3*x^10*e^3 + 2*b^2*c
^2*d*x^9*e^2 + 4/3*a*c^3*d*x^9*e^2 + 9/4*b^2*c^2*d^2*x^8*e + 3/2*a*c^3*d^2*x^8*e + 6/7*b^2*c^2*d^3*x^7 + 4/7*a
*c^3*d^3*x^7 + 4/9*b^3*c*x^9*e^3 + 4/3*a*b*c^2*x^9*e^3 + 3/2*b^3*c*d*x^8*e^2 + 9/2*a*b*c^2*d*x^8*e^2 + 12/7*b^
3*c*d^2*x^7*e + 36/7*a*b*c^2*d^2*x^7*e + 2/3*b^3*c*d^3*x^6 + 2*a*b*c^2*d^3*x^6 + 1/8*b^4*x^8*e^3 + 3/2*a*b^2*c
*x^8*e^3 + 3/4*a^2*c^2*x^8*e^3 + 3/7*b^4*d*x^7*e^2 + 36/7*a*b^2*c*d*x^7*e^2 + 18/7*a^2*c^2*d*x^7*e^2 + 1/2*b^4
*d^2*x^6*e + 6*a*b^2*c*d^2*x^6*e + 3*a^2*c^2*d^2*x^6*e + 1/5*b^4*d^3*x^5 + 12/5*a*b^2*c*d^3*x^5 + 6/5*a^2*c^2*
d^3*x^5 + 4/7*a*b^3*x^7*e^3 + 12/7*a^2*b*c*x^7*e^3 + 2*a*b^3*d*x^6*e^2 + 6*a^2*b*c*d*x^6*e^2 + 12/5*a*b^3*d^2*
x^5*e + 36/5*a^2*b*c*d^2*x^5*e + a*b^3*d^3*x^4 + 3*a^2*b*c*d^3*x^4 + a^2*b^2*x^6*e^3 + 2/3*a^3*c*x^6*e^3 + 18/
5*a^2*b^2*d*x^5*e^2 + 12/5*a^3*c*d*x^5*e^2 + 9/2*a^2*b^2*d^2*x^4*e + 3*a^3*c*d^2*x^4*e + 2*a^2*b^2*d^3*x^3 + 4
/3*a^3*c*d^3*x^3 + 4/5*a^3*b*x^5*e^3 + 3*a^3*b*d*x^4*e^2 + 4*a^3*b*d^2*x^3*e + 2*a^3*b*d^3*x^2 + 1/4*a^4*x^4*e
^3 + a^4*d*x^3*e^2 + 3/2*a^4*d^2*x^2*e + a^4*d^3*x

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Mupad [B]
time = 0.20, size = 630, normalized size = 1.42 \begin {gather*} x^5\,\left (\frac {4\,a^3\,b\,e^3}{5}+\frac {12\,a^3\,c\,d\,e^2}{5}+\frac {18\,a^2\,b^2\,d\,e^2}{5}+\frac {36\,a^2\,b\,c\,d^2\,e}{5}+\frac {6\,a^2\,c^2\,d^3}{5}+\frac {12\,a\,b^3\,d^2\,e}{5}+\frac {12\,a\,b^2\,c\,d^3}{5}+\frac {b^4\,d^3}{5}\right )+x^8\,\left (\frac {3\,a^2\,c^2\,e^3}{4}+\frac {3\,a\,b^2\,c\,e^3}{2}+\frac {9\,a\,b\,c^2\,d\,e^2}{2}+\frac {3\,a\,c^3\,d^2\,e}{2}+\frac {b^4\,e^3}{8}+\frac {3\,b^3\,c\,d\,e^2}{2}+\frac {9\,b^2\,c^2\,d^2\,e}{4}+\frac {b\,c^3\,d^3}{2}\right )+x^6\,\left (\frac {2\,a^3\,c\,e^3}{3}+a^2\,b^2\,e^3+6\,a^2\,b\,c\,d\,e^2+3\,a^2\,c^2\,d^2\,e+2\,a\,b^3\,d\,e^2+6\,a\,b^2\,c\,d^2\,e+2\,a\,b\,c^2\,d^3+\frac {b^4\,d^2\,e}{2}+\frac {2\,b^3\,c\,d^3}{3}\right )+x^7\,\left (\frac {12\,a^2\,b\,c\,e^3}{7}+\frac {18\,a^2\,c^2\,d\,e^2}{7}+\frac {4\,a\,b^3\,e^3}{7}+\frac {36\,a\,b^2\,c\,d\,e^2}{7}+\frac {36\,a\,b\,c^2\,d^2\,e}{7}+\frac {4\,a\,c^3\,d^3}{7}+\frac {3\,b^4\,d\,e^2}{7}+\frac {12\,b^3\,c\,d^2\,e}{7}+\frac {6\,b^2\,c^2\,d^3}{7}\right )+x^4\,\left (\frac {a^4\,e^3}{4}+3\,a^3\,b\,d\,e^2+3\,c\,a^3\,d^2\,e+\frac {9\,a^2\,b^2\,d^2\,e}{2}+3\,c\,a^2\,b\,d^3+a\,b^3\,d^3\right )+x^9\,\left (\frac {4\,b^3\,c\,e^3}{9}+2\,b^2\,c^2\,d\,e^2+\frac {4\,b\,c^3\,d^2\,e}{3}+\frac {4\,a\,b\,c^2\,e^3}{3}+\frac {c^4\,d^3}{9}+\frac {4\,a\,c^3\,d\,e^2}{3}\right )+a^4\,d^3\,x+\frac {c^4\,e^3\,x^{12}}{12}+\frac {a^2\,d\,x^3\,\left (3\,a^2\,e^2+12\,a\,b\,d\,e+4\,c\,a\,d^2+6\,b^2\,d^2\right )}{3}+\frac {c^2\,e\,x^{10}\,\left (6\,b^2\,e^2+12\,b\,c\,d\,e+3\,c^2\,d^2+4\,a\,c\,e^2\right )}{10}+\frac {a^3\,d^2\,x^2\,\left (3\,a\,e+4\,b\,d\right )}{2}+\frac {c^3\,e^2\,x^{11}\,\left (4\,b\,e+3\,c\,d\right )}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^3*(a + b*x + c*x^2)^4,x)

