3.1841 \(\int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx\)

Optimal. Leaf size=111 \[ -\frac{3 c^2 d^2 (d+e x)^7 \left (c d^2-a e^2\right )}{7 e^4}+\frac{c d (d+e x)^6 \left (c d^2-a e^2\right )^2}{2 e^4}-\frac{(d+e x)^5 \left (c d^2-a e^2\right )^3}{5 e^4}+\frac{c^3 d^3 (d+e x)^8}{8 e^4} \]

[Out]

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

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Rubi [A]  time = 0.416931, antiderivative size = 111, 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 \[ -\frac{3 c^2 d^2 (d+e x)^7 \left (c d^2-a e^2\right )}{7 e^4}+\frac{c d (d+e x)^6 \left (c d^2-a e^2\right )^2}{2 e^4}-\frac{(d+e x)^5 \left (c d^2-a e^2\right )^3}{5 e^4}+\frac{c^3 d^3 (d+e x)^8}{8 e^4} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(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)^5)/(5*e^4) + (c*d*(c*d^2 - a*e^2)^2*(d + e*x)^6)/(
2*e^4) - (3*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^7)/(7*e^4) + (c^3*d^3*(d + e*x)^8)
/(8*e^4)

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Rubi in Sympy [A]  time = 57.0543, size = 99, normalized size = 0.89 \[ \frac{c^{3} d^{3} \left (d + e x\right )^{8}}{8 e^{4}} + \frac{3 c^{2} d^{2} \left (d + e x\right )^{7} \left (a e^{2} - c d^{2}\right )}{7 e^{4}} + \frac{c d \left (d + e x\right )^{6} \left (a e^{2} - c d^{2}\right )^{2}}{2 e^{4}} + \frac{\left (d + e x\right )^{5} \left (a e^{2} - c d^{2}\right )^{3}}{5 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.135184, size = 211, normalized size = 1.9 \[ \frac{1}{280} x \left (56 a^3 e^3 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+28 a^2 c d e^2 x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+8 a c^2 d^2 e x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+c^3 d^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(56*a^3*e^3*(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) + 2
8*a^2*c*d*e^2*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
) + 8*a*c^2*d^2*e*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 1
5*e^4*x^4) + c^3*d^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3
+ 35*e^4*x^4)))/280

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Maple [B]  time = 0.002, size = 531, normalized size = 4.8 \[{\frac{{d}^{3}{e}^{4}{c}^{3}{x}^{8}}{8}}+{\frac{ \left ({d}^{4}{e}^{3}{c}^{3}+3\,{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ){d}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ){e}^{2}{c}^{2}+e \left ( a{e}^{3}{d}^{3}{c}^{2}+2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}dec+dec \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( d \left ( a{e}^{3}{d}^{3}{c}^{2}+2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}dec+dec \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) +e \left ( 4\,a{e}^{2}{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) c+ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( d \left ( 4\,a{e}^{2}{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) c+ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) +e \left ( aed \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}aed+{d}^{3}{e}^{3}c{a}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( aed \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}aed+{d}^{3}{e}^{3}c{a}^{2} \right ) +3\,{e}^{3}{a}^{2}{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{3}{a}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) +{a}^{3}{e}^{4}{d}^{3} \right ){x}^{2}}{2}}+{a}^{3}{e}^{3}{d}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.726794, size = 339, normalized size = 3.05 \[ \frac{1}{8} \, c^{3} d^{3} e^{4} x^{8} + a^{3} d^{4} e^{3} x + \frac{1}{7} \,{\left (4 \, c^{3} d^{4} e^{3} + 3 \, a c^{2} d^{2} e^{5}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, c^{3} d^{5} e^{2} + 4 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, c^{3} d^{6} e + 18 \, a c^{2} d^{4} e^{3} + 12 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )} x^{5} + \frac{1}{4} \,{\left (c^{3} d^{7} + 12 \, a c^{2} d^{5} e^{2} + 18 \, a^{2} c d^{3} e^{4} + 4 \, a^{3} d e^{6}\right )} x^{4} +{\left (a c^{2} d^{6} e + 4 \, a^{2} c d^{4} e^{3} + 2 \, a^{3} d^{2} e^{5}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{5} e^{2} + 4 \, a^{3} d^{3} e^{4}\right )} x^{2} \]

