3.1831 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{d+e x} \, dx\)

Optimal. Leaf size=54 \[ \frac{\left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^2 d^2}+\frac{e (a e+c d x)^4}{4 c^2 d^2} \]

[Out]

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

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Rubi [A]  time = 0.111041, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ \frac{\left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^2 d^2}+\frac{e (a e+c d x)^4}{4 c^2 d^2} \]

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),x]

[Out]

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

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Rubi in Sympy [A]  time = 24.9488, size = 48, normalized size = 0.89 \[ \frac{e \left (a e + c d x\right )^{4}}{4 c^{2} d^{2}} - \frac{\left (a e + c d x\right )^{3} \left (a e^{2} - c d^{2}\right )}{3 c^{2} d^{2}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.0229265, size = 54, normalized size = 1. \[ \frac{1}{12} x \left (6 a^2 e^2 (2 d+e x)+4 a c d e x (3 d+2 e x)+c^2 d^2 x^2 (4 d+3 e x)\right ) \]

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),x]

[Out]

(x*(6*a^2*e^2*(2*d + e*x) + 4*a*c*d*e*x*(3*d + 2*e*x) + c^2*d^2*x^2*(4*d + 3*e*x
)))/12

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Maple [A]  time = 0.002, size = 77, normalized size = 1.4 \[{\frac{{c}^{2}{d}^{2}e{x}^{4}}{4}}+{\frac{ \left ( ad{e}^{2}c+cd \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( ae \left ( a{e}^{2}+c{d}^{2} \right ) +c{d}^{2}ae \right ){x}^{2}}{2}}+{a}^{2}d{e}^{2}x \]

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),x)

[Out]

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

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Maxima [A]  time = 0.722565, size = 86, normalized size = 1.59 \[ \frac{1}{4} \, c^{2} d^{2} e x^{4} + a^{2} d e^{2} x + \frac{1}{3} \,{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a c d^{2} e + a^{2} e^{3}\right )} x^{2} \]

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)^2/(e*x + d),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.234023, size = 86, normalized size = 1.59 \[ \frac{1}{4} \, c^{2} d^{2} e x^{4} + a^{2} d e^{2} x + \frac{1}{3} \,{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a c d^{2} e + a^{2} e^{3}\right )} x^{2} \]

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)^2/(e*x + d),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.241112, size = 66, normalized size = 1.22 \[ a^{2} d e^{2} x + \frac{c^{2} d^{2} e x^{4}}{4} + x^{3} \left (\frac{2 a c d e^{2}}{3} + \frac{c^{2} d^{3}}{3}\right ) + x^{2} \left (\frac{a^{2} e^{3}}{2} + a c d^{2} e\right ) \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.211026, size = 97, normalized size = 1.8 \[ \frac{1}{12} \,{\left (3 \, c^{2} d^{2} x^{4} e^{5} + 4 \, c^{2} d^{3} x^{3} e^{4} + 8 \, a c d x^{3} e^{6} + 12 \, a c d^{2} x^{2} e^{5} + 6 \, a^{2} x^{2} e^{7} + 12 \, a^{2} d x e^{6}\right )} e^{\left (-4\right )} \]

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),x, algorithm="giac")

[Out]

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