### 3.2118 $$\int (d+e x)^4 (a+b x+c x^2)^2 \, dx$$

Optimal. Leaf size=156 $\frac{(d+e x)^7 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{7 e^5}-\frac{(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5}+\frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2}{5 e^5}-\frac{c (d+e x)^8 (2 c d-b e)}{4 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}$

[Out]

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

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Rubi [A]  time = 0.249701, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {698} $\frac{(d+e x)^7 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{7 e^5}-\frac{(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5}+\frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2}{5 e^5}-\frac{c (d+e x)^8 (2 c d-b e)}{4 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0797288, size = 283, normalized size = 1.81 $\frac{1}{5} x^5 \left (e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+4 c d^2 e (3 a e+2 b d)+c^2 d^4\right )+\frac{1}{2} d x^4 \left (2 a^2 e^3+6 a b d e^2+4 a c d^2 e+2 b^2 d^2 e+b c d^3\right )+a^2 d^4 x+\frac{1}{7} e^2 x^7 \left (2 c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )+\frac{1}{3} d^2 x^3 \left (8 a b d e+2 a \left (3 a e^2+c d^2\right )+b^2 d^2\right )+\frac{1}{3} e x^6 \left (2 c d e (2 a e+3 b d)+b e^2 (a e+2 b d)+2 c^2 d^3\right )+a d^3 x^2 (2 a e+b d)+\frac{1}{4} c e^3 x^8 (b e+2 c d)+\frac{1}{9} c^2 e^4 x^9$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.038, size = 283, normalized size = 1.8 \begin{align*}{\frac{{e}^{4}{c}^{2}{x}^{9}}{9}}+{\frac{ \left ( 2\,{e}^{4}bc+4\,d{e}^{3}{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{c}^{2}+8\,d{e}^{3}bc+{e}^{4} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,{d}^{3}e{c}^{2}+12\,{d}^{2}{e}^{2}bc+4\,d{e}^{3} \left ( 2\,ac+{b}^{2} \right ) +2\,{e}^{4}ab \right ){x}^{6}}{6}}+{\frac{ \left ({c}^{2}{d}^{4}+8\,{d}^{3}ebc+6\,{d}^{2}{e}^{2} \left ( 2\,ac+{b}^{2} \right ) +8\,d{e}^{3}ab+{a}^{2}{e}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{d}^{4}bc+4\,{d}^{3}e \left ( 2\,ac+{b}^{2} \right ) +12\,{d}^{2}{e}^{2}ab+4\,d{e}^{3}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{4} \left ( 2\,ac+{b}^{2} \right ) +8\,{d}^{3}eab+6\,{d}^{2}{e}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{d}^{3}e{a}^{2}+2\,{d}^{4}ab \right ){x}^{2}}{2}}+{d}^{4}{a}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.975, size = 374, normalized size = 2.4 \begin{align*} \frac{1}{9} \, c^{2} e^{4} x^{9} + \frac{1}{4} \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{7} + a^{2} d^{4} x + \frac{1}{3} \,{\left (2 \, c^{2} d^{3} e + 6 \, b c d^{2} e^{2} + a b e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{4} + 8 \, b c d^{3} e + 8 \, a b d e^{3} + a^{2} e^{4} + 6 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (b c d^{4} + 6 \, a b d^{2} e^{2} + 2 \, a^{2} d e^{3} + 2 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{4} + \frac{1}{3} \,{\left (8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{4}\right )} x^{3} +{\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.7284, size = 756, normalized size = 4.85 \begin{align*} \frac{1}{9} x^{9} e^{4} c^{2} + \frac{1}{2} x^{8} e^{3} d c^{2} + \frac{1}{4} x^{8} e^{4} c b + \frac{6}{7} x^{7} e^{2} d^{2} c^{2} + \frac{8}{7} x^{7} e^{3} d c b + \frac{1}{7} x^{7} e^{4} b^{2} + \frac{2}{7} x^{7} e^{4} c a + \frac{2}{3} x^{6} e d^{3} c^{2} + 2 x^{6} e^{2} d^{2} c b + \frac{2}{3} x^{6} e^{3} d b^{2} + \frac{4}{3} x^{6} e^{3} d c a + \frac{1}{3} x^{6} e^{4} b a + \frac{1}{5} x^{5} d^{4} c^{2} + \frac{8}{5} x^{5} e d^{3} c b + \frac{6}{5} x^{5} e^{2} d^{2} b^{2} + \frac{12}{5} x^{5} e^{2} d^{2} c a + \frac{8}{5} x^{5} e^{3} d b a + \frac{1}{5} x^{5} e^{4} a^{2} + \frac{1}{2} x^{4} d^{4} c b + x^{4} e d^{3} b^{2} + 2 x^{4} e d^{3} c a + 3 x^{4} e^{2} d^{2} b a + x^{4} e^{3} d a^{2} + \frac{1}{3} x^{3} d^{4} b^{2} + \frac{2}{3} x^{3} d^{4} c a + \frac{8}{3} x^{3} e d^{3} b a + 2 x^{3} e^{2} d^{2} a^{2} + x^{2} d^{4} b a + 2 x^{2} e d^{3} a^{2} + x d^{4} a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.109907, size = 337, normalized size = 2.16 \begin{align*} a^{2} d^{4} x + \frac{c^{2} e^{4} x^{9}}{9} + x^{8} \left (\frac{b c e^{4}}{4} + \frac{c^{2} d e^{3}}{2}\right ) + x^{7} \left (\frac{2 a c e^{4}}{7} + \frac{b^{2} e^{4}}{7} + \frac{8 b c d e^{3}}{7} + \frac{6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac{a b e^{4}}{3} + \frac{4 a c d e^{3}}{3} + \frac{2 b^{2} d e^{3}}{3} + 2 b c d^{2} e^{2} + \frac{2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac{a^{2} e^{4}}{5} + \frac{8 a b d e^{3}}{5} + \frac{12 a c d^{2} e^{2}}{5} + \frac{6 b^{2} d^{2} e^{2}}{5} + \frac{8 b c d^{3} e}{5} + \frac{c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + 2 a c d^{3} e + b^{2} d^{3} e + \frac{b c d^{4}}{2}\right ) + x^{3} \left (2 a^{2} d^{2} e^{2} + \frac{8 a b d^{3} e}{3} + \frac{2 a c d^{4}}{3} + \frac{b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{2} d^{3} e + a b d^{4}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.1046, size = 443, normalized size = 2.84 \begin{align*} \frac{1}{9} \, c^{2} x^{9} e^{4} + \frac{1}{2} \, c^{2} d x^{8} e^{3} + \frac{6}{7} \, c^{2} d^{2} x^{7} e^{2} + \frac{2}{3} \, c^{2} d^{3} x^{6} e + \frac{1}{5} \, c^{2} d^{4} x^{5} + \frac{1}{4} \, b c x^{8} e^{4} + \frac{8}{7} \, b c d x^{7} e^{3} + 2 \, b c d^{2} x^{6} e^{2} + \frac{8}{5} \, b c d^{3} x^{5} e + \frac{1}{2} \, b c d^{4} x^{4} + \frac{1}{7} \, b^{2} x^{7} e^{4} + \frac{2}{7} \, a c x^{7} e^{4} + \frac{2}{3} \, b^{2} d x^{6} e^{3} + \frac{4}{3} \, a c d x^{6} e^{3} + \frac{6}{5} \, b^{2} d^{2} x^{5} e^{2} + \frac{12}{5} \, a c d^{2} x^{5} e^{2} + b^{2} d^{3} x^{4} e + 2 \, a c d^{3} x^{4} e + \frac{1}{3} \, b^{2} d^{4} x^{3} + \frac{2}{3} \, a c d^{4} x^{3} + \frac{1}{3} \, a b x^{6} e^{4} + \frac{8}{5} \, a b d x^{5} e^{3} + 3 \, a b d^{2} x^{4} e^{2} + \frac{8}{3} \, a b d^{3} x^{3} e + a b d^{4} x^{2} + \frac{1}{5} \, a^{2} x^{5} e^{4} + a^{2} d x^{4} e^{3} + 2 \, a^{2} d^{2} x^{3} e^{2} + 2 \, a^{2} d^{3} x^{2} e + a^{2} d^{4} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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