### 3.1125 $$\int (b d+2 c d x) (a+b x+c x^2)^2 \, dx$$

Optimal. Leaf size=17 $\frac{1}{3} d \left (a+b x+c x^2\right )^3$

[Out]

(d*(a + b*x + c*x^2)^3)/3

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Rubi [A]  time = 0.0055592, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {629} $\frac{1}{3} d \left (a+b x+c x^2\right )^3$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d*(a + b*x + c*x^2)^3)/3

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

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

Mathematica [B]  time = 0.0099525, size = 37, normalized size = 2.18 $\frac{1}{3} d x (b+c x) \left (3 a^2+3 a x (b+c x)+x^2 (b+c x)^2\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d*x*(b + c*x)*(3*a^2 + 3*a*x*(b + c*x) + x^2*(b + c*x)^2))/3

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Maple [B]  time = 0.039, size = 95, normalized size = 5.6 \begin{align*}{\frac{{c}^{3}d{x}^{6}}{3}}+bd{c}^{2}{x}^{5}+{\frac{ \left ( 2\,{b}^{2}dc+2\,cd \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( bd \left ( 2\,ac+{b}^{2} \right ) +4\,cabd \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,cd{a}^{2}+2\,a{b}^{2}d \right ){x}^{2}}{2}}+bd{a}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16972, size = 20, normalized size = 1.18 \begin{align*} \frac{1}{3} \,{\left (c x^{2} + b x + a\right )}^{3} d \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(c*x^2 + b*x + a)^3*d

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Fricas [B]  time = 1.84776, size = 174, normalized size = 10.24 \begin{align*} \frac{1}{3} x^{6} d c^{3} + x^{5} d c^{2} b + x^{4} d c b^{2} + x^{4} d c^{2} a + \frac{1}{3} x^{3} d b^{3} + 2 x^{3} d c b a + x^{2} d b^{2} a + x^{2} d c a^{2} + x d b a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.085105, size = 80, normalized size = 4.71 \begin{align*} a^{2} b d x + b c^{2} d x^{5} + \frac{c^{3} d x^{6}}{3} + x^{4} \left (a c^{2} d + b^{2} c d\right ) + x^{3} \left (2 a b c d + \frac{b^{3} d}{3}\right ) + x^{2} \left (a^{2} c d + a b^{2} d\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.23593, size = 108, normalized size = 6.35 \begin{align*} \frac{1}{3} \, c^{3} d x^{6} + b c^{2} d x^{5} + b^{2} c d x^{4} + a c^{2} d x^{4} + \frac{1}{3} \, b^{3} d x^{3} + 2 \, a b c d x^{3} + a b^{2} d x^{2} + a^{2} c d x^{2} + a^{2} b d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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