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

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

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Rubi [A]  time = 0.27, 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)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac {(d+e x)^6 (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 )}{6 e^7}+\frac {3 (d+e x)^5 \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 )}{5 e^7}-\frac {3 (d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac {3 c^2 (d+e x)^8 (2 c d-b e)}{8 e^7}+\frac {c^3 (d+e x)^9}{9 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.09, size = 282, normalized size = 1.04 \begin {gather*} a^3 d^2 x+\frac {1}{4} x^4 \left (6 a^2 c d e+6 a b^2 d e+3 a b \left (a e^2+2 c d^2\right )+b^3 d^2\right )+\frac {1}{2} a^2 d x^2 (2 a e+3 b d)+\frac {1}{7} c x^7 \left (3 c e (a e+2 b d)+3 b^2 e^2+c^2 d^2\right )+\frac {1}{3} a x^3 \left (6 a b d e+a \left (a e^2+3 c d^2\right )+3 b^2 d^2\right )+\frac {1}{6} x^6 \left (3 b c \left (2 a e^2+c d^2\right )+6 a c^2 d e+b^3 e^2+6 b^2 c d e\right )+\frac {1}{5} x^5 \left (3 b^2 \left (a e^2+c d^2\right )+12 a b c d e+3 a c \left (a e^2+c d^2\right )+2 b^3 d e\right )+\frac {1}{8} c^2 e x^8 (3 b e+2 c d)+\frac {1}{9} c^3 e^2 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