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*(e*x + d),x, algorithm="maxima")

[Out]

1/8*c^3*d^3*e^4*x^8 + a^3*d^4*e^3*x + 1/7*(4*c^3*d^4*e^3 + 3*a*c^2*d^2*e^5)*x^7
+ 1/2*(2*c^3*d^5*e^2 + 4*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^6 + 1/5*(4*c^3*d^6*e + 1
8*a*c^2*d^4*e^3 + 12*a^2*c*d^2*e^5 + a^3*e^7)*x^5 + 1/4*(c^3*d^7 + 12*a*c^2*d^5*
e^2 + 18*a^2*c*d^3*e^4 + 4*a^3*d*e^6)*x^4 + (a*c^2*d^6*e + 4*a^2*c*d^4*e^3 + 2*a
^3*d^2*e^5)*x^3 + 1/2*(3*a^2*c*d^5*e^2 + 4*a^3*d^3*e^4)*x^2

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Fricas [A]  time = 0.185134, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} e^{4} d^{3} c^{3} + \frac{4}{7} x^{7} e^{3} d^{4} c^{3} + \frac{3}{7} x^{7} e^{5} d^{2} c^{2} a + x^{6} e^{2} d^{5} c^{3} + 2 x^{6} e^{4} d^{3} c^{2} a + \frac{1}{2} x^{6} e^{6} d c a^{2} + \frac{4}{5} x^{5} e d^{6} c^{3} + \frac{18}{5} x^{5} e^{3} d^{4} c^{2} a + \frac{12}{5} x^{5} e^{5} d^{2} c a^{2} + \frac{1}{5} x^{5} e^{7} a^{3} + \frac{1}{4} x^{4} d^{7} c^{3} + 3 x^{4} e^{2} d^{5} c^{2} a + \frac{9}{2} x^{4} e^{4} d^{3} c a^{2} + x^{4} e^{6} d a^{3} + x^{3} e d^{6} c^{2} a + 4 x^{3} e^{3} d^{4} c a^{2} + 2 x^{3} e^{5} d^{2} a^{3} + \frac{3}{2} x^{2} e^{2} d^{5} c a^{2} + 2 x^{2} e^{4} d^{3} a^{3} + x e^{3} d^{4} a^{3} \]

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*(e*x + d),x, algorithm="fricas")

[Out]

1/8*x^8*e^4*d^3*c^3 + 4/7*x^7*e^3*d^4*c^3 + 3/7*x^7*e^5*d^2*c^2*a + x^6*e^2*d^5*
c^3 + 2*x^6*e^4*d^3*c^2*a + 1/2*x^6*e^6*d*c*a^2 + 4/5*x^5*e*d^6*c^3 + 18/5*x^5*e
^3*d^4*c^2*a + 12/5*x^5*e^5*d^2*c*a^2 + 1/5*x^5*e^7*a^3 + 1/4*x^4*d^7*c^3 + 3*x^
4*e^2*d^5*c^2*a + 9/2*x^4*e^4*d^3*c*a^2 + x^4*e^6*d*a^3 + x^3*e*d^6*c^2*a + 4*x^
3*e^3*d^4*c*a^2 + 2*x^3*e^5*d^2*a^3 + 3/2*x^2*e^2*d^5*c*a^2 + 2*x^2*e^4*d^3*a^3
+ x*e^3*d^4*a^3

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Sympy [A]  time = 0.296799, size = 270, normalized size = 2.43 \[ a^{3} d^{4} e^{3} x + \frac{c^{3} d^{3} e^{4} x^{8}}{8} + x^{7} \left (\frac{3 a c^{2} d^{2} e^{5}}{7} + \frac{4 c^{3} d^{4} e^{3}}{7}\right ) + x^{6} \left (\frac{a^{2} c d e^{6}}{2} + 2 a c^{2} d^{3} e^{4} + c^{3} d^{5} e^{2}\right ) + x^{5} \left (\frac{a^{3} e^{7}}{5} + \frac{12 a^{2} c d^{2} e^{5}}{5} + \frac{18 a c^{2} d^{4} e^{3}}{5} + \frac{4 c^{3} d^{6} e}{5}\right ) + x^{4} \left (a^{3} d e^{6} + \frac{9 a^{2} c d^{3} e^{4}}{2} + 3 a c^{2} d^{5} e^{2} + \frac{c^{3} d^{7}}{4}\right ) + x^{3} \left (2 a^{3} d^{2} e^{5} + 4 a^{2} c d^{4} e^{3} + a c^{2} d^{6} e\right ) + x^{2} \left (2 a^{3} d^{3} e^{4} + \frac{3 a^{2} c d^{5} e^{2}}{2}\right ) \]

