3.1508 \(\int (b+2 c x) (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{3} \left (a+b x+c x^2\right )^3 \]

[Out]

(a + b*x + c*x^2)^3/3

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Rubi [A]  time = 0.0044276, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {629} \[ \frac{1}{3} \left (a+b x+c x^2\right )^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [B]  time = 0.0097431, size = 36, normalized size = 2.25 \[ \frac{1}{3} x (b+c x) \left (3 a^2+3 a x (b+c x)+x^2 (b+c x)^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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., size = 86, normalized size = 5.4 \begin{align*}{\frac{{c}^{3}{x}^{6}}{3}}+b{c}^{2}{x}^{5}+{\frac{ \left ( 2\,{b}^{2}c+2\,c \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( b \left ( 2\,ac+{b}^{2} \right ) +4\,abc \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,c{a}^{2}+2\,{b}^{2}a \right ){x}^{2}}{2}}+b{a}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04444, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{3} \,{\left (c x^{2} + b x + a\right )}^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(c*x^2 + b*x + a)^3

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Fricas [B]  time = 1.28025, size = 150, normalized size = 9.38 \begin{align*} \frac{1}{3} x^{6} c^{3} + x^{5} c^{2} b + x^{4} c b^{2} + x^{4} c^{2} a + \frac{1}{3} x^{3} b^{3} + 2 x^{3} c b a + x^{2} b^{2} a + x^{2} c a^{2} + x b a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.076576, size = 65, normalized size = 4.06 \begin{align*} a^{2} b x + b c^{2} x^{5} + \frac{c^{3} x^{6}}{3} + x^{4} \left (a c^{2} + b^{2} c\right ) + x^{3} \left (2 a b c + \frac{b^{3}}{3}\right ) + x^{2} \left (a^{2} c + a b^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.17353, size = 96, normalized size = 6. \begin{align*} \frac{1}{3} \, c^{3} x^{6} + b c^{2} x^{5} + b^{2} c x^{4} + a c^{2} x^{4} + \frac{1}{3} \, b^{3} x^{3} + 2 \, a b c x^{3} + a b^{2} x^{2} + a^{2} c x^{2} + a^{2} b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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