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

Optimal. Leaf size=272 \[ \frac {3 c (d+e x)^8 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{8 e^7}-\frac {(d+e x)^7 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}-\frac {3 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^3}{4 e^7}-\frac {c^2 (d+e x)^9 (2 c d-b e)}{3 e^7}+\frac {c^3 (d+e x)^{10}}{10 e^7} \]

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Rubi [A]  time = 0.37, antiderivative size = 272, 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} \begin {gather*} \frac {3 c (d+e x)^8 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{8 e^7}-\frac {(d+e x)^7 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}-\frac {3 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^3}{4 e^7}-\frac {c^2 (d+e x)^9 (2 c d-b e)}{3 e^7}+\frac {c^3 (d+e x)^{10}}{10 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.13, size = 372, normalized size = 1.37 \begin {gather*} a^3 d^3 x+\frac {1}{4} x^4 \left (a^2 e \left (a e^2+9 c d^2\right )+9 a b^2 d^2 e+3 a b d \left (3 a e^2+2 c d^2\right )+b^3 d^3\right )+\frac {3}{2} a^2 d^2 x^2 (a e+b d)+\frac {1}{7} x^7 \left (9 c^2 d e (a e+b d)+3 b c e^2 (2 a e+3 b d)+b^3 e^3+c^3 d^3\right )+\frac {3}{8} c e x^8 \left (c e (a e+3 b d)+b^2 e^2+c^2 d^2\right )+a d x^3 \left (3 a b d e+a \left (a e^2+c d^2\right )+b^2 d^2\right )+\frac {1}{2} x^6 \left (b^2 \left (a e^3+3 c d^2 e\right )+b c d \left (6 a e^2+c d^2\right )+a c e \left (a e^2+3 c d^2\right )+b^3 d e^2\right )+\frac {3}{5} x^5 \left (b^2 \left (3 a d e^2+c d^3\right )+a b e \left (a e^2+6 c d^2\right )+a c d \left (3 a e^2+c d^2\right )+b^3 d^2 e\right )+\frac {1}{3} c^2 e^2 x^9 (b e+c d)+\frac {1}{10} c^3 e^3 x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.35, size = 479, normalized size = 1.76 \begin {gather*} \frac {1}{10} x^{10} e^{3} c^{3} + \frac {1}{3} x^{9} e^{2} d c^{3} + \frac {1}{3} x^{9} e^{3} c^{2} b + \frac {3}{8} x^{8} e d^{2} c^{3} + \frac {9}{8} x^{8} e^{2} d c^{2} b + \frac {3}{8} x^{8} e^{3} c b^{2} + \frac {3}{8} x^{8} e^{3} c^{2} a + \frac {1}{7} x^{7} d^{3} c^{3} + \frac {9}{7} x^{7} e d^{2} c^{2} b + \frac {9}{7} x^{7} e^{2} d c b^{2} + \frac {1}{7} x^{7} e^{3} b^{3} + \frac {9}{7} x^{7} e^{2} d c^{2} a + \frac {6}{7} x^{7} e^{3} c b a + \frac {1}{2} x^{6} d^{3} c^{2} b + \frac {3}{2} x^{6} e d^{2} c b^{2} + \frac {1}{2} x^{6} e^{2} d b^{3} + \frac {3}{2} x^{6} e d^{2} c^{2} a + 3 x^{6} e^{2} d c b a + \frac {1}{2} x^{6} e^{3} b^{2} a + \frac {1}{2} x^{6} e^{3} c a^{2} + \frac {3}{5} x^{5} d^{3} c b^{2} + \frac {3}{5} x^{5} e d^{2} b^{3} + \frac {3}{5} x^{5} d^{3} c^{2} a + \frac {18}{5} x^{5} e d^{2} c b a + \frac {9}{5} x^{5} e^{2} d b^{2} a + \frac {9}{5} x^{5} e^{2} d c a^{2} + \frac {3}{5} x^{5} e^{3} b a^{2} + \frac {1}{4} x^{4} d^{3} b^{3} + \frac {3}{2} x^{4} d^{3} c b a + \frac {9}{4} x^{4} e d^{2} b^{2} a + \frac {9}{4} x^{4} e d^{2} c a^{2} + \frac {9}{4} x^{4} e^{2} d b a^{2} + \frac {1}{4} x^{4} e^{3} a^{3} + x^{3} d^{3} b^{2} a + x^{3} d^{3} c a^{2} + 3 x^{3} e d^{2} b a^{2} + x^{3} e^{2} d a^{3} + \frac {3}{2} x^{2} d^{3} b a^{2} + \frac {3}{2} x^{2} e d^{2} a^{3} + x d^{3} a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 469, normalized size = 1.72 \begin {gather*} \frac {1}{10} \, c^{3} x^{10} e^{3} + \frac {1}{3} \, c^{3} d x^{9} e^{2} + \frac {3}{8} \, c^{3} d^{2} x^{8} e + \frac {1}{7} \, c^{3} d^{3} x^{7} + \frac {1}{3} \, b c^{2} x^{9} e^{3} + \frac {9}{8} \, b c^{2} d x^{8} e^{2} + \frac {9}{7} \, b c^{2} d^{2} x^{7} e + \frac {1}{2} \, b c^{2} d^{3} x^{6} + \frac {3}{8} \, b^{2} c x^{8} e^{3} + \frac {3}{8} \, a c^{2} x^{8} e^{3} + \frac {9}{7} \, b^{2} c d x^{7} e^{2} + \frac {9}{7} \, a c^{2} d x^{7} e^{2} + \frac {3}{2} \, b^{2} c d^{2} x^{6} e + \frac {3}{2} \, a c^{2} d^{2} x^{6} e + \frac {3}{5} \, b^{2} c d^{3} x^{5} + \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {1}{7} \, b^{3} x^{7} e^{3} + \frac {6}{7} \, a b c x^{7} e^{3} + \frac {1}{2} \, b^{3} d x^{6} e^{2} + 3 \, a b c d x^{6} e^{2} + \frac {3}{5} \, b^{3} d^{2} x^{5} e + \frac {18}{5} \, a b c d^{2} x^{5} e + \frac {1}{4} \, b^{3} d^{3} x^{4} + \frac {3}{2} \, a b c d^{3} x^{4} + \frac {1}{2} \, a b^{2} x^{6} e^{3} + \frac {1}{2} \, a^{2} c x^{6} e^{3} + \frac {9}{5} \, a b^{2} d x^{5} e^{2} + \frac {9}{5} \, a^{2} c d x^{5} e^{2} + \frac {9}{4} \, a b^{2} d^{2} x^{4} e + \frac {9}{4} \, a^{2} c d^{2} x^{4} e + a b^{2} d^{3} x^{3} + a^{2} c d^{3} x^{3} + \frac {3}{5} \, a^{2} b x^{5} e^{3} + \frac {9}{4} \, a^{2} b d x^{4} e^{2} + 3 \, a^{2} b d^{2} x^{3} e + \frac {3}{2} \, a^{2} b d^{3} x^{2} + \frac {1}{4} \, a^{3} x^{4} e^{3} + a^{3} d x^{3} e^{2} + \frac {3}{2} \, a^{3} d^{2} x^{2} e + a^{3} 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)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 495, normalized size = 1.82 \begin {gather*} \frac {c^{3} e^{3} x^{10}}{10}+\frac {\left (3 e^{3} b \,c^{2}+3 d \,e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (9 b \,c^{2} d \,e^{2}+3 c^{3} d^{2} e +\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) e^{3}\right ) x^{8}}{8}+a^{3} d^{3} x +\frac {\left (9 b \,c^{2} d^{2} e +c^{3} d^{3}+3 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) d \,e^{2}+\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) e^{3}\right ) x^{7}}{7}+\frac {\left (3 b \,c^{2} d^{3}+3 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) d^{2} e +3 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) d \,e^{2}+\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) e^{3}\right ) x^{6}}{6}+\frac {\left (3 a^{2} b \,e^{3}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) d^{3}+3 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) d^{2} e +3 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) d \,e^{2}\right ) x^{5}}{5}+\frac {\left (a^{3} e^{3}+9 a^{2} b d \,e^{2}+\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) d^{3}+3 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) d^{2} e \right ) x^{4}}{4}+\frac {\left (3 a^{3} d \,e^{2}+9 a^{2} b \,d^{2} e +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) d^{3}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{3}+3 d^{3} a^{2} b \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.13, size = 367, normalized size = 1.35 \begin {gather*} \frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, {\left (c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + 3 \, b c^{2} d e^{2} + {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} + {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (b