### 3.2121 $$\int (d+e x) (a+b x+c x^2)^2 \, dx$$

Optimal. Leaf size=96 $a^2 d x+\frac{1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{2} a x^2 (a e+2 b d)+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{6} c^2 e x^6$

[Out]

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

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Rubi [A]  time = 0.0748955, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.056, Rules used = {631} $a^2 d x+\frac{1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{2} a x^2 (a e+2 b d)+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{6} c^2 e x^6$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(a + b*x + c*x^2)^2,x]

[Out]

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

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

\begin{align*} \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx &=\int \left (a^2 d+a (2 b d+a e) x+\left (b^2 d+2 a c d+2 a b e\right ) x^2+\left (2 b c d+b^2 e+2 a c e\right ) x^3+c (c d+2 b e) x^4+c^2 e x^5\right ) \, dx\\ &=a^2 d x+\frac{1}{2} a (2 b d+a e) x^2+\frac{1}{3} \left (b^2 d+2 a c d+2 a b e\right ) x^3+\frac{1}{4} \left (2 b c d+b^2 e+2 a c e\right ) x^4+\frac{1}{5} c (c d+2 b e) x^5+\frac{1}{6} c^2 e x^6\\ \end{align*}

Mathematica [A]  time = 0.0204831, size = 96, normalized size = 1. $a^2 d x+\frac{1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{2} a x^2 (a e+2 b d)+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{6} c^2 e x^6$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 91, normalized size = 1. \begin{align*}{\frac{{c}^{2}e{x}^{6}}{6}}+{\frac{ \left ( 2\,bce+{c}^{2}d \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,bcd+e \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( \left ( 2\,ac+{b}^{2} \right ) d+2\,aeb \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{2}e+2\,abd \right ){x}^{2}}{2}}+{a}^{2}dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.967785, size = 122, normalized size = 1.27 \begin{align*} \frac{1}{6} \, c^{2} e x^{6} + \frac{1}{5} \,{\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac{1}{4} \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} x^{4} + a^{2} d x + \frac{1}{3} \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d + a^{2} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.70081, size = 244, normalized size = 2.54 \begin{align*} \frac{1}{6} x^{6} e c^{2} + \frac{1}{5} x^{5} d c^{2} + \frac{2}{5} x^{5} e c b + \frac{1}{2} x^{4} d c b + \frac{1}{4} x^{4} e b^{2} + \frac{1}{2} x^{4} e c a + \frac{1}{3} x^{3} d b^{2} + \frac{2}{3} x^{3} d c a + \frac{2}{3} x^{3} e b a + x^{2} d b a + \frac{1}{2} x^{2} e a^{2} + x d a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.077819, size = 100, normalized size = 1.04 \begin{align*} a^{2} d x + \frac{c^{2} e x^{6}}{6} + x^{5} \left (\frac{2 b c e}{5} + \frac{c^{2} d}{5}\right ) + x^{4} \left (\frac{a c e}{2} + \frac{b^{2} e}{4} + \frac{b c d}{2}\right ) + x^{3} \left (\frac{2 a b e}{3} + \frac{2 a c d}{3} + \frac{b^{2} d}{3}\right ) + x^{2} \left (\frac{a^{2} e}{2} + a b d\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11165, size = 142, normalized size = 1.48 \begin{align*} \frac{1}{6} \, c^{2} x^{6} e + \frac{1}{5} \, c^{2} d x^{5} + \frac{2}{5} \, b c x^{5} e + \frac{1}{2} \, b c d x^{4} + \frac{1}{4} \, b^{2} x^{4} e + \frac{1}{2} \, a c x^{4} e + \frac{1}{3} \, b^{2} d x^{3} + \frac{2}{3} \, a c d x^{3} + \frac{2}{3} \, a b x^{3} e + a b d x^{2} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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