### 3.1840 $$\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^2 \, dx$$

Optimal. Leaf size=77 $-\frac{2 c d (d+e x)^5 \left (c d^2-a e^2\right )}{5 e^3}+\frac{(d+e x)^4 \left (c d^2-a e^2\right )^2}{4 e^3}+\frac{c^2 d^2 (d+e x)^6}{6 e^3}$

[Out]

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

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Rubi [A]  time = 0.0792925, antiderivative size = 77, 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} $-\frac{2 c d (d+e x)^5 \left (c d^2-a e^2\right )}{5 e^3}+\frac{(d+e x)^4 \left (c d^2-a e^2\right )^2}{4 e^3}+\frac{c^2 d^2 (d+e x)^6}{6 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0323462, size = 120, normalized size = 1.56 $\frac{1}{60} x \left (15 a^2 e^2 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+6 a c d e x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+c^2 d^2 x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(15*a^2*e^2*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 6*a*c*d*e*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 +
4*e^3*x^3) + c^2*d^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3)))/60

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Maple [B]  time = 0.039, size = 195, normalized size = 2.5 \begin{align*}{\frac{{d}^{2}{e}^{3}{c}^{2}{x}^{6}}{6}}+{\frac{ \left ({d}^{3}{e}^{2}{c}^{2}+2\,{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) dc \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) ec+e \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\,{e}^{2}ad \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{d}^{2}ae \left ( a{e}^{2}+c{d}^{2} \right ) +{a}^{2}{d}^{2}{e}^{3} \right ){x}^{2}}{2}}+{a}^{2}{d}^{3}{e}^{2}x \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)^2,x)

[Out]

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

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Maxima [A]  time = 1.63677, size = 189, normalized size = 2.45 \begin{align*} \frac{1}{6} \, c^{2} d^{2} e^{3} x^{6} + a^{2} d^{3} e^{2} x + \frac{1}{5} \,{\left (3 \, c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x^{4} + \frac{1}{3} \,{\left (c^{2} d^{5} + 6 \, a c d^{3} e^{2} + 3 \, a^{2} d e^{4}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a c d^{4} e + 3 \, a^{2} d^{2} e^{3}\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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.42418, size = 312, normalized size = 4.05 \begin{align*} \frac{1}{6} x^{6} e^{3} d^{2} c^{2} + \frac{3}{5} x^{5} e^{2} d^{3} c^{2} + \frac{2}{5} x^{5} e^{4} d c a + \frac{3}{4} x^{4} e d^{4} c^{2} + \frac{3}{2} x^{4} e^{3} d^{2} c a + \frac{1}{4} x^{4} e^{5} a^{2} + \frac{1}{3} x^{3} d^{5} c^{2} + 2 x^{3} e^{2} d^{3} c a + x^{3} e^{4} d a^{2} + x^{2} e d^{4} c a + \frac{3}{2} x^{2} e^{3} d^{2} a^{2} + x e^{2} d^{3} a^{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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.107471, size = 150, normalized size = 1.95 \begin{align*} a^{2} d^{3} e^{2} x + \frac{c^{2} d^{2} e^{3} x^{6}}{6} + x^{5} \left (\frac{2 a c d e^{4}}{5} + \frac{3 c^{2} d^{3} e^{2}}{5}\right ) + x^{4} \left (\frac{a^{2} e^{5}}{4} + \frac{3 a c d^{2} e^{3}}{2} + \frac{3 c^{2} d^{4} e}{4}\right ) + x^{3} \left (a^{2} d e^{4} + 2 a c d^{3} e^{2} + \frac{c^{2} d^{5}}{3}\right ) + x^{2} \left (\frac{3 a^{2} d^{2} e^{3}}{2} + a c d^{4} e\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)**2,x)

[Out]

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

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Giac [A]  time = 1.15863, size = 188, normalized size = 2.44 \begin{align*} \frac{1}{6} \, c^{2} d^{2} x^{6} e^{3} + \frac{3}{5} \, c^{2} d^{3} x^{5} e^{2} + \frac{3}{4} \, c^{2} d^{4} x^{4} e + \frac{1}{3} \, c^{2} d^{5} x^{3} + \frac{2}{5} \, a c d x^{5} e^{4} + \frac{3}{2} \, a c d^{2} x^{4} e^{3} + 2 \, a c d^{3} x^{3} e^{2} + a c d^{4} x^{2} e + \frac{1}{4} \, a^{2} x^{4} e^{5} + a^{2} d x^{3} e^{4} + \frac{3}{2} \, a^{2} d^{2} x^{2} e^{3} + a^{2} d^{3} x e^{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)^2,x, algorithm="giac")

[Out]

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