3.2148 \(\int (d+e x)^2 (a+b x+c x^2)^4 \, dx\)

Optimal. Leaf size=441 \[ \frac {(d+e x)^7 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{7 e^9}+\frac {2 c^2 (d+e x)^9 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^9}-\frac {c (d+e x)^8 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9}-\frac {2 (d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac {2 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^4}{3 e^9}-\frac {2 c^3 (d+e x)^{10} (2 c d-b e)}{5 e^9}+\frac {c^4 (d+e x)^{11}}{11 e^9} \]

[Out]

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

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Rubi [A]  time = 0.52, antiderivative size = 441, 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 = {698} \[ \frac {(d+e x)^7 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{7 e^9}+\frac {2 c^2 (d+e x)^9 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^9}-\frac {c (d+e x)^8 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9}-\frac {2 (d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac {2 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^4}{3 e^9}-\frac {2 c^3 (d+e x)^{10} (2 c d-b e)}{5 e^9}+\frac {c^4 (d+e x)^{11}}{11 e^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 698

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

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Mathematica [A]  time = 0.14, size = 428, normalized size = 0.97 \[ a^4 d^2 x+a^3 d x^2 (a e+2 b d)+\frac {1}{3} a^2 x^3 \left (8 a b d e+a \left (a e^2+4 c d^2\right )+6 b^2 d^2\right )+a x^4 \left (2 a^2 c d e+3 a b^2 d e+a b \left (a e^2+3 c d^2\right )+b^3 d^2\right )+\frac {1}{3} x^6 \left (6 a^2 c^2 d e+2 b^3 \left (a e^2+c d^2\right )+12 a b^2 c d e+6 a b c \left (a e^2+c d^2\right )+b^4 d e\right )+\frac {1}{5} x^5 \left (24 a^2 b c d e+2 a^2 c \left (2 a e^2+3 c d^2\right )+8 a b^3 d e+6 a b^2 \left (a e^2+2 c d^2\right )+b^4 d^2\right )+\frac {1}{9} c^2 x^9 \left (4 c e (a e+2 b d)+6 b^2 e^2+c^2 d^2\right )+\frac {1}{2} c x^8 \left (b c \left (3 a e^2+c d^2\right )+2 a c^2 d e+b^3 e^2+3 b^2 c d e\right )+\frac {1}{7} x^7 \left (6 b^2 c \left (2 a e^2+c d^2\right )+24 a b c^2 d e+2 a c^2 \left (3 a e^2+2 c d^2\right )+b^4 e^2+8 b^3 c d e\right )+\frac {1}{5} c^3 e x^{10} (2 b e+c d)+\frac {1}{11} c^4 e^2 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^4*d^2*x + a^3*d*(2*b*d + a*e)*x^2 + (a^2*(6*b^2*d^2 + 8*a*b*d*e + a*(4*c*d^2 + a*e^2))*x^3)/3 + a*(b^3*d^2 +
3*a*b^2*d*e + 2*a^2*c*d*e + a*b*(3*c*d^2 + a*e^2))*x^4 + ((b^4*d^2 + 8*a*b^3*d*e + 24*a^2*b*c*d*e + 6*a*b^2*(
2*c*d^2 + a*e^2) + 2*a^2*c*(3*c*d^2 + 2*a*e^2))*x^5)/5 + ((b^4*d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e + 2*b^3*(c
*d^2 + a*e^2) + 6*a*b*c*(c*d^2 + a*e^2))*x^6)/3 + ((8*b^3*c*d*e + 24*a*b*c^2*d*e + b^4*e^2 + 6*b^2*c*(c*d^2 +
2*a*e^2) + 2*a*c^2*(2*c*d^2 + 3*a*e^2))*x^7)/7 + (c*(3*b^2*c*d*e + 2*a*c^2*d*e + b^3*e^2 + b*c*(c*d^2 + 3*a*e^
2))*x^8)/2 + (c^2*(c^2*d^2 + 6*b^2*e^2 + 4*c*e*(2*b*d + a*e))*x^9)/9 + (c^3*e*(c*d + 2*b*e)*x^10)/5 + (c^4*e^2
*x^11)/11

