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

Optimal. Leaf size=166 \[ \frac{x \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )}{e^4}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{2 d e^4 \left (d+e x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right ) \left (7 c d^2-e (a e+3 b d)\right )}{2 d^{3/2} e^{9/2}}-\frac{2 c x^3 (c d-b e)}{3 e^3}+\frac{c^2 x^5}{5 e^2} \]

[Out]

((3*c^2*d^2 + b^2*e^2 - 2*c*e*(2*b*d - a*e))*x)/e^4 - (2*c*(c*d - b*e)*x^3)/(3*e
^3) + (c^2*x^5)/(5*e^2) + ((c*d^2 - b*d*e + a*e^2)^2*x)/(2*d*e^4*(d + e*x^2)) -
((c*d^2 - b*d*e + a*e^2)*(7*c*d^2 - e*(3*b*d + a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]]
)/(2*d^(3/2)*e^(9/2))

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Rubi [A]  time = 0.535022, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )}{e^4}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{2 d e^4 \left (d+e x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right ) \left (7 c d^2-e (a e+3 b d)\right )}{2 d^{3/2} e^{9/2}}-\frac{2 c x^3 (c d-b e)}{3 e^3}+\frac{c^2 x^5}{5 e^2} \]

Antiderivative was successfully verified.

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

[Out]

((3*c^2*d^2 + b^2*e^2 - 2*c*e*(2*b*d - a*e))*x)/e^4 - (2*c*(c*d - b*e)*x^3)/(3*e
^3) + (c^2*x^5)/(5*e^2) + ((c*d^2 - b*d*e + a*e^2)^2*x)/(2*d*e^4*(d + e*x^2)) -
((c*d^2 - b*d*e + a*e^2)*(7*c*d^2 - e*(3*b*d + a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]]
)/(2*d^(3/2)*e^(9/2))

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Rubi in Sympy [A]  time = 155.121, size = 212, normalized size = 1.28 \[ \frac{c^{2} x^{7}}{5 e \left (d + e x^{2}\right )} + \frac{c x^{3} \left (10 b e - 7 c d\right )}{15 e^{3}} + \frac{x \left (10 a c e^{2} + 5 b^{2} e^{2} - 20 b c d e + 14 c^{2} d^{2}\right )}{5 e^{4}} + \frac{x \left (5 a^{2} e^{4} - 10 a b d e^{3} + 10 a c d^{2} e^{2} + 5 b^{2} d^{2} e^{2} - 10 b c d^{3} e + 7 c^{2} d^{4}\right )}{10 d e^{4} \left (d + e x^{2}\right )} + \frac{\left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 d^{\frac{3}{2}} e^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*x**7/(5*e*(d + e*x**2)) + c*x**3*(10*b*e - 7*c*d)/(15*e**3) + x*(10*a*c*e**
2 + 5*b**2*e**2 - 20*b*c*d*e + 14*c**2*d**2)/(5*e**4) + x*(5*a**2*e**4 - 10*a*b*
d*e**3 + 10*a*c*d**2*e**2 + 5*b**2*d**2*e**2 - 10*b*c*d**3*e + 7*c**2*d**4)/(10*
d*e**4*(d + e*x**2)) + (a*e**2 - b*d*e + c*d**2)*(a*e**2 + 3*b*d*e - 7*c*d**2)*a
tan(sqrt(e)*x/sqrt(d))/(2*d**(3/2)*e**(9/2))

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Mathematica [A]  time = 0.186173, size = 183, normalized size = 1.1 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-e^2 \left (a^2 e^2+2 a b d e-3 b^2 d^2\right )+2 c d^2 e (3 a e-5 b d)+7 c^2 d^4\right )}{2 d^{3/2} e^{9/2}}+\frac{x \left (2 c e (a e-2 b d)+b^2 e^2+3 c^2 d^2\right )}{e^4}+\frac{x \left (e (a e-b d)+c d^2\right )^2}{2 d e^4 \left (d+e x^2\right )}+\frac{2 c x^3 (b e-c d)}{3 e^3}+\frac{c^2 x^5}{5 e^2} \]

Antiderivative was successfully verified.

