3.14.24 \(\int (b+2 c x) (d+e x)^2 (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=411 \[ \frac {(d+e x)^6 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{6 e^8}+\frac {3 c^2 (d+e x)^8 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{8 e^8}-\frac {5 c (d+e x)^7 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}-\frac {3 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^8}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{4 e^8}-\frac {(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac {7 c^3 (d+e x)^9 (2 c d-b e)}{9 e^8}+\frac {c^4 (d+e x)^{10}}{5 e^8} \]

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Rubi [A]  time = 0.45, antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} \frac {(d+e x)^6 \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 )}{6 e^8}+\frac {3 c^2 (d+e x)^8 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{8 e^8}-\frac {5 c (d+e x)^7 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}-\frac {3 (d+e x)^5 (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 )}{5 e^8}+\frac {(d+e x)^4 \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 )}{4 e^8}-\frac {(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac {7 c^3 (d+e x)^9 (2 c d-b e)}{9 e^8}+\frac {c^4 (d+e x)^{10}}{5 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 413, normalized size = 1.00 \begin {gather*} a^3 b d^2 x+\frac {1}{2} a^2 d x^2 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac {1}{3} a x^3 \left (4 a^2 c d e+6 a b^2 d e+a b \left (a e^2+9 c d^2\right )+3 b^3 d^2\right )+\frac {1}{5} x^5 \left (12 a^2 c^2 d e+b^3 \left (3 a e^2+5 c d^2\right )+24 a b^2 c d e+3 a b c \left (3 a e^2+5 c d^2\right )+2 b^4 d e\right )+\frac {1}{4} x^4 \left (18 a^2 b c d e+2 a^2 c \left (a e^2+3 c d^2\right )+6 a b^3 d e+3 a b^2 \left (a e^2+4 c d^2\right )+b^4 d^2\right )+\frac {1}{8} c^2 x^8 \left (2 c e (3 a e+7 b d)+9 b^2 e^2+2 c^2 d^2\right )+\frac {1}{7} c x^7 \left (b c \left (15 a e^2+7 c d^2\right )+12 a c^2 d e+5 b^3 e^2+18 b^2 c d e\right )+\frac {1}{6} x^6 \left (3 b^2 c \left (4 a e^2+3 c d^2\right )+30 a b c^2 d e+6 a c^2 \left (a e^2+c d^2\right )+b^4 e^2+10 b^3 c d e\right )+\frac {1}{9} c^3 e x^9 (7 b e+4 c d)+\frac {1}{5} c^4 e^2 x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*b*d^2*x + (a^2*d*(3*b^2*d + 2*a*c*d + 2*a*b*e)*x^2)/2 + (a*(3*b^3*d^2 + 6*a*b^2*d*e + 4*a^2*c*d*e + a*b*(9