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fricas [A]  time = 0.82, size = 537, normalized size = 1.22 \[ \frac {1}{11} x^{11} e^{2} c^{4} + \frac {1}{5} x^{10} e d c^{4} + \frac {2}{5} x^{10} e^{2} c^{3} b + \frac {1}{9} x^{9} d^{2} c^{4} + \frac {8}{9} x^{9} e d c^{3} b + \frac {2}{3} x^{9} e^{2} c^{2} b^{2} + \frac {4}{9} x^{9} e^{2} c^{3} a + \frac {1}{2} x^{8} d^{2} c^{3} b + \frac {3}{2} x^{8} e d c^{2} b^{2} + \frac {1}{2} x^{8} e^{2} c b^{3} + x^{8} e d c^{3} a + \frac {3}{2} x^{8} e^{2} c^{2} b a + \frac {6}{7} x^{7} d^{2} c^{2} b^{2} + \frac {8}{7} x^{7} e d c b^{3} + \frac {1}{7} x^{7} e^{2} b^{4} + \frac {4}{7} x^{7} d^{2} c^{3} a + \frac {24}{7} x^{7} e d c^{2} b a + \frac {12}{7} x^{7} e^{2} c b^{2} a + \frac {6}{7} x^{7} e^{2} c^{2} a^{2} + \frac {2}{3} x^{6} d^{2} c b^{3} + \frac {1}{3} x^{6} e d b^{4} + 2 x^{6} d^{2} c^{2} b a + 4 x^{6} e d c b^{2} a + \frac {2}{3} x^{6} e^{2} b^{3} a + 2 x^{6} e d c^{2} a^{2} + 2 x^{6} e^{2} c b a^{2} + \frac {1}{5} x^{5} d^{2} b^{4} + \frac {12}{5} x^{5} d^{2} c b^{2} a + \frac {8}{5} x^{5} e d b^{3} a + \frac {6}{5} x^{5} d^{2} c^{2} a^{2} + \frac {24}{5} x^{5} e d c b a^{2} + \frac {6}{5} x^{5} e^{2} b^{2} a^{2} + \frac {4}{5} x^{5} e^{2} c a^{3} + x^{4} d^{2} b^{3} a + 3 x^{4} d^{2} c b a^{2} + 3 x^{4} e d b^{2} a^{2} + 2 x^{4} e d c a^{3} + x^{4} e^{2} b a^{3} + 2 x^{3} d^{2} b^{2} a^{2} + \frac {4}{3} x^{3} d^{2} c a^{3} + \frac {8}{3} x^{3} e d b a^{3} + \frac {1}{3} x^{3} e^{2} a^{4} + 2 x^{2} d^{2} b a^{3} + x^{2} e d a^{4} + x d^{2} a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*e^2*c^4 + 1/5*x^10*e*d*c^4 + 2/5*x^10*e^2*c^3*b + 1/9*x^9*d^2*c^4 + 8/9*x^9*e*d*c^3*b + 2/3*x^9*e^2*
c^2*b^2 + 4/9*x^9*e^2*c^3*a + 1/2*x^8*d^2*c^3*b + 3/2*x^8*e*d*c^2*b^2 + 1/2*x^8*e^2*c*b^3 + x^8*e*d*c^3*a + 3/
2*x^8*e^2*c^2*b*a + 6/7*x^7*d^2*c^2*b^2 + 8/7*x^7*e*d*c*b^3 + 1/7*x^7*e^2*b^4 + 4/7*x^7*d^2*c^3*a + 24/7*x^7*e
*d*c^2*b*a + 12/7*x^7*e^2*c*b^2*a + 6/7*x^7*e^2*c^2*a^2 + 2/3*x^6*d^2*c*b^3 + 1/3*x^6*e*d*b^4 + 2*x^6*d^2*c^2*
b*a + 4*x^6*e*d*c*b^2*a + 2/3*x^6*e^2*b^3*a + 2*x^6*e*d*c^2*a^2 + 2*x^6*e^2*c*b*a^2 + 1/5*x^5*d^2*b^4 + 12/5*x
^5*d^2*c*b^2*a + 8/5*x^5*e*d*b^3*a + 6/5*x^5*d^2*c^2*a^2 + 24/5*x^5*e*d*c*b*a^2 + 6/5*x^5*e^2*b^2*a^2 + 4/5*x^
5*e^2*c*a^3 + x^4*d^2*b^3*a + 3*x^4*d^2*c*b*a^2 + 3*x^4*e*d*b^2*a^2 + 2*x^4*e*d*c*a^3 + x^4*e^2*b*a^3 + 2*x^3*
d^2*b^2*a^2 + 4/3*x^3*d^2*c*a^3 + 8/3*x^3*e*d*b*a^3 + 1/3*x^3*e^2*a^4 + 2*x^2*d^2*b*a^3 + x^2*e*d*a^4 + x*d^2*
a^4

