3.2104 \(\int \frac{a+b x+c x^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=67 \[ -\frac{a e^2-b d e+c d^2}{3 e^3 (d+e x)^3}+\frac{2 c d-b e}{2 e^3 (d+e x)^2}-\frac{c}{e^3 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.106562, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{a e^2-b d e+c d^2}{3 e^3 (d+e x)^3}+\frac{2 c d-b e}{2 e^3 (d+e x)^2}-\frac{c}{e^3 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.514, size = 58, normalized size = 0.87 \[ - \frac{c}{e^{3} \left (d + e x\right )} - \frac{b e - 2 c d}{2 e^{3} \left (d + e x\right )^{2}} - \frac{a e^{2} - b d e + c d^{2}}{3 e^{3} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.036343, size = 50, normalized size = 0.75 \[ -\frac{e (2 a e+b (d+3 e x))+2 c \left (d^2+3 d e x+3 e^2 x^2\right )}{6 e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 63, normalized size = 0.9 \[ -{\frac{be-2\,cd}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{a{e}^{2}-bde+c{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{c}{{e}^{3} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)/(e*x+d)^4,x)

[Out]

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

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Maxima [A]  time = 0.801858, size = 104, normalized size = 1.55 \[ -\frac{6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 2 \, a e^{2} + 3 \,{\left (2 \, c d e + b e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.196374, size = 104, normalized size = 1.55 \[ -\frac{6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 2 \, a e^{2} + 3 \,{\left (2 \, c d e + b e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.0164, size = 82, normalized size = 1.22 \[ - \frac{2 a e^{2} + b d e + 2 c d^{2} + 6 c e^{2} x^{2} + x \left (3 b e^{2} + 6 c d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.202688, size = 68, normalized size = 1.01 \[ -\frac{{\left (6 \, c x^{2} e^{2} + 6 \, c d x e + 2 \, c d^{2} + 3 \, b x e^{2} + b d e + 2 \, a e^{2}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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