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

Optimal. Leaf size=33 \[ \frac {1}{3} x^3 (b e+c d)+\frac {1}{2} b d x^2+\frac {1}{4} c e x^4 \]

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {631} \begin {gather*} \frac {1}{3} x^3 (b e+c d)+\frac {1}{2} b d x^2+\frac {1}{4} c e x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(b*x + c*x^2),x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{12} x^2 (b (6 d+4 e x)+c x (4 d+3 e x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*(b*x + c*x^2),x]

[Out]

(x^2*(c*x*(4*d + 3*e*x) + b*(6*d + 4*e*x)))/12

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.36, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{4} x^{4} e c + \frac {1}{3} x^{3} d c + \frac {1}{3} x^{3} e b + \frac {1}{2} x^{2} d b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 31, normalized size = 0.94 \begin {gather*} \frac {1}{4} \, c x^{4} e + \frac {1}{3} \, c d x^{3} + \frac {1}{3} \, b x^{3} e + \frac {1}{2} \, b d x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 28, normalized size = 0.85 \begin {gather*} \frac {c e \,x^{4}}{4}+\frac {b d \,x^{2}}{2}+\frac {\left (b e +c d \right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{4} \, c e x^{4} + \frac {1}{2} \, b d x^{2} + \frac {1}{3} \, {\left (c d + b e\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 28, normalized size = 0.85 \begin {gather*} \frac {c\,e\,x^4}{4}+\left (\frac {b\,e}{3}+\frac {c\,d}{3}\right )\,x^3+\frac {b\,d\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)*(d + e*x),x)

[Out]

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

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sympy [A]  time = 0.07, size = 29, normalized size = 0.88 \begin {gather*} \frac {b d x^{2}}{2} + \frac {c e x^{4}}{4} + x^{3} \left (\frac {b e}{3} + \frac {c d}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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