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

[Out]

((3*c^2*d^2 + b^2*e^2 + 2*c*e*(-2*b*d + a*e))*x)/e^4 + (2*c*(-(c*d) + b*e)*x^3)/
(3*e^3) + (c^2*x^5)/(5*e^2) + ((c*d^2 + e*(-(b*d) + a*e))^2*x)/(2*d*e^4*(d + e*x
^2)) - ((7*c^2*d^4 + 2*c*d^2*e*(-5*b*d + 3*a*e) - e^2*(-3*b^2*d^2 + 2*a*b*d*e +
a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*e^(9/2))

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Maple [B]  time = 0.015, size = 320, normalized size = 1.9 \[{\frac{{c}^{2}{x}^{5}}{5\,{e}^{2}}}+{\frac{2\,b{x}^{3}c}{3\,{e}^{2}}}-{\frac{2\,{c}^{2}d{x}^{3}}{3\,{e}^{3}}}+2\,{\frac{acx}{{e}^{2}}}+{\frac{{b}^{2}x}{{e}^{2}}}-4\,{\frac{bcdx}{{e}^{3}}}+3\,{\frac{{c}^{2}{d}^{2}x}{{e}^{4}}}+{\frac{{a}^{2}x}{2\,d \left ( e{x}^{2}+d \right ) }}-{\frac{abx}{e \left ( e{x}^{2}+d \right ) }}+{\frac{adxc}{{e}^{2} \left ( e{x}^{2}+d \right ) }}+{\frac{dx{b}^{2}}{2\,{e}^{2} \left ( e{x}^{2}+d \right ) }}-{\frac{xb{d}^{2}c}{{e}^{3} \left ( e{x}^{2}+d \right ) }}+{\frac{{d}^{3}x{c}^{2}}{2\,{e}^{4} \left ( e{x}^{2}+d \right ) }}+{\frac{{a}^{2}}{2\,d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{ab}{e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-3\,{\frac{acd}{{e}^{2}\sqrt{de}}\arctan \left ({\frac{ex}{\sqrt{de}}} \right ) }-{\frac{3\,{b}^{2}d}{2\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+5\,{\frac{bc{d}^{2}}{{e}^{3}\sqrt{de}}\arctan \left ({\frac{ex}{\sqrt{de}}} \right ) }-{\frac{7\,{c}^{2}{d}^{3}}{2\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284966, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, c^{2} d^{5} - 10 \, b c d^{4} e - 2 \, a b d^{2} e^{3} - a^{2} d e^{4} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} +{\left (7 \, c^{2} d^{4} e - 10 \, b c d^{3} e^{2} - 2 \, a b d e^{4} - a^{2} e^{5} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) - 2 \,{\left (6 \, c^{2} d e^{3} x^{7} - 2 \,{\left (7 \, c^{2} d^{2} e^{2} - 10 \, b c d e^{3}\right )} x^{5} + 10 \,{\left (7 \, c^{2} d^{3} e - 10 \, b c d^{2} e^{2} + 3 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{3} + 15 \,{\left (7 \, c^{2} d^{4} - 10 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x\right )} \sqrt{-d e}}{60 \,{\left (d e^{5} x^{2} + d^{2} e^{4}\right )} \sqrt{-d e}}, -\frac{15 \,{\left (7 \, c^{2} d^{5} - 10 \, b c d^{4} e - 2 \, a b d^{2} e^{3} - a^{2} d e^{4} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} +{\left (7 \, c^{2} d^{4} e - 10 \, b c d^{3} e^{2} - 2 \, a b d e^{4} - a^{2} e^{5} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) -{\left (6 \, c^{2} d e^{3} x^{7} - 2 \,{\left (7 \, c^{2} d^{2} e^{2} - 10 \, b c d e^{3}\right )} x^{5} + 10 \,{\left (7 \, c^{2} d^{3} e - 10 \, b c d^{2} e^{2} + 3 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{3} + 15 \,{\left (7 \, c^{2} d^{4} - 10 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x\right )} \sqrt{d e}}{30 \,{\left (d e^{5} x^{2} + d^{2} e^{4}\right )} \sqrt{d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/60*(15*(7*c^2*d^5 - 10*b*c*d^4*e - 2*a*b*d^2*e^3 - a^2*d*e^4 + 3*(b^2 + 2*a*
c)*d^3*e^2 + (7*c^2*d^4*e - 10*b*c*d^3*e^2 - 2*a*b*d*e^4 - a^2*e^5 + 3*(b^2 + 2*
a*c)*d^2*e^3)*x^2)*log((2*d*e*x + (e*x^2 - d)*sqrt(-d*e))/(e*x^2 + d)) - 2*(6*c^
2*d*e^3*x^7 - 2*(7*c^2*d^2*e^2 - 10*b*c*d*e^3)*x^5 + 10*(7*c^2*d^3*e - 10*b*c*d^
2*e^2 + 3*(b^2 + 2*a*c)*d*e^3)*x^3 + 15*(7*c^2*d^4 - 10*b*c*d^3*e - 2*a*b*d*e^3
+ a^2*e^4 + 3*(b^2 + 2*a*c)*d^2*e^2)*x)*sqrt(-d*e))/((d*e^5*x^2 + d^2*e^4)*sqrt(
-d*e)), -1/30*(15*(7*c^2*d^5 - 10*b*c*d^4*e - 2*a*b*d^2*e^3 - a^2*d*e^4 + 3*(b^2
+ 2*a*c)*d^3*e^2 + (7*c^2*d^4*e - 10*b*c*d^3*e^2 - 2*a*b*d*e^4 - a^2*e^5 + 3*(b
^2 + 2*a*c)*d^2*e^3)*x^2)*arctan(sqrt(d*e)*x/d) - (6*c^2*d*e^3*x^7 - 2*(7*c^2*d^
2*e^2 - 10*b*c*d*e^3)*x^5 + 10*(7*c^2*d^3*e - 10*b*c*d^2*e^2 + 3*(b^2 + 2*a*c)*d
*e^3)*x^3 + 15*(7*c^2*d^4 - 10*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + 3*(b^2 + 2*a*
c)*d^2*e^2)*x)*sqrt(d*e))/((d*e^5*x^2 + d^2*e^4)*sqrt(d*e))]

