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

Optimal. Leaf size=55 \[ \frac{1}{4} e x^4 (b e+2 c d)+\frac{1}{3} d x^3 (2 b e+c d)+\frac{1}{2} b d^2 x^2+\frac{1}{5} c e^2 x^5 \]

[Out]

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

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Rubi [A]  time = 0.0376067, antiderivative size = 55, 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 = {698} \[ \frac{1}{4} e x^4 (b e+2 c d)+\frac{1}{3} d x^3 (2 b e+c d)+\frac{1}{2} b d^2 x^2+\frac{1}{5} c e^2 x^5 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(b*x + c*x^2),x]

[Out]

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

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0119075, size = 49, normalized size = 0.89 \[ \frac{1}{60} x^2 \left (15 e x^2 (b e+2 c d)+20 d x (2 b e+c d)+30 b d^2+12 c e^2 x^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 52, normalized size = 1. \begin{align*}{\frac{c{e}^{2}{x}^{5}}{5}}+{\frac{ \left ({e}^{2}b+2\,dec \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,bde+c{d}^{2} \right ){x}^{3}}{3}}+{\frac{b{d}^{2}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10459, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{5} \, c e^{2} x^{5} + \frac{1}{2} \, b d^{2} x^{2} + \frac{1}{4} \,{\left (2 \, c d e + b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{2} + 2 \, b d e\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.47921, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{5} x^{5} e^{2} c + \frac{1}{2} x^{4} e d c + \frac{1}{4} x^{4} e^{2} b + \frac{1}{3} x^{3} d^{2} c + \frac{2}{3} x^{3} e d b + \frac{1}{2} x^{2} d^{2} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.118933, size = 54, normalized size = 0.98 \begin{align*} \frac{b d^{2} x^{2}}{2} + \frac{c e^{2} x^{5}}{5} + x^{4} \left (\frac{b e^{2}}{4} + \frac{c d e}{2}\right ) + x^{3} \left (\frac{2 b d e}{3} + \frac{c d^{2}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.3394, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{5} \, c x^{5} e^{2} + \frac{1}{2} \, c d x^{4} e + \frac{1}{3} \, c d^{2} x^{3} + \frac{1}{4} \, b x^{4} e^{2} + \frac{2}{3} \, b d x^{3} e + \frac{1}{2} \, b d^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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