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

Optimal. Leaf size=17 \[ \frac{c^2 (d+e x)^6}{6 e} \]

[Out]

(c^2*(d + e*x)^6)/(6*e)

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Rubi [A]  time = 0.016419, antiderivative size = 17, 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 \[ \frac{c^2 (d+e x)^6}{6 e} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(d + e*x)^6)/(6*e)

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Rubi in Sympy [A]  time = 16.9579, size = 12, normalized size = 0.71 \[ \frac{c^{2} \left (d + e x\right )^{6}}{6 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*(d + e*x)**6/(6*e)

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Mathematica [A]  time = 0.00406282, size = 17, normalized size = 1. \[ \frac{c^2 (d+e x)^6}{6 e} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(d + e*x)^6)/(6*e)

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Maple [B]  time = 0.002, size = 72, normalized size = 4.2 \[{\frac{{c}^{2}{e}^{5}{x}^{6}}{6}}+d{c}^{2}{e}^{4}{x}^{5}+{\frac{5\,{d}^{2}{c}^{2}{e}^{3}{x}^{4}}{2}}+{\frac{10\,{d}^{3}{c}^{2}{e}^{2}{x}^{3}}{3}}+{\frac{5\,{c}^{2}e{d}^{4}{x}^{2}}{2}}+{c}^{2}{d}^{5}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*c^2*e^5*x^6+d*c^2*e^4*x^5+5/2*d^2*c^2*e^3*x^4+10/3*d^3*c^2*e^2*x^3+5/2*c^2*e
*d^4*x^2+c^2*d^5*x

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Maxima [A]  time = 0.700608, size = 41, normalized size = 2.41 \[ \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{3}}{6 \, c e} \]

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)^2*(e*x + d),x, algorithm="maxima")

[Out]

1/6*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^3/(c*e)

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Fricas [A]  time = 0.199083, size = 1, normalized size = 0.06 \[ \frac{1}{6} x^{6} e^{5} c^{2} + x^{5} e^{4} d c^{2} + \frac{5}{2} x^{4} e^{3} d^{2} c^{2} + \frac{10}{3} x^{3} e^{2} d^{3} c^{2} + \frac{5}{2} x^{2} e d^{4} c^{2} + x d^{5} c^{2} \]

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)^2*(e*x + d),x, algorithm="fricas")

[Out]

1/6*x^6*e^5*c^2 + x^5*e^4*d*c^2 + 5/2*x^4*e^3*d^2*c^2 + 10/3*x^3*e^2*d^3*c^2 + 5
/2*x^2*e*d^4*c^2 + x*d^5*c^2

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Sympy [A]  time = 0.140751, size = 80, normalized size = 4.71 \[ c^{2} d^{5} x + \frac{5 c^{2} d^{4} e x^{2}}{2} + \frac{10 c^{2} d^{3} e^{2} x^{3}}{3} + \frac{5 c^{2} d^{2} e^{3} x^{4}}{2} + c^{2} d e^{4} x^{5} + \frac{c^{2} e^{5} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*d**5*x + 5*c**2*d**4*e*x**2/2 + 10*c**2*d**3*e**2*x**3/3 + 5*c**2*d**2*e**3
*x**4/2 + c**2*d*e**4*x**5 + c**2*e**5*x**6/6

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GIAC/XCAS [A]  time = 0.207573, size = 92, normalized size = 5.41 \[ \frac{1}{6} \, c^{2} x^{6} e^{5} + c^{2} d x^{5} e^{4} + \frac{5}{2} \, c^{2} d^{2} x^{4} e^{3} + \frac{10}{3} \, c^{2} d^{3} x^{3} e^{2} + \frac{5}{2} \, c^{2} d^{4} x^{2} e + c^{2} d^{5} x \]

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)^2*(e*x + d),x, algorithm="giac")

[Out]

1/6*c^2*x^6*e^5 + c^2*d*x^5*e^4 + 5/2*c^2*d^2*x^4*e^3 + 10/3*c^2*d^3*x^3*e^2 + 5
/2*c^2*d^4*x^2*e + c^2*d^5*x