3.1890 \(\int \frac{(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0974425, 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 \[ -\frac{c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac{e}{2 c^2 d^2 (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 23.5338, size = 48, normalized size = 0.89 \[ - \frac{e}{2 c^{2} d^{2} \left (a e + c d x\right )^{2}} + \frac{a e^{2} - c d^{2}}{3 c^{2} d^{2} \left (a e + c d x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

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

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Mathematica [A]  time = 0.0301683, size = 37, normalized size = 0.69 \[ -\frac{a e^2+c d (2 d+3 e x)}{6 c^2 d^2 (a e+c d x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

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Maple [A]  time = 0.008, size = 51, normalized size = 0.9 \[ -{\frac{e}{2\,{c}^{2}{d}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{-a{e}^{2}+c{d}^{2}}{3\,{c}^{2}{d}^{2} \left ( cdx+ae \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^5/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)

[Out]

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

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Maxima [A]  time = 0.743144, size = 100, normalized size = 1.85 \[ -\frac{3 \, c d e x + 2 \, c d^{2} + a e^{2}}{6 \,{\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} e x^{2} + 3 \, a^{2} c^{3} d^{3} e^{2} x + a^{3} c^{2} d^{2} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.207691, size = 100, normalized size = 1.85 \[ -\frac{3 \, c d e x + 2 \, c d^{2} + a e^{2}}{6 \,{\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} e x^{2} + 3 \, a^{2} c^{3} d^{3} e^{2} x + a^{3} c^{2} d^{2} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.63193, size = 80, normalized size = 1.48 \[ - \frac{a e^{2} + 2 c d^{2} + 3 c d e x}{6 a^{3} c^{2} d^{2} e^{3} + 18 a^{2} c^{3} d^{3} e^{2} x + 18 a c^{4} d^{4} e x^{2} + 6 c^{5} d^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

-(a*e**2 + 2*c*d**2 + 3*c*d*e*x)/(6*a**3*c**2*d**2*e**3 + 18*a**2*c**3*d**3*e**2
*x + 18*a*c**4*d**4*e*x**2 + 6*c**5*d**5*x**3)

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GIAC/XCAS [A]  time = 0.27777, size = 1, normalized size = 0.02 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="giac")

[Out]

Done