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

Optimal. Leaf size=17 \[ \frac {1}{2} d \left (a+b x+c x^2\right )^2 \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {629} \begin {gather*} \frac {1}{2} d \left (a+b x+c x^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.29 \begin {gather*} \frac {1}{2} d x (b+c x) (2 a+x (b+c x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*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 d+2 c d x) \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.35, size = 38, normalized size = 2.24 \begin {gather*} \frac {1}{2} x^{4} d c^{2} + x^{3} d c b + \frac {1}{2} x^{2} d b^{2} + x^{2} d c a + x d b a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.15, size = 32, normalized size = 1.88 \begin {gather*} {\left (c d x^{2} + b d x\right )} a + \frac {{\left (c d x^{2} + b d x\right )}^{2}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 39, normalized size = 2.29 \begin {gather*} \frac {c^{2} d \,x^{4}}{2}+b c d \,x^{3}+a b d x +\frac {\left (2 a c d +b^{2} d \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.43, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{2} \, {\left (c x^{2} + b x + a\right )}^{2} d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^2 + b*x + a)^2*d

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mupad [B]  time = 0.05, size = 36, normalized size = 2.12 \begin {gather*} \frac {c^2\,d\,x^4}{2}+\frac {d\,x^2\,\left (b^2+2\,a\,c\right )}{2}+b\,c\,d\,x^3+a\,b\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.07, size = 39, normalized size = 2.29 \begin {gather*} a b d x + b c d x^{3} + \frac {c^{2} d x^{4}}{2} + x^{2} \left (a c d + \frac {b^{2} d}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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