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

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {629} \begin {gather*} \frac {1}{2} \left (a+b x+c x^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(a + b*x + c*x^2),x]

[Out]

(a + b*x + c*x^2)^2/2

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

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.31 \begin {gather*} \frac {1}{2} x (b+c x) (2 a+x (b+c x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b + c*x)*(2*a + x*(b + c*x)))/2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (b+2 c x) \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.37, size = 33, normalized size = 2.06 \begin {gather*} \frac {1}{2} x^{4} c^{2} + x^{3} c b + \frac {1}{2} x^{2} b^{2} + x^{2} c a + x b a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 25, normalized size = 1.56 \begin {gather*} \frac {1}{2} \, {\left (c x^{2} + b x\right )}^{2} + {\left (c x^{2} + b x\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^2 + b*x)^2 + (c*x^2 + b*x)*a

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maple [B]  time = 0.04, size = 33, normalized size = 2.06 \begin {gather*} \frac {c^{2} x^{4}}{2}+b c \,x^{3}+a b x +\frac {\left (2 a c +b^{2}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.60, size = 14, normalized size = 0.88 \begin {gather*} \frac {1}{2} \, {\left (c x^{2} + b x + a\right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^2 + b*x + a)^2

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mupad [B]  time = 0.04, size = 32, normalized size = 2.00 \begin {gather*} x^2\,\left (\frac {b^2}{2}+a\,c\right )+\frac {c^2\,x^4}{2}+a\,b\,x+b\,c\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)*(a + b*x + c*x^2),x)

[Out]

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

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sympy [B]  time = 0.07, size = 31, normalized size = 1.94 \begin {gather*} a b x + b c x^{3} + \frac {c^{2} x^{4}}{2} + x^{2} \left (a c + \frac {b^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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