3.1831 \(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{3} (d+e x)^3 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^4}{4 e^2} \]

[Out]

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

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Rubi [A]  time = 0.033846, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {626, 43} \[ \frac{1}{3} (d+e x)^3 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^4}{4 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 626

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.011793, size = 51, normalized size = 1.31 \[ \frac{1}{12} x \left (4 a e \left (3 d^2+3 d e x+e^2 x^2\right )+c d x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(4*a*e*(3*d^2 + 3*d*e*x + e^2*x^2) + c*d*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)))/12

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Maple [A]  time = 0.039, size = 69, normalized size = 1.8 \begin{align*}{\frac{d{e}^{2}c{x}^{4}}{4}}+{\frac{ \left ({d}^{2}ec+e \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( d \left ( a{e}^{2}+c{d}^{2} \right ) +ad{e}^{2} \right ){x}^{2}}{2}}+a{d}^{2}ex \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02497, size = 73, normalized size = 1.87 \begin{align*} \frac{1}{4} \, c d e^{2} x^{4} + a d^{2} e x + \frac{1}{3} \,{\left (2 \, c d^{2} e + a e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (c d^{3} + 2 \, a d e^{2}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.3799, size = 126, normalized size = 3.23 \begin{align*} \frac{1}{4} x^{4} e^{2} d c + \frac{2}{3} x^{3} e d^{2} c + \frac{1}{3} x^{3} e^{3} a + \frac{1}{2} x^{2} d^{3} c + x^{2} e^{2} d a + x e d^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.1642, size = 56, normalized size = 1.44 \begin{align*} a d^{2} e x + \frac{c d e^{2} x^{4}}{4} + x^{3} \left (\frac{a e^{3}}{3} + \frac{2 c d^{2} e}{3}\right ) + x^{2} \left (a d e^{2} + \frac{c d^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16421, size = 73, normalized size = 1.87 \begin{align*} \frac{1}{4} \, c d x^{4} e^{2} + \frac{2}{3} \, c d^{2} x^{3} e + \frac{1}{2} \, c d^{3} x^{2} + \frac{1}{3} \, a x^{3} e^{3} + a d x^{2} e^{2} + a d^{2} x e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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