3.1518 \(\int (b+2 c x) (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{4} \left (a+b x+c x^2\right )^4 \]

[Out]

(a + b*x + c*x^2)^4/4

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Rubi [A]  time = 0.0045857, 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} \[ \frac{1}{4} \left (a+b x+c x^2\right )^4 \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(a + b*x + c*x^2)^3,x]

[Out]

(a + b*x + c*x^2)^4/4

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

Mathematica [B]  time = 0.0185674, size = 51, normalized size = 3.19 \[ \frac{1}{4} x (b+c x) \left (6 a^2 x (b+c x)+4 a^3+4 a x^2 (b+c x)^2+x^3 (b+c x)^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(a + b*x + c*x^2)^3,x]

[Out]

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

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Maple [B]  time = 0., size = 218, normalized size = 13.6 \begin{align*}{\frac{{c}^{4}{x}^{8}}{4}}+b{c}^{3}{x}^{7}+{\frac{ \left ( 3\,{b}^{2}{c}^{2}+2\,c \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( b \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,c \left ( b \left ( 2\,ac+{b}^{2} \right ) +4\,abc \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( b \left ( b \left ( 2\,ac+{b}^{2} \right ) +4\,abc \right ) +2\,c \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( b \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +6\,c{a}^{2}b \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,c{a}^{3}+3\,{b}^{2}{a}^{2} \right ){x}^{2}}{2}}+b{a}^{3}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)^3,x)

[Out]

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

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Maxima [A]  time = 1.03006, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{4} \,{\left (c x^{2} + b x + a\right )}^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

1/4*(c*x^2 + b*x + a)^4

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Fricas [B]  time = 1.24024, size = 269, normalized size = 16.81 \begin{align*} \frac{1}{4} x^{8} c^{4} + x^{7} c^{3} b + \frac{3}{2} x^{6} c^{2} b^{2} + x^{6} c^{3} a + x^{5} c b^{3} + 3 x^{5} c^{2} b a + \frac{1}{4} x^{4} b^{4} + 3 x^{4} c b^{2} a + \frac{3}{2} x^{4} c^{2} a^{2} + x^{3} b^{3} a + 3 x^{3} c b a^{2} + \frac{3}{2} x^{2} b^{2} a^{2} + x^{2} c a^{3} + x b a^{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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.088608, size = 121, normalized size = 7.56 \begin{align*} a^{3} b x + b c^{3} x^{7} + \frac{c^{4} x^{8}}{4} + x^{6} \left (a c^{3} + \frac{3 b^{2} c^{2}}{2}\right ) + x^{5} \left (3 a b c^{2} + b^{3} c\right ) + x^{4} \left (\frac{3 a^{2} c^{2}}{2} + 3 a b^{2} c + \frac{b^{4}}{4}\right ) + x^{3} \left (3 a^{2} b c + a b^{3}\right ) + x^{2} \left (a^{3} c + \frac{3 a^{2} b^{2}}{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)**3,x)

[Out]

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

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Giac [B]  time = 1.16429, size = 170, normalized size = 10.62 \begin{align*} \frac{1}{4} \, c^{4} x^{8} + b c^{3} x^{7} + \frac{3}{2} \, b^{2} c^{2} x^{6} + a c^{3} x^{6} + b^{3} c x^{5} + 3 \, a b c^{2} x^{5} + \frac{1}{4} \, b^{4} x^{4} + 3 \, a b^{2} c x^{4} + \frac{3}{2} \, a^{2} c^{2} x^{4} + a b^{3} x^{3} + 3 \, a^{2} b c x^{3} + \frac{3}{2} \, a^{2} b^{2} x^{2} + a^{3} c x^{2} + a^{3} 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)^3,x, algorithm="giac")

[Out]

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