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)**3,x)

[Out]

a**3*d**4*e**3*x + c**3*d**3*e**4*x**8/8 + x**7*(3*a*c**2*d**2*e**5/7 + 4*c**3*d
**4*e**3/7) + x**6*(a**2*c*d*e**6/2 + 2*a*c**2*d**3*e**4 + c**3*d**5*e**2) + x**
5*(a**3*e**7/5 + 12*a**2*c*d**2*e**5/5 + 18*a*c**2*d**4*e**3/5 + 4*c**3*d**6*e/5
) + x**4*(a**3*d*e**6 + 9*a**2*c*d**3*e**4/2 + 3*a*c**2*d**5*e**2 + c**3*d**7/4)
+ x**3*(2*a**3*d**2*e**5 + 4*a**2*c*d**4*e**3 + a*c**2*d**6*e) + x**2*(2*a**3*d
**3*e**4 + 3*a**2*c*d**5*e**2/2)

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GIAC/XCAS [A]  time = 0.212088, size = 346, normalized size = 3.12 \[ \frac{1}{8} \, c^{3} d^{3} x^{8} e^{4} + \frac{4}{7} \, c^{3} d^{4} x^{7} e^{3} + c^{3} d^{5} x^{6} e^{2} + \frac{4}{5} \, c^{3} d^{6} x^{5} e + \frac{1}{4} \, c^{3} d^{7} x^{4} + \frac{3}{7} \, a c^{2} d^{2} x^{7} e^{5} + 2 \, a c^{2} d^{3} x^{6} e^{4} + \frac{18}{5} \, a c^{2} d^{4} x^{5} e^{3} + 3 \, a c^{2} d^{5} x^{4} e^{2} + a c^{2} d^{6} x^{3} e + \frac{1}{2} \, a^{2} c d x^{6} e^{6} + \frac{12}{5} \, a^{2} c d^{2} x^{5} e^{5} + \frac{9}{2} \, a^{2} c d^{3} x^{4} e^{4} + 4 \, a^{2} c d^{4} x^{3} e^{3} + \frac{3}{2} \, a^{2} c d^{5} x^{2} e^{2} + \frac{1}{5} \, a^{3} x^{5} e^{7} + a^{3} d x^{4} e^{6} + 2 \, a^{3} d^{2} x^{3} e^{5} + 2 \, a^{3} d^{3} x^{2} e^{4} + a^{3} d^{4} x e^{3} \]

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*(e*x + d),x, algorithm="giac")

[Out]

1/8*c^3*d^3*x^8*e^4 + 4/7*c^3*d^4*x^7*e^3 + c^3*d^5*x^6*e^2 + 4/5*c^3*d^6*x^5*e
+ 1/4*c^3*d^7*x^4 + 3/7*a*c^2*d^2*x^7*e^5 + 2*a*c^2*d^3*x^6*e^4 + 18/5*a*c^2*d^4
*x^5*e^3 + 3*a*c^2*d^5*x^4*e^2 + a*c^2*d^6*x^3*e + 1/2*a^2*c*d*x^6*e^6 + 12/5*a^
2*c*d^2*x^5*e^5 + 9/2*a^2*c*d^3*x^4*e^4 + 4*a^2*c*d^4*x^3*e^3 + 3/2*a^2*c*d^5*x^
2*e^2 + 1/5*a^3*x^5*e^7 + a^3*d*x^4*e^6 + 2*a^3*d^2*x^3*e^5 + 2*a^3*d^3*x^2*e^4
+ a^3*d^4*x*e^3