3.2109 \(\int (d+e x)^2 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=69 \[ \frac{(d+e x)^3 \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac{(d+e x)^4 (2 c d-b e)}{4 e^3}+\frac{c (d+e x)^5}{5 e^3} \]

[Out]

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

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Rubi [A]  time = 0.0538831, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {698} \[ \frac{(d+e x)^3 \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac{(d+e x)^4 (2 c d-b e)}{4 e^3}+\frac{c (d+e x)^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + b*x + c*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.020477, size = 73, normalized size = 1.06 \[ \frac{1}{3} x^3 \left (a e^2+2 b d e+c d^2\right )+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 x+\frac{1}{4} e x^4 (b e+2 c d)+\frac{1}{5} c e^2 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 70, 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 ( a{e}^{2}+2\,bde+c{d}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,ade+{d}^{2}b \right ){x}^{2}}{2}}+a{d}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.99918, size = 93, normalized size = 1.35 \begin{align*} \frac{1}{5} \, c e^{2} x^{5} + \frac{1}{4} \,{\left (2 \, c d e + b e^{2}\right )} x^{4} + a d^{2} x + \frac{1}{3} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b d^{2} + 2 \, a d e\right )} 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+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.77474, size = 180, normalized size = 2.61 \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}{3} x^{3} e^{2} a + \frac{1}{2} x^{2} d^{2} b + x^{2} e d a + x d^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a),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 + 1/3*x^3*e^2*a + 1/2*x^2*d^2*b
+ x^2*e*d*a + x*d^2*a

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Sympy [A]  time = 0.072641, size = 73, normalized size = 1.06 \begin{align*} a d^{2} x + \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{a e^{2}}{3} + \frac{2 b d e}{3} + \frac{c d^{2}}{3}\right ) + x^{2} \left (a d e + \frac{b d^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10094, size = 101, normalized size = 1.46 \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} + \frac{1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a),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 + 1/3*a*x^3*e^2
+ a*d*x^2*e + a*d^2*x