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

Optimal. Leaf size=15 \[ \frac{c (d+e x)^5}{5 e} \]

[Out]

(c*(d + e*x)^5)/(5*e)

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Rubi [A]  time = 0.0160711, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{c (d+e x)^5}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)^5)/(5*e)

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Rubi in Sympy [A]  time = 14.9615, size = 10, normalized size = 0.67 \[ \frac{c \left (d + e x\right )^{5}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(d + e*x)**5/(5*e)

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Mathematica [A]  time = 0.00651645, size = 15, normalized size = 1. \[ \frac{c (d+e x)^5}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)^5)/(5*e)

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Maple [B]  time = 0.001, size = 48, normalized size = 3.2 \[{\frac{{e}^{4}c{x}^{5}}{5}}+d{e}^{3}c{x}^{4}+2\,{d}^{2}c{e}^{2}{x}^{3}+2\,c{d}^{3}e{x}^{2}+c{d}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.69258, size = 63, normalized size = 4.2 \[ \frac{1}{5} \, c e^{4} x^{5} + c d e^{3} x^{4} + 2 \, c d^{2} e^{2} x^{3} + 2 \, c d^{3} e x^{2} + c d^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.189446, size = 1, normalized size = 0.07 \[ \frac{1}{5} x^{5} e^{4} c + x^{4} e^{3} d c + 2 x^{3} e^{2} d^{2} c + 2 x^{2} e d^{3} c + x d^{4} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.116987, size = 51, normalized size = 3.4 \[ c d^{4} x + 2 c d^{3} e x^{2} + 2 c d^{2} e^{2} x^{3} + c d e^{3} x^{4} + \frac{c e^{4} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*d**4*x + 2*c*d**3*e*x**2 + 2*c*d**2*e**2*x**3 + c*d*e**3*x**4 + c*e**4*x**5/5

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GIAC/XCAS [A]  time = 0.209047, size = 61, normalized size = 4.07 \[ \frac{1}{5} \, c x^{5} e^{4} + c d x^{4} e^{3} + 2 \, c d^{2} x^{3} e^{2} + 2 \, c d^{3} x^{2} e + c d^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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