### 3.2120 $$\int (d+e x)^2 (a+b x+c x^2)^2 \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.140405, 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} $\frac{(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5}+\frac{(d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}{3 e^5}-\frac{c (d+e x)^6 (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^7}{7 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0432366, size = 153, normalized size = 0.98 $\frac{1}{3} x^3 \left (a^2 e^2+4 a b d e+2 a c d^2+b^2 d^2\right )+a^2 d^2 x+\frac{1}{5} x^5 \left (2 a c e^2+b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{2} x^4 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+a d x^2 (a e+b d)+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{7} c^2 e^2 x^7$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 155, normalized size = 1. \begin{align*}{\frac{{c}^{2}{e}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{e}^{2}bc+2\,{c}^{2}de \right ){x}^{6}}{6}}+{\frac{ \left ({c}^{2}{d}^{2}+4\,bcde+{e}^{2} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{d}^{2}bc+2\,de \left ( 2\,ac+{b}^{2} \right ) +2\,{e}^{2}ab \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,abde+{a}^{2}{e}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{2}+2\,{d}^{2}ab \right ){x}^{2}}{2}}+{a}^{2}{d}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.955823, size = 197, normalized size = 1.26 \begin{align*} \frac{1}{7} \, c^{2} e^{2} x^{7} + \frac{1}{3} \,{\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{2} + 4 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac{1}{2} \,{\left (b c d^{2} + a b e^{2} +{\left (b^{2} + 2 \, a c\right )} d e\right )} x^{4} + \frac{1}{3} \,{\left (4 \, a b d e + a^{2} e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{3} +{\left (a b d^{2} + a^{2} d e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.72716, size = 412, normalized size = 2.64 \begin{align*} \frac{1}{7} x^{7} e^{2} c^{2} + \frac{1}{3} x^{6} e d c^{2} + \frac{1}{3} x^{6} e^{2} c b + \frac{1}{5} x^{5} d^{2} c^{2} + \frac{4}{5} x^{5} e d c b + \frac{1}{5} x^{5} e^{2} b^{2} + \frac{2}{5} x^{5} e^{2} c a + \frac{1}{2} x^{4} d^{2} c b + \frac{1}{2} x^{4} e d b^{2} + x^{4} e d c a + \frac{1}{2} x^{4} e^{2} b a + \frac{1}{3} x^{3} d^{2} b^{2} + \frac{2}{3} x^{3} d^{2} c a + \frac{4}{3} x^{3} e d b a + \frac{1}{3} x^{3} e^{2} a^{2} + x^{2} d^{2} b a + x^{2} e d a^{2} + x d^{2} a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.089446, size = 173, normalized size = 1.11 \begin{align*} a^{2} d^{2} x + \frac{c^{2} e^{2} x^{7}}{7} + x^{6} \left (\frac{b c e^{2}}{3} + \frac{c^{2} d e}{3}\right ) + x^{5} \left (\frac{2 a c e^{2}}{5} + \frac{b^{2} e^{2}}{5} + \frac{4 b c d e}{5} + \frac{c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac{a b e^{2}}{2} + a c d e + \frac{b^{2} d e}{2} + \frac{b c d^{2}}{2}\right ) + x^{3} \left (\frac{a^{2} e^{2}}{3} + \frac{4 a b d e}{3} + \frac{2 a c d^{2}}{3} + \frac{b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{2} d e + a b d^{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08653, size = 240, normalized size = 1.54 \begin{align*} \frac{1}{7} \, c^{2} x^{7} e^{2} + \frac{1}{3} \, c^{2} d x^{6} e + \frac{1}{5} \, c^{2} d^{2} x^{5} + \frac{1}{3} \, b c x^{6} e^{2} + \frac{4}{5} \, b c d x^{5} e + \frac{1}{2} \, b c d^{2} x^{4} + \frac{1}{5} \, b^{2} x^{5} e^{2} + \frac{2}{5} \, a c x^{5} e^{2} + \frac{1}{2} \, b^{2} d x^{4} e + a c d x^{4} e + \frac{1}{3} \, b^{2} d^{2} x^{3} + \frac{2}{3} \, a c d^{2} x^{3} + \frac{1}{2} \, a b x^{4} e^{2} + \frac{4}{3} \, a b d x^{3} e + a b d^{2} x^{2} + \frac{1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + a^{2} d^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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