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giac [A]  time = 0.16, size = 537, normalized size = 1.22 \[ \frac {1}{11} \, c^{4} x^{11} e^{2} + \frac {1}{5} \, c^{4} d x^{10} e + \frac {1}{9} \, c^{4} d^{2} x^{9} + \frac {2}{5} \, b c^{3} x^{10} e^{2} + \frac {8}{9} \, b c^{3} d x^{9} e + \frac {1}{2} \, b c^{3} d^{2} x^{8} + \frac {2}{3} \, b^{2} c^{2} x^{9} e^{2} + \frac {4}{9} \, a c^{3} x^{9} e^{2} + \frac {3}{2} \, b^{2} c^{2} d x^{8} e + a c^{3} d x^{8} e + \frac {6}{7} \, b^{2} c^{2} d^{2} x^{7} + \frac {4}{7} \, a c^{3} d^{2} x^{7} + \frac {1}{2} \, b^{3} c x^{8} e^{2} + \frac {3}{2} \, a b c^{2} x^{8} e^{2} + \frac {8}{7} \, b^{3} c d x^{7} e + \frac {24}{7} \, a b c^{2} d x^{7} e + \frac {2}{3} \, b^{3} c d^{2} x^{6} + 2 \, a b c^{2} d^{2} x^{6} + \frac {1}{7} \, b^{4} x^{7} e^{2} + \frac {12}{7} \, a b^{2} c x^{7} e^{2} + \frac {6}{7} \, a^{2} c^{2} x^{7} e^{2} + \frac {1}{3} \, b^{4} d x^{6} e + 4 \, a b^{2} c d x^{6} e + 2 \, a^{2} c^{2} d x^{6} e + \frac {1}{5} \, b^{4} d^{2} x^{5} + \frac {12}{5} \, a b^{2} c d^{2} x^{5} + \frac {6}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {2}{3} \, a b^{3} x^{6} e^{2} + 2 \, a^{2} b c x^{6} e^{2} + \frac {8}{5} \, a b^{3} d x^{5} e + \frac {24}{5} \, a^{2} b c d x^{5} e + a b^{3} d^{2} x^{4} + 3 \, a^{2} b c d^{2} x^{4} + \frac {6}{5} \, a^{2} b^{2} x^{5} e^{2} + \frac {4}{5} \, a^{3} c x^{5} e^{2} + 3 \, a^{2} b^{2} d x^{4} e + 2 \, a^{3} c d x^{4} e + 2 \, a^{2} b^{2} d^{2} x^{3} + \frac {4}{3} \, a^{3} c d^{2} x^{3} + a^{3} b x^{4} e^{2} + \frac {8}{3} \, a^{3} b d x^{3} e + 2 \, a^{3} b d^{2} x^{2} + \frac {1}{3} \, a^{4} x^{3} e^{2} + a^{4} d x^{2} e + a^{4} d^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^4*x^11*e^2 + 1/5*c^4*d*x^10*e + 1/9*c^4*d^2*x^9 + 2/5*b*c^3*x^10*e^2 + 8/9*b*c^3*d*x^9*e + 1/2*b*c^3*d^
2*x^8 + 2/3*b^2*c^2*x^9*e^2 + 4/9*a*c^3*x^9*e^2 + 3/2*b^2*c^2*d*x^8*e + a*c^3*d*x^8*e + 6/7*b^2*c^2*d^2*x^7 +
4/7*a*c^3*d^2*x^7 + 1/2*b^3*c*x^8*e^2 + 3/2*a*b*c^2*x^8*e^2 + 8/7*b^3*c*d*x^7*e + 24/7*a*b*c^2*d*x^7*e + 2/3*b
^3*c*d^2*x^6 + 2*a*b*c^2*d^2*x^6 + 1/7*b^4*x^7*e^2 + 12/7*a*b^2*c*x^7*e^2 + 6/7*a^2*c^2*x^7*e^2 + 1/3*b^4*d*x^
6*e + 4*a*b^2*c*d*x^6*e + 2*a^2*c^2*d*x^6*e + 1/5*b^4*d^2*x^5 + 12/5*a*b^2*c*d^2*x^5 + 6/5*a^2*c^2*d^2*x^5 + 2
/3*a*b^3*x^6*e^2 + 2*a^2*b*c*x^6*e^2 + 8/5*a*b^3*d*x^5*e + 24/5*a^2*b*c*d*x^5*e + a*b^3*d^2*x^4 + 3*a^2*b*c*d^
2*x^4 + 6/5*a^2*b^2*x^5*e^2 + 4/5*a^3*c*x^5*e^2 + 3*a^2*b^2*d*x^4*e + 2*a^3*c*d*x^4*e + 2*a^2*b^2*d^2*x^3 + 4/
3*a^3*c*d^2*x^3 + a^3*b*x^4*e^2 + 8/3*a^3*b*d*x^3*e + 2*a^3*b*d^2*x^2 + 1/3*a^4*x^3*e^2 + a^4*d*x^2*e + a^4*d^
2*x

