3.1879 \(\int \frac{(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=35 \[ -\frac{(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0405761, antiderivative size = 35, 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 \[ -\frac{(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.26, size = 27, normalized size = 0.77 \[ \frac{\left (d + e x\right )^{2}}{2 \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0259218, size = 35, normalized size = 1. \[ -\frac{a e^2+c d (d+2 e x)}{2 c^2 d^2 (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.007, size = 51, normalized size = 1.5 \[ -{\frac{e}{{c}^{2}{d}^{2} \left ( cdx+ae \right ) }}-{\frac{-a{e}^{2}+c{d}^{2}}{2\,{c}^{2}{d}^{2} \left ( cdx+ae \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.744066, size = 76, normalized size = 2.17 \[ -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.203153, size = 76, normalized size = 2.17 \[ -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.11969, size = 60, normalized size = 1.71 \[ - \frac{a e^{2} + c d^{2} + 2 c d e x}{2 a^{2} c^{2} d^{2} e^{2} + 4 a c^{3} d^{3} e x + 2 c^{4} d^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 1.53286, size = 1, normalized size = 0.03 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")

[Out]

Done