*c*d^2 + a*e^2))*x^3)/3 + ((b^4*d^2 + 6*a*b^3*d*e + 18*a^2*b*c*d*e + 2*a^2*c*(3*c*d^2 + a*e^2) + 3*a*b^2*(4*c*
d^2 + a*e^2))*x^4)/4 + ((2*b^4*d*e + 24*a*b^2*c*d*e + 12*a^2*c^2*d*e + b^3*(5*c*d^2 + 3*a*e^2) + 3*a*b*c*(5*c*
d^2 + 3*a*e^2))*x^5)/5 + ((10*b^3*c*d*e + 30*a*b*c^2*d*e + b^4*e^2 + 6*a*c^2*(c*d^2 + a*e^2) + 3*b^2*c*(3*c*d^
2 + 4*a*e^2))*x^6)/6 + (c*(18*b^2*c*d*e + 12*a*c^2*d*e + 5*b^3*e^2 + b*c*(7*c*d^2 + 15*a*e^2))*x^7)/7 + (c^2*(
2*c^2*d^2 + 9*b^2*e^2 + 2*c*e*(7*b*d + 3*a*e))*x^8)/8 + (c^3*e*(4*c*d + 7*b*e)*x^9)/9 + (c^4*e^2*x^10)/5

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.38, size = 502, normalized size = 1.22 \begin {gather*} \frac {1}{5} x^{10} e^{2} c^{4} + \frac {4}{9} x^{9} e d c^{4} + \frac {7}{9} x^{9} e^{2} c^{3} b + \frac {1}{4} x^{8} d^{2} c^{4} + \frac {7}{4} x^{8} e d c^{3} b + \frac {9}{8} x^{8} e^{2} c^{2} b^{2} + \frac {3}{4} x^{8} e^{2} c^{3} a + x^{7} d^{2} c^{3} b + \frac {18}{7} x^{7} e d c^{2} b^{2} + \frac {5}{7} x^{7} e^{2} c b^{3} + \frac {12}{7} x^{7} e d c^{3} a + \frac {15}{7} x^{7} e^{2} c^{2} b a + \frac {3}{2} x^{6} d^{2} c^{2} b^{2} + \frac {5}{3} x^{6} e d c b^{3} + \frac {1}{6} x^{6} e^{2} b^{4} + x^{6} d^{2} c^{3} a + 5 x^{6} e d c^{2} b a + 2 x^{6} e^{2} c b^{2} a + x^{6} e^{2} c^{2} a^{2} + x^{5} d^{2} c b^{3} + \frac {2}{5} x^{5} e d b^{4} + 3 x^{5} d^{2} c^{2} b a + \frac {24}{5} x^{5} e d c b^{2} a + \frac {3}{5} x^{5} e^{2} b^{3} a + \frac {12}{5} x^{5} e d c^{2} a^{2} + \frac {9}{5} x^{5} e^{2} c b a^{2} + \frac {1}{4} x^{4} d^{2} b^{4} + 3 x^{4} d^{2} c b^{2} a + \frac {3}{2} x^{4} e d b^{3} a + \frac {3}{2} x^{4} d^{2} c^{2} a^{2} + \frac {9}{2} x^{4} e d c b a^{2} + \frac {3}{4} x^{4} e^{2} b^{2} a^{2} + \frac {1}{2} x^{4} e^{2} c a^{3} + x^{3} d^{2} b^{3} a + 3 x^{3} d^{2} c b a^{2} + 2 x^{3} e d b^{2} a^{2} + \frac {4}{3} x^{3} e d c a^{3} + \frac {1}{3} x^{3} e^{2} b a^{3} + \frac {3}{2} x^{2} d^{2} b^{2} a^{2} + x^{2} d^{2} c a^{3} + x^{2} e d b a^{3} + x d^{2} b a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^10*e^2*c^4 + 4/9*x^9*e*d*c^4 + 7/9*x^9*e^2*c^3*b + 1/4*x^8*d^2*c^4 + 7/4*x^8*e*d*c^3*b + 9/8*x^8*e^2*c^2
*b^2 + 3/4*x^8*e^2*c^3*a + x^7*d^2*c^3*b + 18/7*x^7*e*d*c^2*b^2 + 5/7*x^7*e^2*c*b^3 + 12/7*x^7*e*d*c^3*a + 15/
7*x^7*e^2*c^2*b*a + 3/2*x^6*d^2*c^2*b^2 + 5/3*x^6*e*d*c*b^3 + 1/6*x^6*e^2*b^4 + x^6*d^2*c^3*a + 5*x^6*e*d*c^2*
b*a + 2*x^6*e^2*c*b^2*a + x^6*e^2*c^2*a^2 + x^5*d^2*c*b^3 + 2/5*x^5*e*d*b^4 + 3*x^5*d^2*c^2*b*a + 24/5*x^5*e*d