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

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fricas [A]  time = 0.35, size = 330, normalized size = 1.21 \begin {gather*} \frac {1}{9} x^{9} e^{2} c^{3} + \frac {1}{4} x^{8} e d c^{3} + \frac {3}{8} x^{8} e^{2} c^{2} b + \frac {1}{7} x^{7} d^{2} c^{3} + \frac {6}{7} x^{7} e d c^{2} b + \frac {3}{7} x^{7} e^{2} c b^{2} + \frac {3}{7} x^{7} e^{2} c^{2} a + \frac {1}{2} x^{6} d^{2} c^{2} b + x^{6} e d c b^{2} + \frac {1}{6} x^{6} e^{2} b^{3} + x^{6} e d c^{2} a + x^{6} e^{2} c b a + \frac {3}{5} x^{5} d^{2} c b^{2} + \frac {2}{5} x^{5} e d b^{3} + \frac {3}{5} x^{5} d^{2} c^{2} a + \frac {12}{5} x^{5} e d c b a + \frac {3}{5} x^{5} e^{2} b^{2} a + \frac {3}{5} x^{5} e^{2} c a^{2} + \frac {1}{4} x^{4} d^{2} b^{3} + \frac {3}{2} x^{4} d^{2} c b a + \frac {3}{2} x^{4} e d b^{2} a + \frac {3}{2} x^{4} e d c a^{2} + \frac {3}{4} x^{4} e^{2} b a^{2} + x^{3} d^{2} b^{2} a + x^{3} d^{2} c a^{2} + 2 x^{3} e d b a^{2} + \frac {1}{3} x^{3} e^{2} a^{3} + \frac {3}{2} x^{2} d^{2} b a^{2} + x^{2} e d a^{3} + x d^{2} a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 330, normalized size = 1.21 \begin {gather*} \frac {1}{9} \, c^{3} x^{9} e^{2} + \frac {1}{4} \, c^{3} d x^{8} e + \frac {1}{7} \, c^{3} d^{2} x^{7} + \frac {3}{8} \, b c^{2} x^{8} e^{2} + \frac {6}{7} \, b c^{2} d x^{7} e + \frac {1}{2} \, b c^{2} d^{2} x^{6} + \frac {3}{7} \, b^{2} c x^{7} e^{2} + \frac {3}{7} \, a c^{2} x^{7} e^{2} + b^{2} c d x^{6} e + a c^{2} d x^{6} e + \frac {3}{5} \, b^{2} c d^{2} x^{5} + \frac {3}{5} \, a c^{2} d^{2} x^{5} + \frac {1}{6} \, b^{3} x^{6} e^{2} + a b c x^{6} e^{2} + \frac {2}{5} \, b^{3} d x^{5} e + \frac {12}{5} \, a b c d x^{5} e + \frac {1}{4} \, b^{3} d^{2} x^{4} + \frac {3}{2} \, a b c d^{2} x^{4} + \frac {3}{5} \, a b^{2} x^{5} e^{2} + \frac {3}{5} \, a^{2} c x^{5} e^{2} + \frac {3}{2} \, a b^{2} d x^{4} e + \frac {3}{2} \, a^{2} c d x^{4} e + a b^{2} d^{2} x^{3} + a^{2} c d^{2} x^{3} + \frac {3}{4} \, a^{2} b x^{4} e^{2} + 2 \, a^{2} b d x^{3} e + \frac {3}{2} \, a^{2} b d^{2} x^{2} + \frac {1}{3} \, a^{3} x^{3} e^{2} + a^{3} d x^{2} e + a^{3} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 359, normalized size = 1.32 \begin {gather*} \frac {c^{3} e^{2} x^{9}}{9}+\frac {\left (3 e^{2} b \,c^{2}+2 d e \,c^{3}\right ) x^{8}}{8}+\frac {\left (6 b \,c^{2} d e +c^{3} d^{2}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) e^{2}\right ) x^{7}}{7}+a^{3} d^{2} x +\frac {\left (3 b \,c^{2} d^{2}+2 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) d e +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) e^{2}\right ) x^{6}}{6}+\frac {\left (\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) d^{2}+2 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) d e +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) e^{2}\right ) x^{5}}{5}+\frac {\left (3 a^{2} b \,e^{2}+\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) d^{2}+2 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) d e \right ) x^{4}}{4}+\frac {\left (a^{3} e^{2}+6 a^{2} b d e +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) d^{2}\right ) x^{3}}{3}+\frac {\left (2 d e \,a^{3}+3 d^{2} a^{2} b \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.08, size = 275, normalized size = 1.01 \begin {gather*} \frac {1}{9} \, c^{3} e^{2} x^{9} + \frac {1}{8} \, {\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, b c^{2} d^{2} + 6 \, {\left (b^{2} c + a c^{2}\right )} d e + {\left (b^{3} + 6 \, a b c\right )} e^{2}\right )} x^{6} + a^{3} d^{2} x + \frac {1}{5} \, {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d e + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, a^{2} b e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{2} + 6 \, {\left (a b^{2} + a^{2} c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b d e + a^{3} e^{2} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.10, size = 276, normalized size = 1.01 \begin {gather*} x^3\,\left (\frac {a^3\,e^2}{3}+2\,a^2\,b\,d\,e+c\,a^2\,d^2+a\,b^2\,d^2\right )+x^7\,\left (\frac {3\,b^2\,c\,e^2}{7}+\frac {6\,b\,c^2\,d\,e}{7}+\frac {c^3\,d^2}{7}+\frac {3\,a\,c^2\,e^2}{7}\right )+x^4\,\left (\frac {3\,a^2\,b\,e^2}{4}+\frac {3\,c\,a^2\,d\,e}{2}+\frac {3\,a\,b^2\,d\,e}{2}+\frac {3\,c\,a\,b\,d^2}{2}+\frac {b^3\,d^2}{4}\right )+x^6\,\left (\frac {b^3\,e^2}{6}+b^2\,c\,d\,e+\frac {b\,c^2\,d^2}{2}+a\,b\,c\,e^2+a\,c^2\,d\,e\right )+x^5\,\left (\frac {3\,a^2\,c\,e^2}{5}+\frac {3\,a\,b^2\,e^2}{5}+\frac {12\,a\,b\,c\,d\,e}{5}+\frac {3\,a\,c^2\,d^2}{5}+\frac {2\,b^3\,d\,e}{5}+\frac {3\,b^2\,c\,d^2}{5}\right )+a^3\,d^2\,x+\frac {c^3\,e^2\,x^9}{9}+\frac {a^2\,d\,x^2\,\left (2\,a\,e+3\,b\,d\right )}{2}+\frac {c^2\,e\,x^8\,\left (3\,b\,e+2\,c\,d\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.12, size = 332, normalized size = 1.22 \begin {gather*} a^{3} d^{2} x + \frac {c^{3} e^{2} x^{9}}{9} + x^{8} \left (\frac {3 b c^{2} e^{2}}{8} + \frac {c^{3} d e}{4}\right ) + x^{7} \left (\frac {3 a c^{2} e^{2}}{7} + \frac {3 b^{2} c e^{2}}{7} + \frac {6 b c^{2} d e}{7} + \frac {c^{3} d^{2}}{7}\right ) + x^{6} \left (a b c e^{2} + a c^{2} d e + \frac {b^{3} e^{2}}{6} + b^{2} c d e + \frac {b c^{2} d^{2}}{2}\right ) + x^{5} \left (\frac {3 a^{2} c e^{2}}{5} + \frac {3 a b^{2} e^{2}}{5} + \frac {12 a b c d e}{5} + \frac {3 a c^{2} d^{2}}{5} + \frac {2 b^{3} d e}{5} + \frac {3 b^{2} c d^{2}}{5}\right ) + x^{4} \left (\frac {3 a^{2} b e^{2}}{4} + \frac {3 a^{2} c d e}{2} + \frac {3 a b^{2} d e}{2} + \frac {3 a b c d^{2}}{2} + \frac {b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac {a^{3} e^{2}}{3} + 2 a^{2} b d e + a^{2} c d^{2} + a b^{2} d^{2}\right ) + x^{2} \left (a^{3} d e + \frac {3 a^{2} b d^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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