3.2105 \(\int \frac{a+b x+c x^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=69 \[ -\frac{a e^2-b d e+c d^2}{4 e^3 (d+e x)^4}+\frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{c}{2 e^3 (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.10735, antiderivative size = 69, 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 \[ -\frac{a e^2-b d e+c d^2}{4 e^3 (d+e x)^4}+\frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{c}{2 e^3 (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.6654, size = 61, normalized size = 0.88 \[ - \frac{c}{2 e^{3} \left (d + e x\right )^{2}} - \frac{b e - 2 c d}{3 e^{3} \left (d + e x\right )^{3}} - \frac{a e^{2} - b d e + c d^{2}}{4 e^{3} \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0394622, size = 49, normalized size = 0.71 \[ -\frac{e (3 a e+b (d+4 e x))+c \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 63, normalized size = 0.9 \[ -{\frac{c}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{be-2\,cd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{a{e}^{2}-bde+c{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 0.815371, size = 116, normalized size = 1.68 \[ -\frac{6 \, c e^{2} x^{2} + c d^{2} + b d e + 3 \, a e^{2} + 4 \,{\left (c d e + b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(6*c*e^2*x^2 + c*d^2 + b*d*e + 3*a*e^2 + 4*(c*d*e + b*e^2)*x)/(e^7*x^4 + 4
*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Fricas [A]  time = 0.195367, size = 116, normalized size = 1.68 \[ -\frac{6 \, c e^{2} x^{2} + c d^{2} + b d e + 3 \, a e^{2} + 4 \,{\left (c d e + b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(6*c*e^2*x^2 + c*d^2 + b*d*e + 3*a*e^2 + 4*(c*d*e + b*e^2)*x)/(e^7*x^4 + 4
*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Sympy [A]  time = 5.70777, size = 92, normalized size = 1.33 \[ - \frac{3 a e^{2} + b d e + c d^{2} + 6 c e^{2} x^{2} + x \left (4 b e^{2} + 4 c d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*a*e**2 + b*d*e + c*d**2 + 6*c*e**2*x**2 + x*(4*b*e**2 + 4*c*d*e))/(12*d**4*e
**3 + 48*d**3*e**4*x + 72*d**2*e**5*x**2 + 48*d*e**6*x**3 + 12*e**7*x**4)

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GIAC/XCAS [A]  time = 0.203517, size = 119, normalized size = 1.72 \[ -\frac{1}{12} \,{\left (\frac{6 \, c e}{{\left (x e + d\right )}^{2}} - \frac{8 \, c d e}{{\left (x e + d\right )}^{3}} + \frac{3 \, c d^{2} e}{{\left (x e + d\right )}^{4}} + \frac{4 \, b e^{2}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b d e^{2}}{{\left (x e + d\right )}^{4}} + \frac{3 \, a e^{3}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(6*c*e/(x*e + d)^2 - 8*c*d*e/(x*e + d)^3 + 3*c*d^2*e/(x*e + d)^4 + 4*b*e^2
/(x*e + d)^3 - 3*b*d*e^2/(x*e + d)^4 + 3*a*e^3/(x*e + d)^4)*e^(-4)