3.1842 \(\int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx\)

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 = 0.332433, 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 \[ -\frac{c^2 d^2 (d+e x)^6 \left (c d^2-a e^2\right )}{2 e^4}+\frac{3 c d (d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^4}-\frac{(d+e x)^4 \left (c d^2-a e^2\right )^3}{4 e^4}+\frac{c^3 d^3 (d+e x)^7}{7 e^4} \]

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)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 3 a^{2} d^{2} e^{2} \left (a e^{2} + c d^{2}\right ) \int x\, dx + a d e x^{3} \left (a^{2} e^{4} + 3 a c d^{2} e^{2} + c^{2} d^{4}\right ) + \frac{c^{3} d^{3} e^{3} x^{7}}{7} + \frac{c^{2} d^{2} e^{2} x^{6} \left (a e^{2} + c d^{2}\right )}{2} + \frac{3 c d e x^{5} \left (a^{2} e^{4} + 3 a c d^{2} e^{2} + c^{2} d^{4}\right )}{5} + d^{3} e^{3} \int a^{3}\, dx + \frac{x^{4} \left (a e^{2} + c d^{2}\right ) \left (a^{2} e^{4} + 8 a c d^{2} e^{2} + c^{2} d^{4}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

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Mathematica [A]  time = 0.100951, size = 167, normalized size = 1.5 \[ \frac{1}{140} x \left (35 a^3 e^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+21 a^2 c d e^2 x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+7 a c^2 d^2 e x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+c^3 d^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right ) \]

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*(1
0*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 = 0.001, size = 266, normalized size = 2.4 \[{\frac{{d}^{3}{e}^{3}{c}^{3}{x}^{7}}{7}}+{\frac{ \left ( a{e}^{2}+c{d}^{2} \right ){d}^{2}{e}^{2}{c}^{2}{x}^{6}}{2}}+{\frac{ \left ( a{e}^{3}{d}^{3}{c}^{2}+2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}dec+dec \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,a{e}^{2}{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) c+ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( aed \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}aed+{d}^{3}{e}^{3}c{a}^{2} \right ){x}^{3}}{3}}+{\frac{3\,{a}^{2}{e}^{2}{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ){x}^{2}}{2}}+{a}^{3}{e}^{3}{d}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(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*e^3*d^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*e^2*d^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*e*d*(2*a*
c*d^2*e^2+(a*e^2+c*d^2)^2)+2*(a*e^2+c*d^2)^2*a*e*d+d^3*e^3*c*a^2)*x^3+3/2*a^2*e^
2*d^2*(a*e^2+c*d^2)*x^2+a^3*e^3*d^3*x

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Maxima [A]  time = 0.7259, size = 247, normalized size = 2.23 \[ \frac{1}{7} \, c^{3} d^{3} e^{3} x^{7} + \frac{1}{2} \,{\left (c d^{2} + a e^{2}\right )} c^{2} d^{2} e^{2} x^{6} + a^{3} d^{3} e^{3} x + \frac{3}{5} \,{\left (c d^{2} + a e^{2}\right )}^{2} c d e x^{5} + \frac{1}{2} \,{\left (2 \, c d e x^{3} + 3 \,{\left (c d^{2} + a e^{2}\right )} x^{2}\right )} a^{2} d^{2} e^{2} + \frac{1}{4} \,{\left (c d^{2} + a e^{2}\right )}^{3} x^{4} + \frac{1}{10} \,{\left (6 \, c^{2} d^{2} e^{2} x^{5} + 15 \,{\left (c d^{2} + a e^{2}\right )} c d e x^{4} + 10 \,{\left (c d^{2} + a e^{2}\right )}^{2} x^{3}\right )} a d e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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 [A]  time = 0.187356, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} e^{3} d^{3} c^{3} + \frac{1}{2} x^{6} e^{2} d^{4} c^{3} + \frac{1}{2} x^{6} e^{4} d^{2} c^{2} a + \frac{3}{5} x^{5} e d^{5} c^{3} + \frac{9}{5} x^{5} e^{3} d^{3} c^{2} a + \frac{3}{5} x^{5} e^{5} d c a^{2} + \frac{1}{4} x^{4} d^{6} c^{3} + \frac{9}{4} x^{4} e^{2} d^{4} c^{2} a + \frac{9}{4} x^{4} e^{4} d^{2} c a^{2} + \frac{1}{4} x^{4} e^{6} a^{3} + x^{3} e d^{5} c^{2} a + 3 x^{3} e^{3} d^{3} c a^{2} + x^{3} e^{5} d a^{3} + \frac{3}{2} x^{2} e^{2} d^{4} c a^{2} + \frac{3}{2} x^{2} e^{4} d^{2} a^{3} + x e^{3} d^{3} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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 [A]  time = 0.235231, size = 218, normalized size = 1.96 \[ a^{3} d^{3} e^{3} x + \frac{c^{3} d^{3} e^{3} x^{7}}{7} + x^{6} \left (\frac{a c^{2} d^{2} e^{4}}{2} + \frac{c^{3} d^{4} e^{2}}{2}\right ) + x^{5} \left (\frac{3 a^{2} c d e^{5}}{5} + \frac{9 a c^{2} d^{3} e^{3}}{5} + \frac{3 c^{3} d^{5} e}{5}\right ) + x^{4} \left (\frac{a^{3} e^{6}}{4} + \frac{9 a^{2} c d^{2} e^{4}}{4} + \frac{9 a c^{2} d^{4} e^{2}}{4} + \frac{c^{3} d^{6}}{4}\right ) + x^{3} \left (a^{3} d e^{5} + 3 a^{2} c d^{3} e^{3} + a c^{2} d^{5} e\right ) + x^{2} \left (\frac{3 a^{3} d^{2} e^{4}}{2} + \frac{3 a^{2} c d^{4} e^{2}}{2}\right ) \]

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/XCAS [A]  time = 0.208901, size = 274, normalized size = 2.47 \[ \frac{1}{7} \, c^{3} d^{3} x^{7} e^{3} + \frac{1}{2} \, c^{3} d^{4} x^{6} e^{2} + \frac{3}{5} \, c^{3} d^{5} x^{5} e + \frac{1}{4} \, c^{3} d^{6} x^{4} + \frac{1}{2} \, a c^{2} d^{2} x^{6} e^{4} + \frac{9}{5} \, a c^{2} d^{3} x^{5} e^{3} + \frac{9}{4} \, a c^{2} d^{4} x^{4} e^{2} + a c^{2} d^{5} x^{3} e + \frac{3}{5} \, a^{2} c d x^{5} e^{5} + \frac{9}{4} \, a^{2} c d^{2} x^{4} e^{4} + 3 \, a^{2} c d^{3} x^{3} e^{3} + \frac{3}{2} \, a^{2} c d^{4} x^{2} e^{2} + \frac{1}{4} \, a^{3} x^{4} e^{6} + a^{3} d x^{3} e^{5} + \frac{3}{2} \, a^{3} d^{2} x^{2} e^{4} + a^{3} d^{3} x e^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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