3.1827 \(\int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^6} \, dx\)

Optimal. Leaf size=39 \[ -\frac{a-\frac{c d^2}{e^2}}{4 (d+e x)^4}-\frac{c d}{3 e^2 (d+e x)^3} \]

[Out]

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

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Rubi [A]  time = 0.07208, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{a-\frac{c d^2}{e^2}}{4 (d+e x)^4}-\frac{c d}{3 e^2 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.2963, size = 37, normalized size = 0.95 \[ - \frac{c d}{3 e^{2} \left (d + e x\right )^{3}} - \frac{a e^{2} - c d^{2}}{4 e^{2} \left (d + e x\right )^{4}} \]

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)/(e*x+d)**6,x)

[Out]

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

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Mathematica [A]  time = 0.0234266, size = 30, normalized size = 0.77 \[ -\frac{3 a e^2+c d (d+4 e x)}{12 e^2 (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 40, normalized size = 1. \[ -{\frac{a{e}^{2}-c{d}^{2}}{4\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{cd}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}} \]

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)/(e*x+d)^6,x)

[Out]

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

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Maxima [A]  time = 0.732291, size = 89, normalized size = 2.28 \[ -\frac{4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \]

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)/(e*x + d)^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.21923, size = 89, normalized size = 2.28 \[ -\frac{4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \]

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)/(e*x + d)^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.41742, size = 70, normalized size = 1.79 \[ - \frac{3 a e^{2} + c d^{2} + 4 c d e x}{12 d^{4} e^{2} + 48 d^{3} e^{3} x + 72 d^{2} e^{4} x^{2} + 48 d e^{5} x^{3} + 12 e^{6} x^{4}} \]

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)/(e*x+d)**6,x)

[Out]

-(3*a*e**2 + c*d**2 + 4*c*d*e*x)/(12*d**4*e**2 + 48*d**3*e**3*x + 72*d**2*e**4*x
**2 + 48*d*e**5*x**3 + 12*e**6*x**4)

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GIAC/XCAS [A]  time = 0.207763, size = 65, normalized size = 1.67 \[ -\frac{{\left (4 \, c d x^{2} e^{2} + 5 \, c d^{2} x e + c d^{3} + 3 \, a x e^{3} + 3 \, a d e^{2}\right )} e^{\left (-2\right )}}{12 \,{\left (x e + d\right )}^{5}} \]

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)/(e*x + d)^6,x, algorithm="giac")

[Out]

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