3.1497 \(\int (b+2 c x) (d+e x) (a+b x+c x^2) \, dx\)

Optimal. Leaf size=79 \[ \frac{1}{3} x^3 \left (2 a c e+b^2 e+3 b c d\right )+\frac{1}{2} x^2 \left (a b e+2 a c d+b^2 d\right )+a b d x+\frac{1}{4} c x^4 (3 b e+2 c d)+\frac{2}{5} c^2 e x^5 \]

[Out]

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

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Rubi [A]  time = 0.0642936, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {771} \[ \frac{1}{3} x^3 \left (2 a c e+b^2 e+3 b c d\right )+\frac{1}{2} x^2 \left (a b e+2 a c d+b^2 d\right )+a b d x+\frac{1}{4} c x^4 (3 b e+2 c d)+\frac{2}{5} c^2 e x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0181632, size = 79, normalized size = 1. \[ \frac{1}{3} x^3 \left (2 a c e+b^2 e+3 b c d\right )+\frac{1}{2} x^2 \left (a b e+2 a c d+b^2 d\right )+a b d x+\frac{1}{4} c x^4 (3 b e+2 c d)+\frac{2}{5} c^2 e x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 83, normalized size = 1.1 \begin{align*}{\frac{2\,{c}^{2}e{x}^{5}}{5}}+{\frac{ \left ( \left ( be+2\,cd \right ) c+2\,ceb \right ){x}^{4}}{4}}+{\frac{ \left ( bcd+ \left ( be+2\,cd \right ) b+2\,ace \right ){x}^{3}}{3}}+{\frac{ \left ({b}^{2}d+ \left ( be+2\,cd \right ) a \right ){x}^{2}}{2}}+abdx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.969976, size = 99, normalized size = 1.25 \begin{align*} \frac{2}{5} \, c^{2} e x^{5} + \frac{1}{4} \,{\left (2 \, c^{2} d + 3 \, b c e\right )} x^{4} + a b d x + \frac{1}{3} \,{\left (3 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.41207, size = 196, normalized size = 2.48 \begin{align*} \frac{2}{5} x^{5} e c^{2} + \frac{1}{2} x^{4} d c^{2} + \frac{3}{4} x^{4} e c b + x^{3} d c b + \frac{1}{3} x^{3} e b^{2} + \frac{2}{3} x^{3} e c a + \frac{1}{2} x^{2} d b^{2} + x^{2} d c a + \frac{1}{2} x^{2} e b a + x d b a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.073326, size = 82, normalized size = 1.04 \begin{align*} a b d x + \frac{2 c^{2} e x^{5}}{5} + x^{4} \left (\frac{3 b c e}{4} + \frac{c^{2} d}{2}\right ) + x^{3} \left (\frac{2 a c e}{3} + \frac{b^{2} e}{3} + b c d\right ) + x^{2} \left (\frac{a b e}{2} + a c d + \frac{b^{2} d}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18525, size = 115, normalized size = 1.46 \begin{align*} \frac{2}{5} \, c^{2} x^{5} e + \frac{1}{2} \, c^{2} d x^{4} + \frac{3}{4} \, b c x^{4} e + b c d x^{3} + \frac{1}{3} \, b^{2} x^{3} e + \frac{2}{3} \, a c x^{3} e + \frac{1}{2} \, b^{2} d x^{2} + a c d x^{2} + \frac{1}{2} \, a b x^{2} e + a b d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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