3.2115 \(\int \frac{\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=139 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^5 (d+e x)}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]

[Out]

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

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Rubi [A]  time = 0.321256, antiderivative size = 139, 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 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^5 (d+e x)}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 c \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{5}} + \frac{\int c^{2}\, dx}{e^{4}} - \frac{2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}}{e^{5} \left (d + e x\right )} - \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{e^{5} \left (d + e x\right )^{2}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{2}}{3 e^{5} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.17584, size = 176, normalized size = 1.27 \[ \frac{-e^2 \left (a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )\right )+c e \left (b d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 a e \left (d^2+3 d e x+3 e^2 x^2\right )\right )-6 c (d+e x)^3 (2 c d-b e) \log (d+e x)+c^2 \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )}{3 e^5 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) - e^2*(a^2
*e^2 + a*b*e*(d + 3*e*x) + b^2*(d^2 + 3*d*e*x + 3*e^2*x^2)) + c*e*(-2*a*e*(d^2 +
3*d*e*x + 3*e^2*x^2) + b*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) - 6*c*(2*c*d - b*e
)*(d + e*x)^3*Log[d + e*x])/(3*e^5*(d + e*x)^3)

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Maple [B]  time = 0.013, size = 279, normalized size = 2. \[{\frac{{c}^{2}x}{{e}^{4}}}-{\frac{ab}{{e}^{2} \left ( ex+d \right ) ^{2}}}+2\,{\frac{acd}{{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{c{d}^{2}b}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{{a}^{2}}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{2\,bda}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{2\,ac{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{2}{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{2\,{d}^{3}bc}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}{d}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+2\,{\frac{c\ln \left ( ex+d \right ) b}{{e}^{4}}}-4\,{\frac{{c}^{2}d\ln \left ( ex+d \right ) }{{e}^{5}}}-2\,{\frac{ac}{{e}^{3} \left ( ex+d \right ) }}-{\frac{{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+6\,{\frac{bcd}{{e}^{4} \left ( ex+d \right ) }}-6\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.821291, size = 262, normalized size = 1.88 \[ -\frac{13 \, c^{2} d^{4} - 11 \, b c d^{3} e + a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 3 \,{\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac{c^{2} x}{e^{4}} - \frac{2 \,{\left (2 \, c^{2} d - b c e\right )} \log \left (e x + d\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(13*c^2*d^4 - 11*b*c*d^3*e + a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2 +
3*(6*c^2*d^2*e^2 - 6*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 3*(10*c^2*d^3*e - 9*b*
c*d^2*e^2 + a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6
*x + d^3*e^5) + c^2*x/e^4 - 2*(2*c^2*d - b*c*e)*log(e*x + d)/e^5

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Fricas [A]  time = 0.21621, size = 381, normalized size = 2.74 \[ \frac{3 \, c^{2} e^{4} x^{4} + 9 \, c^{2} d e^{3} x^{3} - 13 \, c^{2} d^{4} + 11 \, b c d^{3} e - a b d e^{3} - a^{2} e^{4} -{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 3 \,{\left (3 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 3 \,{\left (9 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{4} - b c d^{3} e +{\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 3 \,{\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} x^{2} + 3 \,{\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*c^2*e^4*x^4 + 9*c^2*d*e^3*x^3 - 13*c^2*d^4 + 11*b*c*d^3*e - a*b*d*e^3 - a
^2*e^4 - (b^2 + 2*a*c)*d^2*e^2 - 3*(3*c^2*d^2*e^2 - 6*b*c*d*e^3 + (b^2 + 2*a*c)*
e^4)*x^2 - 3*(9*c^2*d^3*e - 9*b*c*d^2*e^2 + a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x - 6
*(2*c^2*d^4 - b*c*d^3*e + (2*c^2*d*e^3 - b*c*e^4)*x^3 + 3*(2*c^2*d^2*e^2 - b*c*d
*e^3)*x^2 + 3*(2*c^2*d^3*e - b*c*d^2*e^2)*x)*log(e*x + d))/(e^8*x^3 + 3*d*e^7*x^
2 + 3*d^2*e^6*x + d^3*e^5)

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Sympy [A]  time = 26.8745, size = 218, normalized size = 1.57 \[ \frac{c^{2} x}{e^{4}} + \frac{2 c \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{a^{2} e^{4} + a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 11 b c d^{3} e + 13 c^{2} d^{4} + x^{2} \left (6 a c e^{4} + 3 b^{2} e^{4} - 18 b c d e^{3} + 18 c^{2} d^{2} e^{2}\right ) + x \left (3 a b e^{4} + 6 a c d e^{3} + 3 b^{2} d e^{3} - 27 b c d^{2} e^{2} + 30 c^{2} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*x/e**4 + 2*c*(b*e - 2*c*d)*log(d + e*x)/e**5 - (a**2*e**4 + a*b*d*e**3 + 2*
a*c*d**2*e**2 + b**2*d**2*e**2 - 11*b*c*d**3*e + 13*c**2*d**4 + x**2*(6*a*c*e**4
+ 3*b**2*e**4 - 18*b*c*d*e**3 + 18*c**2*d**2*e**2) + x*(3*a*b*e**4 + 6*a*c*d*e*
*3 + 3*b**2*d*e**3 - 27*b*c*d**2*e**2 + 30*c**2*d**3*e))/(3*d**3*e**5 + 9*d**2*e
**6*x + 9*d*e**7*x**2 + 3*e**8*x**3)

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GIAC/XCAS [A]  time = 0.204778, size = 230, normalized size = 1.65 \[ c^{2} x e^{\left (-4\right )} - 2 \,{\left (2 \, c^{2} d - b c e\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (13 \, c^{2} d^{4} - 11 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + a b d e^{3} + 3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4} + 2 \, a c e^{4}\right )} x^{2} + a^{2} e^{4} + 3 \,{\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + b^{2} d e^{3} + 2 \, a c d e^{3} + a b e^{4}\right )} x\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^2*x*e^(-4) - 2*(2*c^2*d - b*c*e)*e^(-5)*ln(abs(x*e + d)) - 1/3*(13*c^2*d^4 - 1
1*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 + a*b*d*e^3 + 3*(6*c^2*d^2*e^2 - 6*b*c
*d*e^3 + b^2*e^4 + 2*a*c*e^4)*x^2 + a^2*e^4 + 3*(10*c^2*d^3*e - 9*b*c*d^2*e^2 +
b^2*d*e^3 + 2*a*c*d*e^3 + a*b*e^4)*x)*e^(-5)/(x*e + d)^3