3.262 \(\int \frac{\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx\)

Optimal. Leaf size=201 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e^2 \left (3 a^2 e^2+2 a b d e+3 b^2 d^2\right )-6 c d^2 e (5 b d-a e)+35 c^2 d^4\right )}{8 d^{5/2} e^{9/2}}-\frac{x \left (-3 a e^2-5 b d e+13 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}-\frac{c x (3 c d-2 b e)}{e^4}+\frac{c^2 x^3}{3 e^3} \]

[Out]

-((c*(3*c*d - 2*b*e)*x)/e^4) + (c^2*x^3)/(3*e^3) + ((c*d^2 - b*d*e + a*e^2)^2*x)
/(4*d*e^4*(d + e*x^2)^2) - ((13*c*d^2 - 5*b*d*e - 3*a*e^2)*(c*d^2 - b*d*e + a*e^
2)*x)/(8*d^2*e^4*(d + e*x^2)) + ((35*c^2*d^4 - 6*c*d^2*e*(5*b*d - a*e) + e^2*(3*
b^2*d^2 + 2*a*b*d*e + 3*a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^(9/2
))

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Rubi [A]  time = 0.826522, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e^2 \left (3 a^2 e^2+2 a b d e+3 b^2 d^2\right )-6 c d^2 e (5 b d-a e)+35 c^2 d^4\right )}{8 d^{5/2} e^{9/2}}-\frac{x \left (-3 a e^2-5 b d e+13 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}-\frac{c x (3 c d-2 b e)}{e^4}+\frac{c^2 x^3}{3 e^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^2/(d + e*x^2)^3,x]

[Out]

-((c*(3*c*d - 2*b*e)*x)/e^4) + (c^2*x^3)/(3*e^3) + ((c*d^2 - b*d*e + a*e^2)^2*x)
/(4*d*e^4*(d + e*x^2)^2) - ((13*c*d^2 - 5*b*d*e - 3*a*e^2)*(c*d^2 - b*d*e + a*e^
2)*x)/(8*d^2*e^4*(d + e*x^2)) + ((35*c^2*d^4 - 6*c*d^2*e*(5*b*d - a*e) + e^2*(3*
b^2*d^2 + 2*a*b*d*e + 3*a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^(9/2
))

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Rubi in Sympy [A]  time = 170.48, size = 279, normalized size = 1.39 \[ \frac{c^{2} x^{7}}{3 e \left (d + e x^{2}\right )^{2}} + \frac{c x \left (6 b e - 7 c d\right )}{3 e^{4}} + \frac{x \left (3 a^{2} e^{4} - 6 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} - 6 b c d^{3} e + 7 c^{2} d^{4}\right )}{12 d e^{4} \left (d + e x^{2}\right )^{2}} + \frac{x \left (3 a^{2} e^{4} + 2 a b d e^{3} - 10 a c d^{2} e^{2} - 5 b^{2} d^{2} e^{2} + 18 b c d^{3} e - 21 c^{2} d^{4}\right )}{8 d^{2} e^{4} \left (d + e x^{2}\right )} + \frac{\left (3 a^{2} e^{4} + 2 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} - 30 b c d^{3} e + 35 c^{2} d^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 d^{\frac{5}{2}} e^{\frac{9}{2}}} \]

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)**3,x)

[Out]

c**2*x**7/(3*e*(d + e*x**2)**2) + c*x*(6*b*e - 7*c*d)/(3*e**4) + x*(3*a**2*e**4
- 6*a*b*d*e**3 + 6*a*c*d**2*e**2 + 3*b**2*d**2*e**2 - 6*b*c*d**3*e + 7*c**2*d**4
)/(12*d*e**4*(d + e*x**2)**2) + x*(3*a**2*e**4 + 2*a*b*d*e**3 - 10*a*c*d**2*e**2
- 5*b**2*d**2*e**2 + 18*b*c*d**3*e - 21*c**2*d**4)/(8*d**2*e**4*(d + e*x**2)) +
(3*a**2*e**4 + 2*a*b*d*e**3 + 6*a*c*d**2*e**2 + 3*b**2*d**2*e**2 - 30*b*c*d**3*
e + 35*c**2*d**4)*atan(sqrt(e)*x/sqrt(d))/(8*d**(5/2)*e**(9/2))

