### 3.1853 $$\int (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx$$

Optimal. Leaf size=111 $-\frac{c^2 d^2 (d+e x)^6 \left (c d^2-a e^2\right )}{2 e^4}+\frac{3 c d (d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^4}-\frac{(d+e x)^4 \left (c d^2-a e^2\right )^3}{4 e^4}+\frac{c^3 d^3 (d+e x)^7}{7 e^4}$

[Out]

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

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Rubi [A]  time = 0.148613, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.074, Rules used = {610, 43} $-\frac{c^2 d^2 (d+e x)^6 \left (c d^2-a e^2\right )}{2 e^4}+\frac{3 c d (d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^4}-\frac{(d+e x)^4 \left (c d^2-a e^2\right )^3}{4 e^4}+\frac{c^3 d^3 (d+e x)^7}{7 e^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 610

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[1/c^p, Int[Simp[
b/2 - q/2 + c*x, x]^p*Simp[b/2 + q/2 + c*x, x]^p, x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGt
Q[p, 0] && PerfectSquareQ[b^2 - 4*a*c]

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

Mathematica [A]  time = 0.0485335, size = 167, normalized size = 1.5 $\frac{1}{140} x \left (21 a^2 c d e^2 x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+35 a^3 e^3 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+7 a c^2 d^2 e x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+c^3 d^3 x^3 \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(35*a^3*e^3*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 21*a^2*c*d*e^2*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*
x^2 + 4*e^3*x^3) + 7*a*c^2*d^2*e*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + c^3*d^3*x^3*(35*d^3 +
84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3)))/140

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Maple [B]  time = 0.038, size = 266, normalized size = 2.4 \begin{align*}{\frac{{d}^{3}{e}^{3}{c}^{3}{x}^{7}}{7}}+{\frac{ \left ( a{e}^{2}+c{d}^{2} \right ){d}^{2}{e}^{2}{c}^{2}{x}^{6}}{2}}+{\frac{ \left ( a{d}^{3}{e}^{3}{c}^{2}+2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}dec+dec \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,a{d}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) c+ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( ade \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}ade+{d}^{3}{e}^{3}c{a}^{2} \right ){x}^{3}}{3}}+{\frac{3\,{a}^{2}{d}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{3}{e}^{3}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10195, size = 247, normalized size = 2.23 \begin{align*} \frac{1}{7} \, c^{3} d^{3} e^{3} x^{7} + \frac{1}{2} \,{\left (c d^{2} + a e^{2}\right )} c^{2} d^{2} e^{2} x^{6} + a^{3} d^{3} e^{3} x + \frac{3}{5} \,{\left (c d^{2} + a e^{2}\right )}^{2} c d e x^{5} + \frac{1}{2} \,{\left (2 \, c d e x^{3} + 3 \,{\left (c d^{2} + a e^{2}\right )} x^{2}\right )} a^{2} d^{2} e^{2} + \frac{1}{4} \,{\left (c d^{2} + a e^{2}\right )}^{3} x^{4} + \frac{1}{10} \,{\left (6 \, c^{2} d^{2} e^{2} x^{5} + 15 \,{\left (c d^{2} + a e^{2}\right )} c d e x^{4} + 10 \,{\left (c d^{2} + a e^{2}\right )}^{2} x^{3}\right )} a d e \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c^3*d^3*e^3*x^7 + 1/2*(c*d^2 + a*e^2)*c^2*d^2*e^2*x^6 + a^3*d^3*e^3*x + 3/5*(c*d^2 + a*e^2)^2*c*d*e*x^5 +
1/2*(2*c*d*e*x^3 + 3*(c*d^2 + a*e^2)*x^2)*a^2*d^2*e^2 + 1/4*(c*d^2 + a*e^2)^3*x^4 + 1/10*(6*c^2*d^2*e^2*x^5 +
15*(c*d^2 + a*e^2)*c*d*e*x^4 + 10*(c*d^2 + a*e^2)^2*x^3)*a*d*e