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maple [A]  time = 0.04, size = 545, normalized size = 1.24 \[ \frac {c^{4} e^{2} x^{11}}{11}+\frac {\left (4 e^{2} b \,c^{3}+2 d e \,c^{4}\right ) x^{10}}{10}+\frac {\left (8 b \,c^{3} d e +c^{4} d^{2}+\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) e^{2}\right ) x^{9}}{9}+\frac {\left (4 b \,c^{3} d^{2}+2 \left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) d e +\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) e^{2}\right ) x^{8}}{8}+a^{4} d^{2} x +\frac {\left (\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) d^{2}+2 \left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) d e +\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) e^{2}\right ) x^{7}}{7}+\frac {\left (\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) d^{2}+2 \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) d e +\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) e^{2}\right ) x^{6}}{6}+\frac {\left (\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) d^{2}+2 \left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) d e +\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) e^{2}\right ) x^{5}}{5}+\frac {\left (4 a^{3} b \,e^{2}+\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) d^{2}+2 \left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) d e \right ) x^{4}}{4}+\frac {\left (a^{4} e^{2}+8 a^{3} b d e +\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) d^{2}\right ) x^{3}}{3}+\frac {\left (2 d e \,a^{4}+4 d^{2} a^{3} b \right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.09, size = 436, normalized size = 0.99 \[ \frac {1}{11} \, c^{4} e^{2} x^{11} + \frac {1}{5} \, {\left (c^{4} d e + 2 \, b c^{3} e^{2}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{2} + 8 \, b c^{3} d e + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} x^{9} + \frac {1}{2} \, {\left (b c^{3} d^{2} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e + {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} + 8 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{2}\right )} x^{7} + a^{4} d^{2} x + \frac {1}{3} \, {\left (2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e + 2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} + 8 