3.189 \(\int (\text{b1}+\text{c1} x) (a+2 b x+c x^2) \, dx\)

Optimal. Leaf size=44 \[ \frac{1}{2} x^2 (a \text{c1}+2 b \text{b1})+a \text{b1} x+\frac{1}{3} x^3 (2 b \text{c1}+\text{b1} c)+\frac{1}{4} c \text{c1} x^4 \]

[Out]

a*b1*x + ((2*b*b1 + a*c1)*x^2)/2 + ((b1*c + 2*b*c1)*x^3)/3 + (c*c1*x^4)/4

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Rubi [A]  time = 0.0365853, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {631} \[ \frac{1}{2} x^2 (a \text{c1}+2 b \text{b1})+a \text{b1} x+\frac{1}{3} x^3 (2 b \text{c1}+\text{b1} c)+\frac{1}{4} c \text{c1} x^4 \]

Antiderivative was successfully verified.

[In]

Int[(b1 + c1*x)*(a + 2*b*x + c*x^2),x]

[Out]

a*b1*x + ((2*b*b1 + a*c1)*x^2)/2 + ((b1*c + 2*b*c1)*x^3)/3 + (c*c1*x^4)/4

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 (\text{b1}+\text{c1} x) \left (a+2 b x+c x^2\right ) \, dx &=\int \left (a \text{b1}+(2 b \text{b1}+a \text{c1}) x+(\text{b1} c+2 b \text{c1}) x^2+c \text{c1} x^3\right ) \, dx\\ &=a \text{b1} x+\frac{1}{2} (2 b \text{b1}+a \text{c1}) x^2+\frac{1}{3} (\text{b1} c+2 b \text{c1}) x^3+\frac{1}{4} c \text{c1} x^4\\ \end{align*}

Mathematica [A]  time = 0.0116867, size = 41, normalized size = 0.93 \[ \frac{1}{12} x (6 a (2 \text{b1}+\text{c1} x)+x (4 b (3 \text{b1}+2 \text{c1} x)+c x (4 \text{b1}+3 \text{c1} x))) \]

Antiderivative was successfully verified.

[In]

Integrate[(b1 + c1*x)*(a + 2*b*x + c*x^2),x]

[Out]

(x*(6*a*(2*b1 + c1*x) + x*(4*b*(3*b1 + 2*c1*x) + c*x*(4*b1 + 3*c1*x))))/12

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Maple [A]  time = 0.003, size = 39, normalized size = 0.9 \begin{align*} a{\it b1}\,x+{\frac{ \left ( a{\it c1}+2\,b{\it b1} \right ){x}^{2}}{2}}+{\frac{ \left ( 2\,b{\it c1}+{\it b1}\,c \right ){x}^{3}}{3}}+{\frac{c{\it c1}\,{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c1*x+b1)*(c*x^2+2*b*x+a),x)

[Out]

a*b1*x+1/2*(a*c1+2*b*b1)*x^2+1/3*(2*b*c1+b1*c)*x^3+1/4*c*c1*x^4

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Maxima [A]  time = 0.925517, size = 51, normalized size = 1.16 \begin{align*} \frac{1}{4} \, c c_{1} x^{4} + \frac{1}{3} \,{\left (b_{1} c + 2 \, b c_{1}\right )} x^{3} + a b_{1} x + \frac{1}{2} \,{\left (2 \, b b_{1} + a c_{1}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x^2+2*b*x+a),x, algorithm="maxima")

[Out]

1/4*c*c1*x^4 + 1/3*(b1*c + 2*b*c1)*x^3 + a*b1*x + 1/2*(2*b*b1 + a*c1)*x^2

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Fricas [A]  time = 1.47087, size = 107, normalized size = 2.43 \begin{align*} \frac{1}{4} x^{4} c_{1} c + \frac{1}{3} x^{3} c b_{1} + \frac{2}{3} x^{3} c_{1} b + x^{2} b_{1} b + \frac{1}{2} x^{2} c_{1} a + x b_{1} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x^2+2*b*x+a),x, algorithm="fricas")

[Out]

1/4*x^4*c1*c + 1/3*x^3*c*b1 + 2/3*x^3*c1*b + x^2*b1*b + 1/2*x^2*c1*a + x*b1*a

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Sympy [A]  time = 0.058321, size = 39, normalized size = 0.89 \begin{align*} a b_{1} x + \frac{c c_{1} x^{4}}{4} + x^{3} \left (\frac{2 b c_{1}}{3} + \frac{b_{1} c}{3}\right ) + x^{2} \left (\frac{a c_{1}}{2} + b b_{1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x**2+2*b*x+a),x)

[Out]

a*b1*x + c*c1*x**4/4 + x**3*(2*b*c1/3 + b1*c/3) + x**2*(a*c1/2 + b*b1)

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Giac [A]  time = 1.0673, size = 53, normalized size = 1.2 \begin{align*} \frac{1}{4} \, c c_{1} x^{4} + \frac{1}{3} \, b_{1} c x^{3} + \frac{2}{3} \, b c_{1} x^{3} + b b_{1} x^{2} + \frac{1}{2} \, a c_{1} x^{2} + a b_{1} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x^2+2*b*x+a),x, algorithm="giac")

[Out]

1/4*c*c1*x^4 + 1/3*b1*c*x^3 + 2/3*b*c1*x^3 + b*b1*x^2 + 1/2*a*c1*x^2 + a*b1*x