3.1826 \(\int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0725779, 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 \[ -\frac{a-\frac{c d^2}{e^2}}{3 (d+e x)^3}-\frac{c d}{2 e^2 (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.2023, size = 37, normalized size = 0.95 \[ - \frac{c d}{2 e^{2} \left (d + e x\right )^{2}} - \frac{a e^{2} - c d^{2}}{3 e^{2} \left (d + e x\right )^{3}} \]

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)**5,x)

[Out]

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

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Mathematica [A]  time = 0.021772, size = 30, normalized size = 0.77 \[ -\frac{2 a e^2+c d (d+3 e x)}{6 e^2 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 40, normalized size = 1. \[ -{\frac{a{e}^{2}-c{d}^{2}}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{cd}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}} \]

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)^5,x)

[Out]

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

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Maxima [A]  time = 0.728895, size = 74, normalized size = 1.9 \[ -\frac{3 \, c d e x + c d^{2} + 2 \, a e^{2}}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \]

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)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.204486, size = 74, normalized size = 1.9 \[ -\frac{3 \, c d e x + c d^{2} + 2 \, a e^{2}}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \]

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)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.07486, size = 58, normalized size = 1.49 \[ - \frac{2 a e^{2} + c d^{2} + 3 c d e x}{6 d^{3} e^{2} + 18 d^{2} e^{3} x + 18 d e^{4} x^{2} + 6 e^{5} x^{3}} \]

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)**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.210819, size = 57, normalized size = 1.46 \[ -\frac{c d e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} + \frac{c d^{2} e^{\left (-2\right )}}{3 \,{\left (x e + d\right )}^{3}} - \frac{a}{3 \,{\left (x e + d\right )}^{3}} \]

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)^5,x, algorithm="giac")

[Out]

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