### 3.1839 $$\int (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^2 \, dx$$

Optimal. Leaf size=77 $-\frac{c d (d+e x)^6 \left (c d^2-a e^2\right )}{3 e^3}+\frac{(d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^3}+\frac{c^2 d^2 (d+e x)^7}{7 e^3}$

[Out]

((c*d^2 - a*e^2)^2*(d + e*x)^5)/(5*e^3) - (c*d*(c*d^2 - a*e^2)*(d + e*x)^6)/(3*e^3) + (c^2*d^2*(d + e*x)^7)/(7
*e^3)

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Rubi [A]  time = 0.131867, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.057, Rules used = {626, 43} $-\frac{c d (d+e x)^6 \left (c d^2-a e^2\right )}{3 e^3}+\frac{(d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^3}+\frac{c^2 d^2 (d+e x)^7}{7 e^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2*(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)^5)/(5*e^3) - (c*d*(c*d^2 - a*e^2)*(d + e*x)^6)/(3*e^3) + (c^2*d^2*(d + e*x)^7)/(7
*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)^2 \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)^4 \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2 (d+e x)^4}{e^2}-\frac{2 c d \left (c d^2-a e^2\right ) (d+e x)^5}{e^2}+\frac{c^2 d^2 (d+e x)^6}{e^2}\right ) \, dx\\ &=\frac{\left (c d^2-a e^2\right )^2 (d+e x)^5}{5 e^3}-\frac{c d \left (c d^2-a e^2\right ) (d+e x)^6}{3 e^3}+\frac{c^2 d^2 (d+e x)^7}{7 e^3}\\ \end{align*}

Mathematica [B]  time = 0.0315438, size = 160, normalized size = 2.08 $a^2 \left (2 d^2 e^4 x^3+2 d^3 e^3 x^2+d^4 e^2 x+d e^5 x^4+\frac{e^6 x^5}{5}\right )+\frac{1}{15} a c d e x^2 \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+\frac{1}{105} c^2 d^2 x^3 \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a*c*d*e*x^2*(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))/15 + (c^2*d^2*x^3*(35*d^4 + 10
5*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4))/105 + a^2*(d^4*e^2*x + 2*d^3*e^3*x^2 + 2*d^2*e^4*x^3
+ d*e^5*x^4 + (e^6*x^5)/5)

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Maple [B]  time = 0.041, size = 295, normalized size = 3.8 \begin{align*}{\frac{{e}^{4}{d}^{2}{c}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{d}^{3}{e}^{3}{c}^{2}+2\,{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) dc \right ){x}^{6}}{6}}+{\frac{ \left ({d}^{4}{e}^{2}{c}^{2}+4\,{d}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) c+{e}^{2} \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) ec+2\,de \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\,{e}^{3}ad \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{2} \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +4\,{d}^{2}{e}^{2}a \left ( a{e}^{2}+c{d}^{2} \right ) +{e}^{4}{a}^{2}{d}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{d}^{3}ae \left ( a{e}^{2}+c{d}^{2} \right ) +2\,{d}^{3}{e}^{3}{a}^{2} \right ){x}^{2}}{2}}+{d}^{4}{a}^{2}{e}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*e^4*d^2*c^2*x^7+1/6*(2*d^3*e^3*c^2+2*e^3*(a*e^2+c*d^2)*d*c)*x^6+1/5*(d^4*e^2*c^2+4*d^2*e^2*(a*e^2+c*d^2)*c
+e^2*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2))*x^5+1/4*(2*d^3*(a*e^2+c*d^2)*e*c+2*d*e*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2)+2
*e^3*a*d*(a*e^2+c*d^2))*x^4+1/3*(d^2*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2)+4*d^2*e^2*a*(a*e^2+c*d^2)+e^4*a^2*d^2)*x^
3+1/2*(2*d^3*a*e*(a*e^2+c*d^2)+2*d^3*e^3*a^2)*x^2+d^4*a^2*e^2*x

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Maxima [B]  time = 1.13418, size = 232, normalized size = 3.01 \begin{align*} \frac{1}{7} \, c^{2} d^{2} e^{4} x^{7} + a^{2} d^{4} e^{2} x + \frac{1}{3} \,{\left (2 \, c^{2} d^{3} e^{3} + a c d e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, c^{2} d^{4} e^{2} + 8 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{5} +{\left (c^{2} d^{5} e + 3 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x^{4} + \frac{1}{3} \,{\left (c^{2} d^{6} + 8 \, a c d^{4} e^{2} + 6 \, a^{2} d^{2} e^{4}\right )} x^{3} +{\left (a c d^{5} e + 2 \, a^{2} d^{3} e^{3}\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c^2*d^2*e^4*x^7 + a^2*d^4*e^2*x + 1/3*(2*c^2*d^3*e^3 + a*c*d*e^5)*x^6 + 1/5*(6*c^2*d^4*e^2 + 8*a*c*d^2*e^4
+ a^2*e^6)*x^5 + (c^2*d^5*e + 3*a*c*d^3*e^3 + a^2*d*e^5)*x^4 + 1/3*(c^2*d^6 + 8*a*c*d^4*e^2 + 6*a^2*d^2*e^4)*
x^3 + (a*c*d^5*e + 2*a^2*d^3*e^3)*x^2