c^{2} d^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e + {\left (b^{3} + 6 \, a b c\right )} d e^{2} + {\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (a^{2} b e^{3} + {\left (b^{2} c + a c^{2}\right )} d^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (9 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{3} + 9 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{4} + {\left (3 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b d^{3} + a^{3} 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)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.12, size = 381, normalized size = 1.40 \begin {gather*} x^4\,\left (\frac {a^3\,e^3}{4}+\frac {9\,a^2\,b\,d\,e^2}{4}+\frac {9\,c\,a^2\,d^2\,e}{4}+\frac {9\,a\,b^2\,d^2\,e}{4}+\frac {3\,c\,a\,b\,d^3}{2}+\frac {b^3\,d^3}{4}\right )+x^7\,\left (\frac {b^3\,e^3}{7}+\frac {9\,b^2\,c\,d\,e^2}{7}+\frac {9\,b\,c^2\,d^2\,e}{7}+\frac {6\,a\,b\,c\,e^3}{7}+\frac {c^3\,d^3}{7}+\frac {9\,a\,c^2\,d\,e^2}{7}\right )+x^5\,\left (\frac {3\,a^2\,b\,e^3}{5}+\frac {9\,a^2\,c\,d\,e^2}{5}+\frac {9\,a\,b^2\,d\,e^2}{5}+\frac {18\,a\,b\,c\,d^2\,e}{5}+\frac {3\,a\,c^2\,d^3}{5}+\frac {3\,b^3\,d^2\,e}{5}+\frac {3\,b^2\,c\,d^3}{5}\right )+x^6\,\left (\frac {a^2\,c\,e^3}{2}+\frac {a\,b^2\,e^3}{2}+3\,a\,b\,c\,d\,e^2+\frac {3\,a\,c^2\,d^2\,e}{2}+\frac {b^3\,d\,e^2}{2}+\frac {3\,b^2\,c\,d^2\,e}{2}+\frac {b\,c^2\,d^3}{2}\right )+a^3\,d^3\,x+\frac {c^3\,e^3\,x^{10}}{10}+\frac {3\,a^2\,d^2\,x^2\,\left (a\,e+b\,d\right )}{2}+\frac {c^2\,e^2\,x^9\,\left (b\,e+c\,d\right )}{3}+a\,d\,x^3\,\left (a^2\,e^2+3\,a\,b\,d\,e+c\,a\,d^2+b^2\,d^2\right )+\frac {3\,c\,e\,x^8\,\left (b^2\,e^2+3\,b\,c\,d\,e+c^2\,d^2+a\,c\,e^2\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.14, size = 484, normalized size = 1.78 \begin {gather*} a^{3} d^{3} x + \frac {c^{3} e^{3} x^{10}}{10} + x^{9} \left (\frac {b c^{2} e^{3}}{3} + \frac {c^{3} d e^{2}}{3}\right ) + x^{8} \left (\frac {3 a c^{2} e^{3}}{8} + \frac {3 b^{2} c e^{3}}{8} + \frac {9 b c^{2} d e^{2}}{8} + \frac {3 c^{3} d^{2} e}{8}\right ) + x^{7} \left (\frac {6 a b c e^{3}}{7} + \frac {9 a c^{2} d e^{2}}{7} + \frac {b^{3} e^{3}}{7} + \frac {9 b^{2} c d e^{2}}{7} + \frac {9 b c^{2} d^{2} e}{7} + \frac {c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac {a^{2} c e^{3}}{2} + \frac {a b^{2} e^{3}}{2} + 3 a b c d e^{2} + \frac {3 a c^{2} d^{2} e}{2} + \frac {b^{3} d e^{2}}{2} + \frac {3 b^{2} c d^{2} e}{2} + \frac {b c^{2} d^{3}}{2}\right ) + x^{5} \left (\frac {3 a^{2} b e^{3}}{5} + \frac {9 a^{2} c d e^{2}}{5} + \frac {9 a b^{2} d e^{2}}{5} + \frac {18 a b c d^{2} e}{5} + \frac {3 a c^{2} d^{3}}{5} + \frac {3 b^{3} d^{2} e}{5} + \frac {3 b^{2} c d^{3}}{5}\right ) + x^{4} \left (\frac {a^{3} e^{3}}{4} + \frac {9 a^{2} b d e^{2}}{4} + \frac {9 a^{2} c d^{2} e}{4} + \frac {9 a b^{2} d^{2} e}{4} + \frac {3 a b c d^{3}}{2} + \frac {b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{3} d e^{2} + 3 a^{2} b d^{2} e + a^{2} c d^{3} + a b^{2} d^{3}\right ) + x^{2} \left (\frac {3 a^{3} d^{2} e}{2} + \frac {3 a^{2} b d^{3}}{2}\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)**3,x)

[Out]

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

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