*c*b^2*a + 3/5*x^5*e^2*b^3*a + 12/5*x^5*e*d*c^2*a^2 + 9/5*x^5*e^2*c*b*a^2 + 1/4*x^4*d^2*b^4 + 3*x^4*d^2*c*b^2*
a + 3/2*x^4*e*d*b^3*a + 3/2*x^4*d^2*c^2*a^2 + 9/2*x^4*e*d*c*b*a^2 + 3/4*x^4*e^2*b^2*a^2 + 1/2*x^4*e^2*c*a^3 +
x^3*d^2*b^3*a + 3*x^3*d^2*c*b*a^2 + 2*x^3*e*d*b^2*a^2 + 4/3*x^3*e*d*c*a^3 + 1/3*x^3*e^2*b*a^3 + 3/2*x^2*d^2*b^
2*a^2 + x^2*d^2*c*a^3 + x^2*e*d*b*a^3 + x*d^2*b*a^3

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giac [A]  time = 0.16, size = 502, normalized size = 1.22 \begin {gather*} \frac {1}{5} \, c^{4} x^{10} e^{2} + \frac {4}{9} \, c^{4} d x^{9} e + \frac {1}{4} \, c^{4} d^{2} x^{8} + \frac {7}{9} \, b c^{3} x^{9} e^{2} + \frac {7}{4} \, b c^{3} d x^{8} e + b c^{3} d^{2} x^{7} + \frac {9}{8} \, b^{2} c^{2} x^{8} e^{2} + \frac {3}{4} \, a c^{3} x^{8} e^{2} + \frac {18}{7} \, b^{2} c^{2} d x^{7} e + \frac {12}{7} \, a c^{3} d x^{7} e + \frac {3}{2} \, b^{2} c^{2} d^{2} x^{6} + a c^{3} d^{2} x^{6} + \frac {5}{7} \, b^{3} c x^{7} e^{2} + \frac {15}{7} \, a b c^{2} x^{7} e^{2} + \frac {5}{3} \, b^{3} c d x^{6} e + 5 \, a b c^{2} d x^{6} e + b^{3} c d^{2} x^{5} + 3 \, a b c^{2} d^{2} x^{5} + \frac {1}{6} \, b^{4} x^{6} e^{2} + 2 \, a b^{2} c x^{6} e^{2} + a^{2} c^{2} x^{6} e^{2} + \frac {2}{5} \, b^{4} d x^{5} e + \frac {24}{5} \, a b^{2} c d x^{5} e + \frac {12}{5} \, a^{2} c^{2} d x^{5} e + \frac {1}{4} \, b^{4} d^{2} x^{4} + 3 \, a b^{2} c d^{2} x^{4} + \frac {3}{2} \, a^{2} c^{2} d^{2} x^{4} + \frac {3}{5} \, a b^{3} x^{5} e^{2} + \frac {9}{5} \, a^{2} b c x^{5} e^{2} + \frac {3}{2} \, a b^{3} d x^{4} e + \frac {9}{2} \, a^{2} b c d x^{4} e + a b^{3} d^{2} x^{3} + 3 \, a^{2} b c d^{2} x^{3} + \frac {3}{4} \, a^{2} b^{2} x^{4} e^{2} + \frac {1}{2} \, a^{3} c x^{4} e^{2} + 2 \, a^{2} b^{2} d x^{3} e + \frac {4}{3} \, a^{3} c d x^{3} e + \frac {3}{2} \, a^{2} b^{2} d^{2} x^{2} + a^{3} c d^{2} x^{2} + \frac {1}{3} \, a^{3} b x^{3} e^{2} + a^{3} b d x^{2} e + a^{3} b d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^4*x^10*e^2 + 4/9*c^4*d*x^9*e + 1/4*c^4*d^2*x^8 + 7/9*b*c^3*x^9*e^2 + 7/4*b*c^3*d*x^8*e + b*c^3*d^2*x^7 +
9/8*b^2*c^2*x^8*e^2 + 3/4*a*c^3*x^8*e^2 + 18/7*b^2*c^2*d*x^7*e + 12/7*a*c^3*d*x^7*e + 3/2*b^2*c^2*d^2*x^6 + a
*c^3*d^2*x^6 + 5/7*b^3*c*x^7*e^2 + 15/7*a*b*c^2*x^7*e^2 + 5/3*b^3*c*d*x^6*e + 5*a*b*c^2*d*x^6*e + b^3*c*d^2*x^
5 + 3*a*b*c^2*d^2*x^5 + 1/6*b^4*x^6*e^2 + 2*a*b^2*c*x^6*e^2 + a^2*c^2*x^6*e^2 + 2/5*b^4*d*x^5*e + 24/5*a*b^2*c
