3.1819 \(\int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{4} (d+e x)^4 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^5}{5 e^2} \]

[Out]

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

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Rubi [A]  time = 0.065713, antiderivative size = 39, 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 \[ \frac{1}{4} (d+e x)^4 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^5}{5 e^2} \]

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),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.8165, size = 36, normalized size = 0.92 \[ \frac{c d \left (d + e x\right )^{5}}{5 e^{2}} + \frac{\left (d + e x\right )^{4} \left (a e^{2} - c d^{2}\right )}{4 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0383957, size = 73, normalized size = 1.87 \[ \frac{1}{20} x \left (5 a e \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+c d x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right ) \]

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),x]

[Out]

(x*(5*a*e*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + c*d*x*(10*d^3 + 20*d^2*e
*x + 15*d*e^2*x^2 + 4*e^3*x^3)))/20

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Maple [B]  time = 0.001, size = 112, normalized size = 2.9 \[{\frac{d{e}^{3}c{x}^{5}}{5}}+{\frac{ \left ( 2\,{d}^{2}{e}^{2}c+{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( c{d}^{3}e+2\,de \left ( a{e}^{2}+c{d}^{2} \right ) +ad{e}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ({d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) +2\,a{d}^{2}{e}^{2} \right ){x}^{2}}{2}}+{d}^{3}aex \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.718356, size = 101, normalized size = 2.59 \[ \frac{1}{5} \, c d e^{3} x^{5} + a d^{3} e x + \frac{1}{4} \,{\left (3 \, c d^{2} e^{2} + a e^{4}\right )} x^{4} +{\left (c d^{3} e + a d e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (c d^{4} + 3 \, a d^{2} e^{2}\right )} x^{2} \]

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)*(e*x + d)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.178018, size = 1, normalized size = 0.03 \[ \frac{1}{5} x^{5} e^{3} d c + \frac{3}{4} x^{4} e^{2} d^{2} c + \frac{1}{4} x^{4} e^{4} a + x^{3} e d^{3} c + x^{3} e^{3} d a + \frac{1}{2} x^{2} d^{4} c + \frac{3}{2} x^{2} e^{2} d^{2} a + x e d^{3} a \]

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)*(e*x + d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.138646, size = 80, normalized size = 2.05 \[ a d^{3} e x + \frac{c d e^{3} x^{5}}{5} + x^{4} \left (\frac{a e^{4}}{4} + \frac{3 c d^{2} e^{2}}{4}\right ) + x^{3} \left (a d e^{3} + c d^{3} e\right ) + x^{2} \left (\frac{3 a d^{2} e^{2}}{2} + \frac{c d^{4}}{2}\right ) \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.208626, size = 101, normalized size = 2.59 \[ \frac{1}{5} \, c d x^{5} e^{3} + \frac{3}{4} \, c d^{2} x^{4} e^{2} + c d^{3} x^{3} e + \frac{1}{2} \, c d^{4} x^{2} + \frac{1}{4} \, a x^{4} e^{4} + a d x^{3} e^{3} + \frac{3}{2} \, a d^{2} x^{2} e^{2} + a d^{3} x e \]

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)*(e*x + d)^2,x, algorithm="giac")

[Out]

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