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Mathematica [A]  time = 0.213285, size = 217, normalized size = 1.08 \[ -\frac{x \left (e^2 \left (-3 a^2 e^2-2 a b d e+5 b^2 d^2\right )-2 c d^2 e (9 b d-5 a e)+13 c^2 d^4\right )}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e^2 \left (3 a^2 e^2+2 a b d e+3 b^2 d^2\right )+6 c d^2 e (a e-5 b d)+35 c^2 d^4\right )}{8 d^{5/2} e^{9/2}}+\frac{x \left (e (a e-b d)+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}+\frac{c x (2 b e-3 c d)}{e^4}+\frac{c^2 x^3}{3 e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^2/(d + e*x^2)^3,x]

[Out]

(c*(-3*c*d + 2*b*e)*x)/e^4 + (c^2*x^3)/(3*e^3) + ((c*d^2 + e*(-(b*d) + a*e))^2*x
)/(4*d*e^4*(d + e*x^2)^2) - ((13*c^2*d^4 - 2*c*d^2*e*(9*b*d - 5*a*e) + e^2*(5*b^
2*d^2 - 2*a*b*d*e - 3*a^2*e^2))*x)/(8*d^2*e^4*(d + e*x^2)) + ((35*c^2*d^4 + 6*c*
d^2*e*(-5*b*d + a*e) + e^2*(3*b^2*d^2 + 2*a*b*d*e + 3*a^2*e^2))*ArcTan[(Sqrt[e]*
x)/Sqrt[d]])/(8*d^(5/2)*e^(9/2))

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Maple [B]  time = 0.016, size = 402, normalized size = 2. \[{\frac{{c}^{2}{x}^{3}}{3\,{e}^{3}}}+2\,{\frac{bcx}{{e}^{3}}}-3\,{\frac{{c}^{2}dx}{{e}^{4}}}+{\frac{3\,{a}^{2}e{x}^{3}}{8\, \left ( e{x}^{2}+d \right ) ^{2}{d}^{2}}}+{\frac{ab{x}^{3}}{4\, \left ( e{x}^{2}+d \right ) ^{2}d}}-{\frac{5\,{x}^{3}ac}{4\,e \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{5\,{b}^{2}{x}^{3}}{8\,e \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{9\,b{x}^{3}cd}{4\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{13\,{d}^{2}{x}^{3}{c}^{2}}{8\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{5\,{a}^{2}x}{8\, \left ( e{x}^{2}+d \right ) ^{2}d}}-{\frac{abx}{4\,e \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{3\,adxc}{4\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{3\,{b}^{2}dx}{8\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{7\,bc{d}^{2}x}{4\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{11\,{d}^{3}x{c}^{2}}{8\,{e}^{4} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{3\,{a}^{2}}{8\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{ab}{4\,de}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,ac}{4\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,{b}^{2}}{8\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{15\,bcd}{4\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{35\,{c}^{2}{d}^{2}}{8\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^2/(e*x^2+d)^3,x)

[Out]

1/3*c^2*x^3/e^3+2*c/e^3*b*x-3*c^2*d*x/e^4+3/8*e/(e*x^2+d)^2/d^2*x^3*a^2+1/4/(e*x
^2+d)^2/d*x^3*a*b-5/4/e/(e*x^2+d)^2*x^3*a*c-5/8/e/(e*x^2+d)^2*x^3*b^2+9/4/e^2/(e
*x^2+d)^2*x^3*b*c*d-13/8/e^3/(e*x^2+d)^2*d^2*x^3*c^2+5/8/(e*x^2+d)^2/d*x*a^2-1/4
/e/(e*x^2+d)^2*a*b*x-3/4/e^2/(e*x^2+d)^2*d*x*a*c-3/8/e^2/(e*x^2+d)^2*b^2*d*x+7/4
/e^3/(e*x^2+d)^2*b*c*d^2*x-11/8/e^4/(e*x^2+d)^2*d^3*x*c^2+3/8/d^2/(d*e)^(1/2)*ar
ctan(x*e/(d*e)^(1/2))*a^2+1/4/e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a*b+3/4/e^
2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a*c+3/8/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(
1/2))*b^2-15/4/e^3*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b*c+35/8/e^4*d^2/(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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295759, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/(e*x^2 + d)^3,x, algorithm="fricas")

