3.1889 \(\int \frac{(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx\)

Optimal. Leaf size=35 \[ -\frac{(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \]

[Out]

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

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Rubi [A]  time = 0.0402574, 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 \[ -\frac{(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.8446, size = 27, normalized size = 0.77 \[ \frac{\left (d + e x\right )^{3}}{3 \left (a e + c d x\right )^{3} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

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

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Mathematica [A]  time = 0.0618198, size = 65, normalized size = 1.86 \[ -\frac{a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 c^3 d^3 (a e+c d x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

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Maple [B]  time = 0.009, size = 96, normalized size = 2.7 \[ -{\frac{{e}^{2}}{{c}^{3}{d}^{3} \left ( cdx+ae \right ) }}+{\frac{e \left ( a{e}^{2}-c{d}^{2} \right ) }{{c}^{3}{d}^{3} \left ( cdx+ae \right ) ^{2}}}-{\frac{{a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{3\,{c}^{3}{d}^{3} \left ( cdx+ae \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^6/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)

[Out]

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

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Maxima [A]  time = 0.741168, size = 153, normalized size = 4.37 \[ -\frac{3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} e x^{2} + 3 \, a^{2} c^{4} d^{4} e^{2} x + a^{3} c^{3} d^{3} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.203471, size = 153, normalized size = 4.37 \[ -\frac{3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} e x^{2} + 3 \, a^{2} c^{4} d^{4} e^{2} x + a^{3} c^{3} d^{3} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.99797, size = 121, normalized size = 3.46 \[ - \frac{a^{2} e^{4} + a c d^{2} e^{2} + c^{2} d^{4} + 3 c^{2} d^{2} e^{2} x^{2} + x \left (3 a c d e^{3} + 3 c^{2} d^{3} e\right )}{3 a^{3} c^{3} d^{3} e^{3} + 9 a^{2} c^{4} d^{4} e^{2} x + 9 a c^{5} d^{5} e x^{2} + 3 c^{6} d^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

-(a**2*e**4 + a*c*d**2*e**2 + c**2*d**4 + 3*c**2*d**2*e**2*x**2 + x*(3*a*c*d*e**
3 + 3*c**2*d**3*e))/(3*a**3*c**3*d**3*e**3 + 9*a**2*c**4*d**4*e**2*x + 9*a*c**5*
d**5*e*x**2 + 3*c**6*d**6*x**3)

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GIAC/XCAS [A]  time = 57.9054, size = 1, normalized size = 0.03 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="giac")

[Out]

Done