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Sympy [A]  time = 12.4832, size = 479, normalized size = 2.89 \[ \frac{c^{2} x^{5}}{5 e^{2}} + \frac{x \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{2 d^{2} e^{4} + 2 d e^{5} x^{2}} - \frac{\sqrt{- \frac{1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right ) \log{\left (- \frac{d^{2} e^{4} \sqrt{- \frac{1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right )}{a^{2} e^{4} + 2 a b d e^{3} - 6 a c d^{2} e^{2} - 3 b^{2} d^{2} e^{2} + 10 b c d^{3} e - 7 c^{2} d^{4}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right ) \log{\left (\frac{d^{2} e^{4} \sqrt{- \frac{1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right )}{a^{2} e^{4} + 2 a b d e^{3} - 6 a c d^{2} e^{2} - 3 b^{2} d^{2} e^{2} + 10 b c d^{3} e - 7 c^{2} d^{4}} + x \right )}}{4} + \frac{x^{3} \left (2 b c e - 2 c^{2} d\right )}{3 e^{3}} + \frac{x \left (2 a c e^{2} + b^{2} e^{2} - 4 b c d e + 3 c^{2} d^{2}\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.265461, size = 279, normalized size = 1.68 \[ \frac{1}{15} \,{\left (3 \, c^{2} x^{5} e^{8} - 10 \, c^{2} d x^{3} e^{7} + 10 \, b c x^{3} e^{8} + 45 \, c^{2} d^{2} x e^{6} - 60 \, b c d x e^{7} + 15 \, b^{2} x e^{8} + 30 \, a c x e^{8}\right )} e^{\left (-10\right )} - \frac{{\left (7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} - 2 \, a b d e^{3} - a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{2 \, d^{\frac{3}{2}}} + \frac{{\left (c^{2} d^{4} x - 2 \, b c d^{3} x e + b^{2} d^{2} x e^{2} + 2 \, a c d^{2} x e^{2} - 2 \, a b d x e^{3} + a^{2} x e^{4}\right )} e^{\left (-4\right )}}{2 \,{\left (x^{2} e + d\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*c^2*x^5*e^8 - 10*c^2*d*x^3*e^7 + 10*b*c*x^3*e^8 + 45*c^2*d^2*x*e^6 - 60*
b*c*d*x*e^7 + 15*b^2*x*e^8 + 30*a*c*x*e^8)*e^(-10) - 1/2*(7*c^2*d^4 - 10*b*c*d^3
*e + 3*b^2*d^2*e^2 + 6*a*c*d^2*e^2 - 2*a*b*d*e^3 - a^2*e^4)*arctan(x*e^(1/2)/sqr
t(d))*e^(-9/2)/d^(3/2) + 1/2*(c^2*d^4*x - 2*b*c*d^3*x*e + b^2*d^2*x*e^2 + 2*a*c*
d^2*x*e^2 - 2*a*b*d*x*e^3 + a^2*x*e^4)*e^(-4)/((x^2*e + d)*d)