3.2355 \(\int x (d+e x) (a+b x+c x^2) \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{3} x^3 (a e+b d)+\frac{1}{2} a d x^2+\frac{1}{4} x^4 (b e+c d)+\frac{1}{5} c e x^5 \]

[Out]

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

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Rubi [A]  time = 0.0373681, antiderivative size = 47, 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 = {765} \[ \frac{1}{3} x^3 (a e+b d)+\frac{1}{2} a d x^2+\frac{1}{4} x^4 (b e+c d)+\frac{1}{5} c e x^5 \]

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x)*(a + b*x + c*x^2),x]

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0143395, size = 41, normalized size = 0.87 \[ \frac{1}{60} x^2 \left (20 x (a e+b d)+30 a d+15 x^2 (b e+c d)+12 c e x^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(30*a*d + 20*(b*d + a*e)*x + 15*(c*d + b*e)*x^2 + 12*c*e*x^3))/60

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Maple [A]  time = 0.001, size = 40, normalized size = 0.9 \begin{align*}{\frac{ad{x}^{2}}{2}}+{\frac{ \left ( ae+bd \right ){x}^{3}}{3}}+{\frac{ \left ( be+cd \right ){x}^{4}}{4}}+{\frac{ce{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08952, size = 53, normalized size = 1.13 \begin{align*} \frac{1}{5} \, c e x^{5} + \frac{1}{4} \,{\left (c d + b e\right )} x^{4} + \frac{1}{2} \, a d x^{2} + \frac{1}{3} \,{\left (b d + a e\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.12767, size = 112, normalized size = 2.38 \begin{align*} \frac{1}{5} x^{5} e c + \frac{1}{4} x^{4} d c + \frac{1}{4} x^{4} e b + \frac{1}{3} x^{3} d b + \frac{1}{3} x^{3} e a + \frac{1}{2} x^{2} d a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.062983, size = 42, normalized size = 0.89 \begin{align*} \frac{a d x^{2}}{2} + \frac{c e x^{5}}{5} + x^{4} \left (\frac{b e}{4} + \frac{c d}{4}\right ) + x^{3} \left (\frac{a e}{3} + \frac{b d}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13966, size = 62, normalized size = 1.32 \begin{align*} \frac{1}{5} \, c x^{5} e + \frac{1}{4} \, c d x^{4} + \frac{1}{4} \, b x^{4} e + \frac{1}{3} \, b d x^{3} + \frac{1}{3} \, a x^{3} e + \frac{1}{2} \, a d x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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