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Fricas [B]  time = 1.37993, size = 452, normalized size = 4.07 \begin{align*} \frac{1}{7} x^{7} e^{3} d^{3} c^{3} + \frac{1}{2} x^{6} e^{2} d^{4} c^{3} + \frac{1}{2} x^{6} e^{4} d^{2} c^{2} a + \frac{3}{5} x^{5} e d^{5} c^{3} + \frac{9}{5} x^{5} e^{3} d^{3} c^{2} a + \frac{3}{5} x^{5} e^{5} d c a^{2} + \frac{1}{4} x^{4} d^{6} c^{3} + \frac{9}{4} x^{4} e^{2} d^{4} c^{2} a + \frac{9}{4} x^{4} e^{4} d^{2} c a^{2} + \frac{1}{4} x^{4} e^{6} a^{3} + x^{3} e d^{5} c^{2} a + 3 x^{3} e^{3} d^{3} c a^{2} + x^{3} e^{5} d a^{3} + \frac{3}{2} x^{2} e^{2} d^{4} c a^{2} + \frac{3}{2} x^{2} e^{4} d^{2} a^{3} + x e^{3} d^{3} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*e^3*d^3*c^3 + 1/2*x^6*e^2*d^4*c^3 + 1/2*x^6*e^4*d^2*c^2*a + 3/5*x^5*e*d^5*c^3 + 9/5*x^5*e^3*d^3*c^2*a
+ 3/5*x^5*e^5*d*c*a^2 + 1/4*x^4*d^6*c^3 + 9/4*x^4*e^2*d^4*c^2*a + 9/4*x^4*e^4*d^2*c*a^2 + 1/4*x^4*e^6*a^3 + x^
3*e*d^5*c^2*a + 3*x^3*e^3*d^3*c*a^2 + x^3*e^5*d*a^3 + 3/2*x^2*e^2*d^4*c*a^2 + 3/2*x^2*e^4*d^2*a^3 + x*e^3*d^3*
a^3

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Sympy [B]  time = 0.145143, size = 218, normalized size = 1.96 \begin{align*} a^{3} d^{3} e^{3} x + \frac{c^{3} d^{3} e^{3} x^{7}}{7} + x^{6} \left (\frac{a c^{2} d^{2} e^{4}}{2} + \frac{c^{3} d^{4} e^{2}}{2}\right ) + x^{5} \left (\frac{3 a^{2} c d e^{5}}{5} + \frac{9 a c^{2} d^{3} e^{3}}{5} + \frac{3 c^{3} d^{5} e}{5}\right ) + x^{4} \left (\frac{a^{3} e^{6}}{4} + \frac{9 a^{2} c d^{2} e^{4}}{4} + \frac{9 a c^{2} d^{4} e^{2}}{4} + \frac{c^{3} d^{6}}{4}\right ) + x^{3} \left (a^{3} d e^{5} + 3 a^{2} c d^{3} e^{3} + a c^{2} d^{5} e\right ) + x^{2} \left (\frac{3 a^{3} d^{2} e^{4}}{2} + \frac{3 a^{2} c d^{4} e^{2}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*d**3*e**3*x + c**3*d**3*e**3*x**7/7 + x**6*(a*c**2*d**2*e**4/2 + c**3*d**4*e**2/2) + x**5*(3*a**2*c*d*e**
5/5 + 9*a*c**2*d**3*e**3/5 + 3*c**3*d**5*e/5) + x**4*(a**3*e**6/4 + 9*a**2*c*d**2*e**4/4 + 9*a*c**2*d**4*e**2/
4 + c**3*d**6/4) + x**3*(a**3*d*e**5 + 3*a**2*c*d**3*e**3 + a*c**2*d**5*e) + x**2*(3*a**3*d**2*e**4/2 + 3*a**2
*c*d**4*e**2/2)

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Giac [A]  time = 1.16336, size = 274, normalized size = 2.47 \begin{align*} \frac{1}{7} \, c^{3} d^{3} x^{7} e^{3} + \frac{1}{2} \, c^{3} d^{4} x^{6} e^{2} + \frac{3}{5} \, c^{3} d^{5} x^{5} e + \frac{1}{4} \, c^{3} d^{6} x^{4} + \frac{1}{2} \, a c^{2} d^{2} x^{6} e^{4} + \frac{9}{5} \, a c^{2} d^{3} x^{5} e^{3} + \frac{9}{4} \, a c^{2} d^{4} x^{4} e^{2} + a c^{2} d^{5} x^{3} e + \frac{3}{5} \, a^{2} c d x^{5} e^{5} + \frac{9}{4} \, a^{2} c d^{2} x^{4} e^{4} + 3 \, a^{2} c d^{3} x^{3} e^{3} + \frac{3}{2} \, a^{2} c d^{4} x^{2} e^{2} + \frac{1}{4} \, a^{3} x^{4} e^{6} + a^{3} d x^{3} e^{5} + \frac{3}{2} \, a^{3} d^{2} x^{2} e^{4} + a^{3} d^{3} x e^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c^3*d^3*x^7*e^3 + 1/2*c^3*d^4*x^6*e^2 + 3/5*c^3*d^5*x^5*e + 1/4*c^3*d^6*x^4 + 1/2*a*c^2*d^2*x^6*e^4 + 9/5*
a*c^2*d^3*x^5*e^3 + 9/4*a*c^2*d^4*x^4*e^2 + a*c^2*d^5*x^3*e + 3/5*a^2*c*d*x^5*e^5 + 9/4*a^2*c*d^2*x^4*e^4 + 3*
a^2*c*d^3*x^3*e^3 + 3/2*a^2*c*d^4*x^2*e^2 + 1/4*a^3*x^4*e^6 + a^3*d*x^3*e^5 + 3/2*a^3*d^2*x^2*e^4 + a^3*d^3*x*
e^3