*d*x^5*e + 12/5*a^2*c^2*d*x^5*e + 1/4*b^4*d^2*x^4 + 3*a*b^2*c*d^2*x^4 + 3/2*a^2*c^2*d^2*x^4 + 3/5*a*b^3*x^5*e^
2 + 9/5*a^2*b*c*x^5*e^2 + 3/2*a*b^3*d*x^4*e + 9/2*a^2*b*c*d*x^4*e + a*b^3*d^2*x^3 + 3*a^2*b*c*d^2*x^3 + 3/4*a^
2*b^2*x^4*e^2 + 1/2*a^3*c*x^4*e^2 + 2*a^2*b^2*d*x^3*e + 4/3*a^3*c*d*x^3*e + 3/2*a^2*b^2*d^2*x^2 + a^3*c*d^2*x^
2 + 1/3*a^3*b*x^3*e^2 + a^3*b*d*x^2*e + a^3*b*d^2*x

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maple [A]  time = 0.04, size = 608, normalized size = 1.48 \begin {gather*} \frac {c^{4} e^{2} x^{10}}{5}+\frac {\left (6 b \,c^{3} e^{2}+\left (b \,e^{2}+4 c d e \right ) c^{3}\right ) x^{9}}{9}+\frac {\left (3 \left (b \,e^{2}+4 c d e \right ) b \,c^{2}+\left (2 b d e +2 c \,d^{2}\right ) c^{3}+2 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) c \,e^{2}\right ) x^{8}}{8}+a^{3} b \,d^{2} x +\frac {\left (b \,c^{3} d^{2}+3 \left (2 b d e +2 c \,d^{2}\right ) b \,c^{2}+2 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) c \,e^{2}+\left (b \,e^{2}+4 c d e \right ) \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right )\right ) x^{7}}{7}+\frac {\left (3 b^{2} c^{2} d^{2}+2 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) c \,e^{2}+\left (2 b d e +2 c \,d^{2}\right ) \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right )+\left (b \,e^{2}+4 c d e \right ) \left (4 a b c +\left (2 a c +b^{2}\right ) b \right )\right ) x^{6}}{6}+\frac {\left (6 a^{2} b c \,e^{2}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b \,d^{2}+\left (2 b d e +2 c \,d^{2}\right ) \left (4 a b c +\left (2 a c +b^{2}\right ) b \right )+\left (b \,e^{2}+4 c d e \right ) \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right )\right ) x^{5}}{5}+\frac {\left (2 a^{3} c \,e^{2}+3 \left (b \,e^{2}+4 c d e \right ) a^{2} b +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b \,d^{2}+\left (2 b d e +2 c \,d^{2}\right ) \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right )\right ) x^{4}}{4}+\frac {\left (\left (b \,e^{2}+4 c d e \right ) a^{3}+3 \left (2 b d e +2 c \,d^{2}\right ) a^{2} b +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b \,d^{2}\right ) x^{3}}{3}+\frac {\left (3 a^{2} b^{2} d^{2}+\left (2 b d e +2 c \,d^{2}\right ) a^{3}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 417, normalized size = 1.01 \begin {gather*} \frac {1}{5} \, c^{4} e^{2} x^{10} + \frac {1}{9} \, {\left (4 \, c^{4} d e + 7 \, b