3.1832 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^2} \, dx\)

Optimal. Leaf size=20 \[ \frac{(a e+c d x)^3}{3 c d} \]

[Out]

(a*e + c*d*x)^3/(3*c*d)

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Rubi [A]  time = 0.024343, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ \frac{(a e+c d x)^3}{3 c d} \]

Antiderivative was successfully verified.

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

[Out]

(a*e + c*d*x)^3/(3*c*d)

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Rubi in Sympy [A]  time = 10.3494, size = 14, normalized size = 0.7 \[ \frac{\left (a e + c d x\right )^{3}}{3 c d} \]

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)**2/(e*x+d)**2,x)

[Out]

(a*e + c*d*x)**3/(3*c*d)

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Mathematica [A]  time = 0.00420074, size = 20, normalized size = 1. \[ \frac{(a e+c d x)^3}{3 c d} \]

Antiderivative was successfully verified.

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

[Out]

(a*e + c*d*x)^3/(3*c*d)

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Maple [A]  time = 0.002, size = 19, normalized size = 1. \[{\frac{ \left ( cdx+ae \right ) ^{3}}{3\,cd}} \]

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)^2/(e*x+d)^2,x)

[Out]

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

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Maxima [A]  time = 0.721325, size = 38, normalized size = 1.9 \[ \frac{1}{3} \, c^{2} d^{2} x^{3} + a c d e x^{2} + a^{2} e^{2} x \]

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)^2/(e*x + d)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.22752, size = 38, normalized size = 1.9 \[ \frac{1}{3} \, c^{2} d^{2} x^{3} + a c d e x^{2} + a^{2} e^{2} x \]

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)^2/(e*x + d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.23434, size = 29, normalized size = 1.45 \[ a^{2} e^{2} x + a c d e x^{2} + \frac{c^{2} d^{2} x^{3}}{3} \]

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)**2/(e*x+d)**2,x)

[Out]

a**2*e**2*x + a*c*d*e*x**2 + c**2*d**2*x**3/3

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GIAC/XCAS [A]  time = 0.214053, size = 116, normalized size = 5.8 \[ \frac{1}{3} \,{\left (c^{2} d^{2} - \frac{3 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} e^{\left (-1\right )}}{x e + d} + \frac{3 \,{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-3\right )} \]

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)^2/(e*x + d)^2,x, algorithm="giac")

[Out]

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