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{2}\right )} x^{5} + {\left (a^{3} b e^{2} + {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (8 \, a^{3} b d e + a^{4} e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.78, size = 445, normalized size = 1.01 \[ x^5\,\left (\frac {4\,a^3\,c\,e^2}{5}+\frac {6\,a^2\,b^2\,e^2}{5}+\frac {24\,a^2\,b\,c\,d\,e}{5}+\frac {6\,a^2\,c^2\,d^2}{5}+\frac {8\,a\,b^3\,d\,e}{5}+\frac {12\,a\,b^2\,c\,d^2}{5}+\frac {b^4\,d^2}{5}\right )+x^3\,\left (\frac {a^4\,e^2}{3}+\frac {8\,a^3\,b\,d\,e}{3}+\frac {4\,c\,a^3\,d^2}{3}+2\,a^2\,b^2\,d^2\right )+x^7\,\left (\frac {6\,a^2\,c^2\,e^2}{7}+\frac {12\,a\,b^2\,c\,e^2}{7}+\frac {24\,a\,b\,c^2\,d\,e}{7}+\frac {4\,a\,c^3\,d^2}{7}+\frac {b^4\,e^2}{7}+\frac {8\,b^3\,c\,d\,e}{7}+\frac {6\,b^2\,c^2\,d^2}{7}\right )+x^9\,\left (\frac {2\,b^2\,c^2\,e^2}{3}+\frac {8\,b\,c^3\,d\,e}{9}+\frac {c^4\,d^2}{9}+\frac {4\,a\,c^3\,e^2}{9}\right )+x^6\,\left (2\,a^2\,b\,c\,e^2+2\,a^2\,c^2\,d\,e+\frac {2\,a\,b^3\,e^2}{3}+4\,a\,b^2\,c\,d\,e+2\,a\,b\,c^2\,d^2+\frac {b^4\,d\,e}{3}+\frac {2\,b^3\,c\,d^2}{3}\right )+x^4\,\left (a^3\,b\,e^2+2\,c\,a^3\,d\,e+3\,a^2\,b^2\,d\,e+3\,c\,a^2\,b\,d^2+a\,b^3\,d^2\right )+x^8\,\left (\frac {b^3\,c\,e^2}{2}+\frac {3\,b^2\,c^2\,d\,e}{2}+\frac {b\,c^3\,d^2}{2}+\frac {3\,a\,b\,c^2\,e^2}{2}+a\,c^3\,d\,e\right )+a^4\,d^2\,x+\frac {c^4\,e^2\,x^{11}}{11}+a^3\,d\,x^2\,\left (a\,e+2\,b\,d\right )+\frac {c^3\,e\,x^{10}\,\left (2\,b\,e+c\,d\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^5*((b^4*d^2)/5 + (4*a^3*c*e^2)/5 + (6*a^2*b^2*e^2)/5 + (6*a^2*c^2*d^2)/5 + (8*a*b^3*d*e)/5 + (12*a*b^2*c*d^2
)/5 + (24*a^2*b*c*d*e)/5) + x^3*((a^4*e^2)/3 + (4*a^3*c*d^2)/3 + 2*a^2*b^2*d^2 + (8*a^3*b*d*e)/3) + x^7*((b^4*
e^2)/7 + (4*a*c^3*d^2)/7 + (6*a^2*c^2*e^2)/7 + (6*b^2*c^2*d^2)/7 + (8*b^3*c*d*e)/7 + (12*a*b^2*c*e^2)/7 + (24*
a*b*c^2*d*e)/7) + x^9*((c^4*d^2)/9 + (4*a*c^3*e^2)/9 + (2*b^2*c^2*e^2)/3 + (8*b*c^3*d*e)/9) + x^6*((2*a*b^3*e^
2)/3 + (2*b^3*c*d^2)/3 + (b^4*d*e)/3 + 2*a*b*c^2*d^2 + 2*a^2*b*c*e^2 + 2*a^2*c^2*d*e + 4*a*b^2*c*d*e) + x^4*(a
*b^3*d^2 + a^3*b*e^2 + 2*a^3*c*d*e + 3*a^2*b*c*d^2 + 3*a^2*b^2*d*e) + x^8*((b*c^3*d^2)/2 + (b^3*c*e^2)/2 + a*c
^3*d*e + (3*a*b*c^2*e^2)/2 + (3*b^2*c^2*d*e)/2) + a^4*d^2*x + (c^4*e^2*x^11)/11 + a^3*d*x^2*(a*e + 2*b*d) + (c
^3*e*x^10*(2*b*e + c*d))/5