[Out]

[1/48*(3*(35*c^2*d^6 - 30*b*c*d^5*e + 2*a*b*d^3*e^3 + 3*a^2*d^2*e^4 + 3*(b^2 + 2
*a*c)*d^4*e^2 + (35*c^2*d^4*e^2 - 30*b*c*d^3*e^3 + 2*a*b*d*e^5 + 3*a^2*e^6 + 3*(
b^2 + 2*a*c)*d^2*e^4)*x^4 + 2*(35*c^2*d^5*e - 30*b*c*d^4*e^2 + 2*a*b*d^2*e^4 + 3
*a^2*d*e^5 + 3*(b^2 + 2*a*c)*d^3*e^3)*x^2)*log((2*d*e*x + (e*x^2 - d)*sqrt(-d*e)
)/(e*x^2 + d)) + 2*(8*c^2*d^2*e^3*x^7 - 8*(7*c^2*d^3*e^2 - 6*b*c*d^2*e^3)*x^5 -
(175*c^2*d^4*e - 150*b*c*d^3*e^2 - 6*a*b*d*e^4 - 9*a^2*e^5 + 15*(b^2 + 2*a*c)*d^
2*e^3)*x^3 - 3*(35*c^2*d^5 - 30*b*c*d^4*e + 2*a*b*d^2*e^3 - 5*a^2*d*e^4 + 3*(b^2
+ 2*a*c)*d^3*e^2)*x)*sqrt(-d*e))/((d^2*e^6*x^4 + 2*d^3*e^5*x^2 + d^4*e^4)*sqrt(
-d*e)), 1/24*(3*(35*c^2*d^6 - 30*b*c*d^5*e + 2*a*b*d^3*e^3 + 3*a^2*d^2*e^4 + 3*(
b^2 + 2*a*c)*d^4*e^2 + (35*c^2*d^4*e^2 - 30*b*c*d^3*e^3 + 2*a*b*d*e^5 + 3*a^2*e^
6 + 3*(b^2 + 2*a*c)*d^2*e^4)*x^4 + 2*(35*c^2*d^5*e - 30*b*c*d^4*e^2 + 2*a*b*d^2*
e^4 + 3*a^2*d*e^5 + 3*(b^2 + 2*a*c)*d^3*e^3)*x^2)*arctan(sqrt(d*e)*x/d) + (8*c^2
*d^2*e^3*x^7 - 8*(7*c^2*d^3*e^2 - 6*b*c*d^2*e^3)*x^5 - (175*c^2*d^4*e - 150*b*c*
d^3*e^2 - 6*a*b*d*e^4 - 9*a^2*e^5 + 15*(b^2 + 2*a*c)*d^2*e^3)*x^3 - 3*(35*c^2*d^
5 - 30*b*c*d^4*e + 2*a*b*d^2*e^3 - 5*a^2*d*e^4 + 3*(b^2 + 2*a*c)*d^3*e^2)*x)*sqr
t(d*e))/((d^2*e^6*x^4 + 2*d^3*e^5*x^2 + d^4*e^4)*sqrt(d*e))]

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Sympy [A]  time = 53.1541, size = 396, normalized size = 1.97 \[ \frac{c^{2} x^{3}}{3 e^{3}} - \frac{\sqrt{- \frac{1}{d^{5} e^{9}}} \left (3 a^{2} e^{4} + 2 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} - 30 b c d^{3} e + 35 c^{2} d^{4}\right ) \log{\left (- d^{3} e^{4} \sqrt{- \frac{1}{d^{5} e^{9}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{d^{5} e^{9}}} \left (3 a^{2} e^{4} + 2 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} - 30 b c d^{3} e + 35 c^{2} d^{4}\right ) \log{\left (d^{3} e^{4} \sqrt{- \frac{1}{d^{5} e^{9}}} + x \right )}}{16} + \frac{x^{3} \left (3 a^{2} e^{5} + 2 a b d e^{4} - 10 a c d^{2} e^{3} - 5 b^{2} d^{2} e^{3} + 18 b c d^{3} e^{2} - 13 c^{2} d^{4} e\right ) + x \left (5 a^{2} d e^{4} - 2 a b d^{2} e^{3} - 6 a c d^{3} e^{2} - 3 b^{2} d^{3} e^{2} + 14 b c d^{4} e - 11 c^{2} d^{5}\right )}{8 d^{4} e^{4} + 16 d^{3} e^{5} x^{2} + 8 d^{2} e^{6} x^{4}} + \frac{x \left (2 b c e - 3 c^{2} d\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**2/(e*x**2+d)**3,x)

