3.2117 \(\int \frac{\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx\)

Optimal. Leaf size=151 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^5 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}-\frac{c^2}{e^5 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.314703, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^5 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}-\frac{c^2}{e^5 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.4876, size = 141, normalized size = 0.93 \[ - \frac{c^{2}}{e^{5} \left (d + e x\right )} - \frac{c \left (b e - 2 c d\right )}{e^{5} \left (d + e x\right )^{2}} - \frac{2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}}{3 e^{5} \left (d + e x\right )^{3}} - \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{2 e^{5} \left (d + e x\right )^{4}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{2}}{5 e^{5} \left (d + e x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.141108, size = 160, normalized size = 1.06 \[ -\frac{e^2 \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )+c e \left (2 a e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+6 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )}{30 e^5 (d+e x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-(6*c^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + e^2*(6*a
^2*e^2 + 3*a*b*e*(d + 5*e*x) + b^2*(d^2 + 5*d*e*x + 10*e^2*x^2)) + c*e*(2*a*e*(d
^2 + 5*d*e*x + 10*e^2*x^2) + 3*b*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)))
/(30*e^5*(d + e*x)^5)

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Maple [A]  time = 0.01, size = 195, normalized size = 1.3 \[ -{\frac{c \left ( be-2\,cd \right ) }{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{2\,ac{e}^{2}+{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{2}{e}^{4}-2\,d{e}^{3}ab+2\,ac{d}^{2}{e}^{2}+{b}^{2}{d}^{2}{e}^{2}-2\,{d}^{3}ebc+{c}^{2}{d}^{4}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{2\,ab{e}^{3}-4\,ad{e}^{2}c-2\,{b}^{2}d{e}^{2}+6\,bc{d}^{2}e-4\,{c}^{2}{d}^{3}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.814578, size = 296, normalized size = 1.96 \[ -\frac{30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 3 \, a b d e^{3} + 6 \, a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 30 \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \,{\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 5 \,{\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{30 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(30*c^2*e^4*x^4 + 6*c^2*d^4 + 3*b*c*d^3*e + 3*a*b*d*e^3 + 6*a^2*e^4 + (b^2
+ 2*a*c)*d^2*e^2 + 30*(2*c^2*d*e^3 + b*c*e^4)*x^3 + 10*(6*c^2*d^2*e^2 + 3*b*c*d
*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 5*(6*c^2*d^3*e + 3*b*c*d^2*e^2 + 3*a*b*e^4 + (b^
2 + 2*a*c)*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 +
5*d^4*e^6*x + d^5*e^5)

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Fricas [A]  time = 0.206298, size = 296, normalized size = 1.96 \[ -\frac{30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 3 \, a b d e^{3} + 6 \, a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 30 \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \,{\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 5 \,{\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{30 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(30*c^2*e^4*x^4 + 6*c^2*d^4 + 3*b*c*d^3*e + 3*a*b*d*e^3 + 6*a^2*e^4 + (b^2
+ 2*a*c)*d^2*e^2 + 30*(2*c^2*d*e^3 + b*c*e^4)*x^3 + 10*(6*c^2*d^2*e^2 + 3*b*c*d
*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 5*(6*c^2*d^3*e + 3*b*c*d^2*e^2 + 3*a*b*e^4 + (b^
2 + 2*a*c)*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 +
5*d^4*e^6*x + d^5*e^5)

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Sympy [A]  time = 117.528, size = 250, normalized size = 1.66 \[ - \frac{6 a^{2} e^{4} + 3 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} + 3 b c d^{3} e + 6 c^{2} d^{4} + 30 c^{2} e^{4} x^{4} + x^{3} \left (30 b c e^{4} + 60 c^{2} d e^{3}\right ) + x^{2} \left (20 a c e^{4} + 10 b^{2} e^{4} + 30 b c d e^{3} + 60 c^{2} d^{2} e^{2}\right ) + x \left (15 a b e^{4} + 10 a c d e^{3} + 5 b^{2} d e^{3} + 15 b c d^{2} e^{2} + 30 c^{2} d^{3} e\right )}{30 d^{5} e^{5} + 150 d^{4} e^{6} x + 300 d^{3} e^{7} x^{2} + 300 d^{2} e^{8} x^{3} + 150 d e^{9} x^{4} + 30 e^{10} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(6*a**2*e**4 + 3*a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 + 3*b*c*d**3*e +
6*c**2*d**4 + 30*c**2*e**4*x**4 + x**3*(30*b*c*e**4 + 60*c**2*d*e**3) + x**2*(2
0*a*c*e**4 + 10*b**2*e**4 + 30*b*c*d*e**3 + 60*c**2*d**2*e**2) + x*(15*a*b*e**4
+ 10*a*c*d*e**3 + 5*b**2*d*e**3 + 15*b*c*d**2*e**2 + 30*c**2*d**3*e))/(30*d**5*e
**5 + 150*d**4*e**6*x + 300*d**3*e**7*x**2 + 300*d**2*e**8*x**3 + 150*d*e**9*x**
4 + 30*e**10*x**5)

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GIAC/XCAS [A]  time = 0.205568, size = 242, normalized size = 1.6 \[ -\frac{{\left (30 \, c^{2} x^{4} e^{4} + 60 \, c^{2} d x^{3} e^{3} + 60 \, c^{2} d^{2} x^{2} e^{2} + 30 \, c^{2} d^{3} x e + 6 \, c^{2} d^{4} + 30 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 15 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 10 \, b^{2} x^{2} e^{4} + 20 \, a c x^{2} e^{4} + 5 \, b^{2} d x e^{3} + 10 \, a c d x e^{3} + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 15 \, a b x e^{4} + 3 \, a b d e^{3} + 6 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{30 \,{\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(30*c^2*x^4*e^4 + 60*c^2*d*x^3*e^3 + 60*c^2*d^2*x^2*e^2 + 30*c^2*d^3*x*e +
6*c^2*d^4 + 30*b*c*x^3*e^4 + 30*b*c*d*x^2*e^3 + 15*b*c*d^2*x*e^2 + 3*b*c*d^3*e
+ 10*b^2*x^2*e^4 + 20*a*c*x^2*e^4 + 5*b^2*d*x*e^3 + 10*a*c*d*x*e^3 + b^2*d^2*e^2
+ 2*a*c*d^2*e^2 + 15*a*b*x*e^4 + 3*a*b*d*e^3 + 6*a^2*e^4)*e^(-5)/(x*e + d)^5