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sympy [A]  time = 0.15, size = 537, normalized size = 1.22 \[ a^{4} d^{2} x + \frac {c^{4} e^{2} x^{11}}{11} + x^{10} \left (\frac {2 b c^{3} e^{2}}{5} + \frac {c^{4} d e}{5}\right ) + x^{9} \left (\frac {4 a c^{3} e^{2}}{9} + \frac {2 b^{2} c^{2} e^{2}}{3} + \frac {8 b c^{3} d e}{9} + \frac {c^{4} d^{2}}{9}\right ) + x^{8} \left (\frac {3 a b c^{2} e^{2}}{2} + a c^{3} d e + \frac {b^{3} c e^{2}}{2} + \frac {3 b^{2} c^{2} d e}{2} + \frac {b c^{3} d^{2}}{2}\right ) + x^{7} \left (\frac {6 a^{2} c^{2} e^{2}}{7} + \frac {12 a b^{2} c e^{2}}{7} + \frac {24 a b c^{2} d e}{7} + \frac {4 a c^{3} d^{2}}{7} + \frac {b^{4} e^{2}}{7} + \frac {8 b^{3} c d e}{7} + \frac {6 b^{2} c^{2} d^{2}}{7}\right ) + x^{6} \left (2 a^{2} b c e^{2} + 2 a^{2} c^{2} d e + \frac {2 a b^{3} e^{2}}{3} + 4 a b^{2} c d e + 2 a b c^{2} d^{2} + \frac {b^{4} d e}{3} + \frac {2 b^{3} c d^{2}}{3}\right ) + x^{5} \left (\frac {4 a^{3} c e^{2}}{5} + \frac {6 a^{2} b^{2} e^{2}}{5} + \frac {24 a^{2} b c d e}{5} + \frac {6 a^{2} c^{2} d^{2}}{5} + \frac {8 a b^{3} d e}{5} + \frac {12 a b^{2} c d^{2}}{5} + \frac {b^{4} d^{2}}{5}\right ) + x^{4} \left (a^{3} b e^{2} + 2 a^{3} c d e + 3 a^{2} b^{2} d e + 3 a^{2} b c d^{2} + a b^{3} d^{2}\right ) + x^{3} \left (\frac {a^{4} e^{2}}{3} + \frac {8 a^{3} b d e}{3} + \frac {4 a^{3} c d^{2}}{3} + 2 a^{2} b^{2} d^{2}\right ) + x^{2} \left (a^{4} d e + 2 a^{3} b d^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*d**2*x + c**4*e**2*x**11/11 + x**10*(2*b*c**3*e**2/5 + c**4*d*e/5) + x**9*(4*a*c**3*e**2/9 + 2*b**2*c**2*
e**2/3 + 8*b*c**3*d*e/9 + c**4*d**2/9) + x**8*(3*a*b*c**2*e**2/2 + a*c**3*d*e + b**3*c*e**2/2 + 3*b**2*c**2*d*
e/2 + b*c**3*d**2/2) + x**7*(6*a**2*c**2*e**2/7 + 12*a*b**2*c*e**2/7 + 24*a*b*c**2*d*e/7 + 4*a*c**3*d**2/7 + b
**4*e**2/7 + 8*b**3*c*d*e/7 + 6*b**2*c**2*d**2/7) + x**6*(2*a**2*b*c*e**2 + 2*a**2*c**2*d*e + 2*a*b**3*e**2/3
+ 4*a*b**2*c*d*e + 2*a*b*c**2*d**2 + b**4*d*e/3 + 2*b**3*c*d**2/3) + x**5*(4*a**3*c*e**2/5 + 6*a**2*b**2*e**2/
5 + 24*a**2*b*c*d*e/5 + 6*a**2*c**2*d**2/5 + 8*a*b**3*d*e/5 + 12*a*b**2*c*d**2/5 + b**4*d**2/5) + x**4*(a**3*b
*e**2 + 2*a**3*c*d*e + 3*a**2*b**2*d*e + 3*a**2*b*c*d**2 + a*b**3*d**2) + x**3*(a**4*e**2/3 + 8*a**3*b*d*e/3 +
4*a**3*c*d**2/3 + 2*a**2*b**2*d**2) + x**2*(a**4*d*e + 2*a**3*b*d**2)

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