### 3.1829 $$\int (d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2) \, dx$$

Optimal. Leaf size=39 $\frac{1}{5} (d+e x)^5 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^6}{6 e^2}$

[Out]

((a - (c*d^2)/e^2)*(d + e*x)^5)/5 + (c*d*(d + e*x)^6)/(6*e^2)

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Rubi [A]  time = 0.0180514, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.061, Rules used = {626, 43} $\frac{1}{5} (d+e x)^5 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^6}{6 e^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3*(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)^5)/5 + (c*d*(d + e*x)^6)/(6*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)^3 \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)^4 \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right ) (d+e x)^4}{e}+\frac{c d (d+e x)^5}{e}\right ) \, dx\\ &=\frac{1}{5} \left (a-\frac{c d^2}{e^2}\right ) (d+e x)^5+\frac{c d (d+e x)^6}{6 e^2}\\ \end{align*}

Mathematica [B]  time = 0.0190197, size = 95, normalized size = 2.44 $\frac{1}{30} x \left (6 a e \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+c d x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(x*(6*a*e*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + c*d*x*(15*d^4 + 40*d^3*e*x + 45*d^2*
e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4)))/30

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Maple [B]  time = 0.04, size = 155, normalized size = 4. \begin{align*}{\frac{{e}^{4}dc{x}^{6}}{6}}+{\frac{ \left ( 3\,{d}^{2}{e}^{3}c+{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{d}^{3}{e}^{2}c+3\,d{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) +{e}^{4}ad \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{4}ec+3\,{d}^{2}e \left ( a{e}^{2}+c{d}^{2} \right ) +3\,{d}^{2}{e}^{3}a \right ){x}^{3}}{3}}+{\frac{ \left ({d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) +3\,{d}^{3}{e}^{2}a \right ){x}^{2}}{2}}+{d}^{4}aex \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

1/6*e^4*d*c*x^6+1/5*(3*d^2*e^3*c+e^3*(a*e^2+c*d^2))*x^5+1/4*(3*d^3*e^2*c+3*d*e^2*(a*e^2+c*d^2)+e^4*a*d)*x^4+1/
3*(d^4*e*c+3*d^2*e*(a*e^2+c*d^2)+3*d^2*e^3*a)*x^3+1/2*(d^3*(a*e^2+c*d^2)+3*d^3*e^2*a)*x^2+d^4*a*e*x

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Maxima [B]  time = 1.01701, size = 138, normalized size = 3.54 \begin{align*} \frac{1}{6} \, c d e^{4} x^{6} + a d^{4} e x + \frac{1}{5} \,{\left (4 \, c d^{2} e^{3} + a e^{5}\right )} x^{5} + \frac{1}{2} \,{\left (3 \, c d^{3} e^{2} + 2 \, a d e^{4}\right )} x^{4} + \frac{2}{3} \,{\left (2 \, c d^{4} e + 3 \, a d^{2} e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (c d^{5} + 4 \, a d^{3} e^{2}\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxima")

[Out]

1/6*c*d*e^4*x^6 + a*d^4*e*x + 1/5*(4*c*d^2*e^3 + a*e^5)*x^5 + 1/2*(3*c*d^3*e^2 + 2*a*d*e^4)*x^4 + 2/3*(2*c*d^4
*e + 3*a*d^2*e^3)*x^3 + 1/2*(c*d^5 + 4*a*d^3*e^2)*x^2

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Fricas [B]  time = 1.27457, size = 228, normalized size = 5.85 \begin{align*} \frac{1}{6} x^{6} e^{4} d c + \frac{4}{5} x^{5} e^{3} d^{2} c + \frac{1}{5} x^{5} e^{5} a + \frac{3}{2} x^{4} e^{2} d^{3} c + x^{4} e^{4} d a + \frac{4}{3} x^{3} e d^{4} c + 2 x^{3} e^{3} d^{2} a + \frac{1}{2} x^{2} d^{5} c + 2 x^{2} e^{2} d^{3} a + x e d^{4} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas")

[Out]

1/6*x^6*e^4*d*c + 4/5*x^5*e^3*d^2*c + 1/5*x^5*e^5*a + 3/2*x^4*e^2*d^3*c + x^4*e^4*d*a + 4/3*x^3*e*d^4*c + 2*x^
3*e^3*d^2*a + 1/2*x^2*d^5*c + 2*x^2*e^2*d^3*a + x*e*d^4*a

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Sympy [B]  time = 0.092746, size = 107, normalized size = 2.74 \begin{align*} a d^{4} e x + \frac{c d e^{4} x^{6}}{6} + x^{5} \left (\frac{a e^{5}}{5} + \frac{4 c d^{2} e^{3}}{5}\right ) + x^{4} \left (a d e^{4} + \frac{3 c d^{3} e^{2}}{2}\right ) + x^{3} \left (2 a d^{2} e^{3} + \frac{4 c d^{4} e}{3}\right ) + x^{2} \left (2 a d^{3} e^{2} + \frac{c d^{5}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

a*d**4*e*x + c*d*e**4*x**6/6 + x**5*(a*e**5/5 + 4*c*d**2*e**3/5) + x**4*(a*d*e**4 + 3*c*d**3*e**2/2) + x**3*(2
*a*d**2*e**3 + 4*c*d**4*e/3) + x**2*(2*a*d**3*e**2 + c*d**5/2)

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Giac [B]  time = 1.23442, size = 132, normalized size = 3.38 \begin{align*} \frac{1}{6} \, c d x^{6} e^{4} + \frac{4}{5} \, c d^{2} x^{5} e^{3} + \frac{3}{2} \, c d^{3} x^{4} e^{2} + \frac{4}{3} \, c d^{4} x^{3} e + \frac{1}{2} \, c d^{5} x^{2} + \frac{1}{5} \, a x^{5} e^{5} + a d x^{4} e^{4} + 2 \, a d^{2} x^{3} e^{3} + 2 \, a d^{3} x^{2} e^{2} + a d^{4} x e \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

[Out]

1/6*c*d*x^6*e^4 + 4/5*c*d^2*x^5*e^3 + 3/2*c*d^3*x^4*e^2 + 4/3*c*d^4*x^3*e + 1/2*c*d^5*x^2 + 1/5*a*x^5*e^5 + a*
d*x^4*e^4 + 2*a*d^2*x^3*e^3 + 2*a*d^3*x^2*e^2 + a*d^4*x*e