c^{3} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (2 \, c^{4} d^{2} + 14 \, b c^{3} d e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (7 \, b c^{3} d^{2} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{2}\right )} x^{7} + a^{3} b d^{2} x + \frac {1}{6} \, {\left (3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} + 10 \, {\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^{6} + \frac {1}{5} \, {\left (5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} + 2 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (a^{3} b e^{2} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a^{3} b d e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.89, size = 414, normalized size = 1.01 \begin {gather*} x^4\,\left (\frac {a^3\,c\,e^2}{2}+\frac {3\,a^2\,b^2\,e^2}{4}+\frac {9\,a^2\,b\,c\,d\,e}{2}+\frac {3\,a^2\,c^2\,d^2}{2}+\frac {3\,a\,b^3\,d\,e}{2}+3\,a\,b^2\,c\,d^2+\frac {b^4\,d^2}{4}\right )+x^6\,\left (a^2\,c^2\,e^2+2\,a\,b^2\,c\,e^2+5\,a\,b\,c^2\,d\,e+a\,c^3\,d^2+\frac {b^4\,e^2}{6}+\frac {5\,b^3\,c\,d\,e}{3}+\frac {3\,b^2\,c^2\,d^2}{2}\right )+x^8\,\left (\frac {9\,b^2\,c^2\,e^2}{8}+\frac {7\,b\,c^3\,d\,e}{4}+\frac {c^4\,d^2}{4}+\frac {3\,a\,c^3\,e^2}{4}\right )+x^5\,\left (\frac {9\,a^2\,b\,c\,e^2}{5}+\frac {12\,a^2\,c^2\,d\,e}{5}+\frac {3\,a\,b^3\,e^2}{5}+\frac {24\,a\,b^2\,c\,d\,e}{5}+3\,a\,b\,c^2\,d^2+\frac {2\,b^4\,d\,e}{5}+b^3\,c\,d^2\right )+x^3\,\left (\frac {a^3\,b\,e^2}{3}+\frac {4\,c\,a^3\,d\,e}{3}+2\,a^2\,b^2\,d\,e+3\,c\,a^2\,b\,d^2+a\,b^3\,d^2\right )+x^7\,\left (\frac {5\,b^3\,c\,e^2}{7}+\frac {18\,b^2\,c^2\,d\,e}{7}+b\,c^3\,d^2+\frac {15\,a\,b\,c^2\,e^2}{7}+\frac {12\,a\,c^3\,d\,e}{7}\right )+\frac {c^4\,e^2\,x^{10}}{5}+\frac {c^3\,e\,x^9\,\left (7\,b\,e+4\,c\,d\right )}{9}+\frac {a^2\,d\,x^2\,\left (3\,d\,b^2+2\,a\,e\,b+2\,a\,c\,d\right )}{2}+a^3\,b\,d^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*((b^4*d^2)/4 + (a^3*c*e^2)/2 + (3*a^2*b^2*e^2)/4 + (3*a^2*c^2*d^2)/2 + (3*a*b^3*d*e)/2 + 3*a*b^2*c*d^2 + (
9*a^2*b*c*d*e)/2) + x^6*((b^4*e^2)/6 + a*c^3*d^2 + a^2*c^2*e^2 + (3*b^2*c^2*d^2)/2 + (5*b^3*c*d*e)/3 + 2*a*b^2
*c*e^2 + 5*a*b*c^2*d*e) + x^8*((c^4*d^2)/4 + (3*a*c^3*e^2)/4 + (9*b^2*c^2*e^2)/8 + (7*b*c^3*d*e)/4) + x^5*((3*
a*b^3*e^2)/5 + b^3*c*d^2 + (2*b^4*d*e)/5 + 3*a*b*c^2*d^2 + (9*a^2*b*c*e^2)/5 + (12*a^2*c^2*d*e)/5 + (24*a*b^2*
c*d*e)/5) + x^3*(a*b^3*d^2 + (a^3*b*e^2)/3 + (4*a^3*c*d*e)/3 + 3*a^2*b*c*d^2 + 2*a^2*b^2*d*e) + x^7*(b*c^3*d^2