[Out]

c**2*x**3/(3*e**3) - sqrt(-1/(d**5*e**9))*(3*a**2*e**4 + 2*a*b*d*e**3 + 6*a*c*d*
*2*e**2 + 3*b**2*d**2*e**2 - 30*b*c*d**3*e + 35*c**2*d**4)*log(-d**3*e**4*sqrt(-
1/(d**5*e**9)) + x)/16 + sqrt(-1/(d**5*e**9))*(3*a**2*e**4 + 2*a*b*d*e**3 + 6*a*
c*d**2*e**2 + 3*b**2*d**2*e**2 - 30*b*c*d**3*e + 35*c**2*d**4)*log(d**3*e**4*sqr
t(-1/(d**5*e**9)) + x)/16 + (x**3*(3*a**2*e**5 + 2*a*b*d*e**4 - 10*a*c*d**2*e**3
- 5*b**2*d**2*e**3 + 18*b*c*d**3*e**2 - 13*c**2*d**4*e) + x*(5*a**2*d*e**4 - 2*
a*b*d**2*e**3 - 6*a*c*d**3*e**2 - 3*b**2*d**3*e**2 + 14*b*c*d**4*e - 11*c**2*d**
5))/(8*d**4*e**4 + 16*d**3*e**5*x**2 + 8*d**2*e**6*x**4) + x*(2*b*c*e - 3*c**2*d
)/e**4

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GIAC/XCAS [A]  time = 0.267756, size = 329, normalized size = 1.64 \[ \frac{1}{3} \,{\left (c^{2} x^{3} e^{6} - 9 \, c^{2} d x e^{5} + 6 \, b c x e^{6}\right )} e^{\left (-9\right )} + \frac{{\left (35 \, c^{2} d^{4} - 30 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} + 2 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{8 \, d^{\frac{5}{2}}} - \frac{{\left (13 \, c^{2} d^{4} x^{3} e - 18 \, b c d^{3} x^{3} e^{2} + 11 \, c^{2} d^{5} x + 5 \, b^{2} d^{2} x^{3} e^{3} + 10 \, a c d^{2} x^{3} e^{3} - 14 \, b c d^{4} x e - 2 \, a b d x^{3} e^{4} + 3 \, b^{2} d^{3} x e^{2} + 6 \, a c d^{3} x e^{2} - 3 \, a^{2} x^{3} e^{5} + 2 \, a b d^{2} x e^{3} - 5 \, a^{2} d x e^{4}\right )} e^{\left (-4\right )}}{8 \,{\left (x^{2} e + d\right )}^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/(e*x^2 + d)^3,x, algorithm="giac")

[Out]

1/3*(c^2*x^3*e^6 - 9*c^2*d*x*e^5 + 6*b*c*x*e^6)*e^(-9) + 1/8*(35*c^2*d^4 - 30*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 + 3*a^2*e^4)*arctan(x*e^(1
/2)/sqrt(d))*e^(-9/2)/d^(5/2) - 1/8*(13*c^2*d^4*x^3*e - 18*b*c*d^3*x^3*e^2 + 11*
c^2*d^5*x + 5*b^2*d^2*x^3*e^3 + 10*a*c*d^2*x^3*e^3 - 14*b*c*d^4*x*e - 2*a*b*d*x^
3*e^4 + 3*b^2*d^3*x*e^2 + 6*a*c*d^3*x*e^2 - 3*a^2*x^3*e^5 + 2*a*b*d^2*x*e^3 - 5*
a^2*d*x*e^4)*e^(-4)/((x^2*e + d)^2*d^2)