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Fricas [B]  time = 1.38587, size = 390, normalized size = 5.06 \begin{align*} \frac{1}{7} x^{7} e^{4} d^{2} c^{2} + \frac{2}{3} x^{6} e^{3} d^{3} c^{2} + \frac{1}{3} x^{6} e^{5} d c a + \frac{6}{5} x^{5} e^{2} d^{4} c^{2} + \frac{8}{5} x^{5} e^{4} d^{2} c a + \frac{1}{5} x^{5} e^{6} a^{2} + x^{4} e d^{5} c^{2} + 3 x^{4} e^{3} d^{3} c a + x^{4} e^{5} d a^{2} + \frac{1}{3} x^{3} d^{6} c^{2} + \frac{8}{3} x^{3} e^{2} d^{4} c a + 2 x^{3} e^{4} d^{2} a^{2} + x^{2} e d^{5} c a + 2 x^{2} e^{3} d^{3} a^{2} + x e^{2} d^{4} a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*e^4*d^2*c^2 + 2/3*x^6*e^3*d^3*c^2 + 1/3*x^6*e^5*d*c*a + 6/5*x^5*e^2*d^4*c^2 + 8/5*x^5*e^4*d^2*c*a + 1/
5*x^5*e^6*a^2 + x^4*e*d^5*c^2 + 3*x^4*e^3*d^3*c*a + x^4*e^5*d*a^2 + 1/3*x^3*d^6*c^2 + 8/3*x^3*e^2*d^4*c*a + 2*
x^3*e^4*d^2*a^2 + x^2*e*d^5*c*a + 2*x^2*e^3*d^3*a^2 + x*e^2*d^4*a^2

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Sympy [B]  time = 0.135621, size = 185, normalized size = 2.4 \begin{align*} a^{2} d^{4} e^{2} x + \frac{c^{2} d^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac{a c d e^{5}}{3} + \frac{2 c^{2} d^{3} e^{3}}{3}\right ) + x^{5} \left (\frac{a^{2} e^{6}}{5} + \frac{8 a c d^{2} e^{4}}{5} + \frac{6 c^{2} d^{4} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{5} + 3 a c d^{3} e^{3} + c^{2} d^{5} e\right ) + x^{3} \left (2 a^{2} d^{2} e^{4} + \frac{8 a c d^{4} e^{2}}{3} + \frac{c^{2} d^{6}}{3}\right ) + x^{2} \left (2 a^{2} d^{3} e^{3} + a c d^{5} e\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d**4*e**2*x + c**2*d**2*e**4*x**7/7 + x**6*(a*c*d*e**5/3 + 2*c**2*d**3*e**3/3) + x**5*(a**2*e**6/5 + 8*a*
c*d**2*e**4/5 + 6*c**2*d**4*e**2/5) + x**4*(a**2*d*e**5 + 3*a*c*d**3*e**3 + c**2*d**5*e) + x**3*(2*a**2*d**2*e
**4 + 8*a*c*d**4*e**2/3 + c**2*d**6/3) + x**2*(2*a**2*d**3*e**3 + a*c*d**5*e)

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Giac [B]  time = 1.20042, size = 238, normalized size = 3.09 \begin{align*} \frac{1}{7} \, c^{2} d^{2} x^{7} e^{4} + \frac{2}{3} \, c^{2} d^{3} x^{6} e^{3} + \frac{6}{5} \, c^{2} d^{4} x^{5} e^{2} + c^{2} d^{5} x^{4} e + \frac{1}{3} \, c^{2} d^{6} x^{3} + \frac{1}{3} \, a c d x^{6} e^{5} + \frac{8}{5} \, a c d^{2} x^{5} e^{4} + 3 \, a c d^{3} x^{4} e^{3} + \frac{8}{3} \, a c d^{4} x^{3} e^{2} + a c d^{5} x^{2} e + \frac{1}{5} \, a^{2} x^{5} e^{6} + a^{2} d x^{4} e^{5} + 2 \, a^{2} d^{2} x^{3} e^{4} + 2 \, a^{2} d^{3} x^{2} e^{3} + a^{2} d^{4} x e^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c^2*d^2*x^7*e^4 + 2/3*c^2*d^3*x^6*e^3 + 6/5*c^2*d^4*x^5*e^2 + c^2*d^5*x^4*e + 1/3*c^2*d^6*x^3 + 1/3*a*c*d*
x^6*e^5 + 8/5*a*c*d^2*x^5*e^4 + 3*a*c*d^3*x^4*e^3 + 8/3*a*c*d^4*x^3*e^2 + a*c*d^5*x^2*e + 1/5*a^2*x^5*e^6 + a^
2*d*x^4*e^5 + 2*a^2*d^2*x^3*e^4 + 2*a^2*d^3*x^2*e^3 + a^2*d^4*x*e^2