+ (5*b^3*c*e^2)/7 + (12*a*c^3*d*e)/7 + (15*a*b*c^2*e^2)/7 + (18*b^2*c^2*d*e)/7) + (c^4*e^2*x^10)/5 + (c^3*e*x
^9*(7*b*e + 4*c*d))/9 + (a^2*d*x^2*(3*b^2*d + 2*a*b*e + 2*a*c*d))/2 + a^3*b*d^2*x

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sympy [A]  time = 0.15, size = 503, normalized size = 1.22 \begin {gather*} a^{3} b d^{2} x + \frac {c^{4} e^{2} x^{10}}{5} + x^{9} \left (\frac {7 b c^{3} e^{2}}{9} + \frac {4 c^{4} d e}{9}\right ) + x^{8} \left (\frac {3 a c^{3} e^{2}}{4} + \frac {9 b^{2} c^{2} e^{2}}{8} + \frac {7 b c^{3} d e}{4} + \frac {c^{4} d^{2}}{4}\right ) + x^{7} \left (\frac {15 a b c^{2} e^{2}}{7} + \frac {12 a c^{3} d e}{7} + \frac {5 b^{3} c e^{2}}{7} + \frac {18 b^{2} c^{2} d e}{7} + b c^{3} d^{2}\right ) + x^{6} \left (a^{2} c^{2} e^{2} + 2 a b^{2} c e^{2} + 5 a b c^{2} d e + a c^{3} d^{2} + \frac {b^{4} e^{2}}{6} + \frac {5 b^{3} c d e}{3} + \frac {3 b^{2} c^{2} d^{2}}{2}\right ) + x^{5} \left (\frac {9 a^{2} b c e^{2}}{5} + \frac {12 a^{2} c^{2} d e}{5} + \frac {3 a b^{3} e^{2}}{5} + \frac {24 a b^{2} c d e}{5} + 3 a b c^{2} d^{2} + \frac {2 b^{4} d e}{5} + b^{3} c d^{2}\right ) + x^{4} \left (\frac {a^{3} c e^{2}}{2} + \frac {3 a^{2} b^{2} e^{2}}{4} + \frac {9 a^{2} b c d e}{2} + \frac {3 a^{2} c^{2} d^{2}}{2} + \frac {3 a b^{3} d e}{2} + 3 a b^{2} c d^{2} + \frac {b^{4} d^{2}}{4}\right ) + x^{3} \left (\frac {a^{3} b e^{2}}{3} + \frac {4 a^{3} c d e}{3} + 2 a^{2} b^{2} d e + 3 a^{2} b c d^{2} + a b^{3} d^{2}\right ) + x^{2} \left (a^{3} b d e + a^{3} c d^{2} + \frac {3 a^{2} b^{2} d^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*b*d**2*x + c**4*e**2*x**10/5 + x**9*(7*b*c**3*e**2/9 + 4*c**4*d*e/9) + x**8*(3*a*c**3*e**2/4 + 9*b**2*c**
2*e**2/8 + 7*b*c**3*d*e/4 + c**4*d**2/4) + x**7*(15*a*b*c**2*e**2/7 + 12*a*c**3*d*e/7 + 5*b**3*c*e**2/7 + 18*b
**2*c**2*d*e/7 + b*c**3*d**2) + x**6*(a**2*c**2*e**2 + 2*a*b**2*c*e**2 + 5*a*b*c**2*d*e + a*c**3*d**2 + b**4*e
**2/6 + 5*b**3*c*d*e/3 + 3*b**2*c**2*d**2/2) + x**5*(9*a**2*b*c*e**2/5 + 12*a**2*c**2*d*e/5 + 3*a*b**3*e**2/5
+ 24*a*b**2*c*d*e/5 + 3*a*b*c**2*d**2 + 2*b**4*d*e/5 + b**3*c*d**2) + x**4*(a**3*c*e**2/2 + 3*a**2*b**2*e**2/4
+ 9*a**2*b*c*d*e/2 + 3*a**2*c**2*d**2/2 + 3*a*b**3*d*e/2 + 3*a*b**2*c*d**2 + b**4*d**2/4) + x**3*(a**3*b*e**2
/3 + 4*a**3*c*d*e/3 + 2*a**2*b**2*d*e + 3*a**2*b*c*d**2 + a*b**3*d**2) + x**2*(a**3*b*d*e + a**3*c*d**2 + 3*a*
*2*b**2*d**2/2)

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