[Out]

x^5*((b^4*d^3)/5 + (4*a^3*b*e^3)/5 + (6*a^2*c^2*d^3)/5 + (18*a^2*b^2*d*e^2)/5 + (12*a*b^2*c*d^3)/5 + (12*a*b^3
*d^2*e)/5 + (12*a^3*c*d*e^2)/5 + (36*a^2*b*c*d^2*e)/5) + x^8*((b^4*e^3)/8 + (b*c^3*d^3)/2 + (3*a^2*c^2*e^3)/4
+ (9*b^2*c^2*d^2*e)/4 + (3*a*b^2*c*e^3)/2 + (3*a*c^3*d^2*e)/2 + (3*b^3*c*d*e^2)/2 + (9*a*b*c^2*d*e^2)/2) + x^6
*((2*a^3*c*e^3)/3 + (2*b^3*c*d^3)/3 + (b^4*d^2*e)/2 + a^2*b^2*e^3 + 3*a^2*c^2*d^2*e + 2*a*b*c^2*d^3 + 2*a*b^3*
d*e^2 + 6*a*b^2*c*d^2*e + 6*a^2*b*c*d*e^2) + x^7*((4*a*b^3*e^3)/7 + (4*a*c^3*d^3)/7 + (3*b^4*d*e^2)/7 + (6*b^2
*c^2*d^3)/7 + (18*a^2*c^2*d*e^2)/7 + (12*a^2*b*c*e^3)/7 + (12*b^3*c*d^2*e)/7 + (36*a*b*c^2*d^2*e)/7 + (36*a*b^
2*c*d*e^2)/7) + x^4*((a^4*e^3)/4 + a*b^3*d^3 + (9*a^2*b^2*d^2*e)/2 + 3*a^2*b*c*d^3 + 3*a^3*b*d*e^2 + 3*a^3*c*d
^2*e) + x^9*((c^4*d^3)/9 + (4*b^3*c*e^3)/9 + 2*b^2*c^2*d*e^2 + (4*a*b*c^2*e^3)/3 + (4*a*c^3*d*e^2)/3 + (4*b*c^
3*d^2*e)/3) + a^4*d^3*x + (c^4*e^3*x^12)/12 + (a^2*d*x^3*(3*a^2*e^2 + 6*b^2*d^2 + 4*a*c*d^2 + 12*a*b*d*e))/3 +
(c^2*e*x^10*(6*b^2*e^2 + 3*c^2*d^2 + 4*a*c*e^2 + 12*b*c*d*e))/10 + (a^3*d^2*x^2*(3*a*e + 4*b*d))/2 + (c^3*e^2
*x^11*(4*b*e + 3*c*d))/11

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