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

Optimal. Leaf size=250 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )-2 c d^2 e (a e+5 b d)+35 c^2 d^4\right )}{16 d^{7/2} e^{9/2}}+\frac{x \left (e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )-2 c d^2 e (11 b d-a e)+29 c^2 d^4\right )}{16 d^3 e^4 \left (d+e x^2\right )}-\frac{x \left (-5 a e^2-7 b d e+19 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{6 d e^4 \left (d+e x^2\right )^3}+\frac{c^2 x}{e^4} \]

[Out]

(c^2*x)/e^4 + ((c*d^2 - b*d*e + a*e^2)^2*x)/(6*d*e^4*(d + e*x^2)^3) - ((19*c*d^2
- 7*b*d*e - 5*a*e^2)*(c*d^2 - b*d*e + a*e^2)*x)/(24*d^2*e^4*(d + e*x^2)^2) + ((
29*c^2*d^4 - 2*c*d^2*e*(11*b*d - a*e) + e^2*(b^2*d^2 + 2*a*b*d*e + 5*a^2*e^2))*x
)/(16*d^3*e^4*(d + e*x^2)) - ((35*c^2*d^4 - 2*c*d^2*e*(5*b*d + a*e) - e^2*(b^2*d
^2 + 2*a*b*d*e + 5*a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(16*d^(7/2)*e^(9/2))

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Rubi [A]  time = 0.951392, antiderivative size = 250, 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 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )-2 c d^2 e (a e+5 b d)+35 c^2 d^4\right )}{16 d^{7/2} e^{9/2}}+\frac{x \left (e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )-2 c d^2 e (11 b d-a e)+29 c^2 d^4\right )}{16 d^3 e^4 \left (d+e x^2\right )}-\frac{x \left (-5 a e^2-7 b d e+19 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{6 d e^4 \left (d+e x^2\right )^3}+\frac{c^2 x}{e^4} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*x)/e^4 + ((c*d^2 - b*d*e + a*e^2)^2*x)/(6*d*e^4*(d + e*x^2)^3) - ((19*c*d^2
- 7*b*d*e - 5*a*e^2)*(c*d^2 - b*d*e + a*e^2)*x)/(24*d^2*e^4*(d + e*x^2)^2) + ((
29*c^2*d^4 - 2*c*d^2*e*(11*b*d - a*e) + e^2*(b^2*d^2 + 2*a*b*d*e + 5*a^2*e^2))*x
)/(16*d^3*e^4*(d + e*x^2)) - ((35*c^2*d^4 - 2*c*d^2*e*(5*b*d + a*e) - e^2*(b^2*d
^2 + 2*a*b*d*e + 5*a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(16*d^(7/2)*e^(9/2))

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Rubi in Sympy [A]  time = 162.457, size = 332, normalized size = 1.33 \[ \frac{c^{2} x^{7}}{e \left (d + e x^{2}\right )^{3}} + \frac{x \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + 7 c^{2} d^{4}\right )}{6 d e^{4} \left (d + e x^{2}\right )^{3}} + \frac{x \left (5 a^{2} e^{4} + 2 a b d e^{3} - 14 a c d^{2} e^{2} - 7 b^{2} d^{2} e^{2} + 26 b c d^{3} e - 91 c^{2} d^{4}\right )}{24 d^{2} e^{4} \left (d + e x^{2}\right )^{2}} + \frac{x \left (5 a^{2} e^{4} + 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 22 b c d^{3} e + 77 c^{2} d^{4}\right )}{16 d^{3} e^{4} \left (d + e x^{2}\right )} + \frac{\left (5 a^{2} e^{4} + 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} + 10 b c d^{3} e - 35 c^{2} d^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{16 d^{\frac{7}{2}} e^{\frac{9}{2}}} \]

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)**4,x)

[Out]

c**2*x**7/(e*(d + e*x**2)**3) + 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 + 7*c**2*d**4)/(6*d*e**4*(d + e*x**2)**3) + x*(5*a
**2*e**4 + 2*a*b*d*e**3 - 14*a*c*d**2*e**2 - 7*b**2*d**2*e**2 + 26*b*c*d**3*e -
91*c**2*d**4)/(24*d**2*e**4*(d + e*x**2)**2) + x*(5*a**2*e**4 + 2*a*b*d*e**3 + 2
*a*c*d**2*e**2 + b**2*d**2*e**2 - 22*b*c*d**3*e + 77*c**2*d**4)/(16*d**3*e**4*(d
+ e*x**2)) + (5*a**2*e**4 + 2*a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 + 1
0*b*c*d**3*e - 35*c**2*d**4)*atan(sqrt(e)*x/sqrt(d))/(16*d**(7/2)*e**(9/2))

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Mathematica [A]  time = 0.280701, size = 267, normalized size = 1.07 \[ -\frac{x \left (e^2 \left (-5 a^2 e^2-2 a b d e+7 b^2 d^2\right )+2 c d^2 e (7 a e-13 b d)+19 c^2 d^4\right )}{24 d^2 e^4 \left (d+e x^2\right )^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )-2 c d^2 e (a e+5 b d)+35 c^2 d^4\right )}{16 d^{7/2} e^{9/2}}+\frac{x \left (e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )+2 c d^2 e (a e-11 b d)+29 c^2 d^4\right )}{16 d^3 e^4 \left (d+e x^2\right )}+\frac{x \left (e (a e-b d)+c d^2\right )^2}{6 d e^4 \left (d+e x^2\right )^3}+\frac{c^2 x}{e^4} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*x)/e^4 + ((c*d^2 + e*(-(b*d) + a*e))^2*x)/(6*d*e^4*(d + e*x^2)^3) - ((19*c^
2*d^4 + 2*c*d^2*e*(-13*b*d + 7*a*e) + e^2*(7*b^2*d^2 - 2*a*b*d*e - 5*a^2*e^2))*x
)/(24*d^2*e^4*(d + e*x^2)^2) + ((29*c^2*d^4 + 2*c*d^2*e*(-11*b*d + a*e) + e^2*(b
^2*d^2 + 2*a*b*d*e + 5*a^2*e^2))*x)/(16*d^3*e^4*(d + e*x^2)) - ((35*c^2*d^4 - 2*
c*d^2*e*(5*b*d + a*e) - e^2*(b^2*d^2 + 2*a*b*d*e + 5*a^2*e^2))*ArcTan[(Sqrt[e]*x
)/Sqrt[d]])/(16*d^(7/2)*e^(9/2))

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Maple [B]  time = 0.018, size = 506, normalized size = 2. \[{\frac{{c}^{2}x}{{e}^{4}}}+{\frac{{b}^{2}{x}^{5}}{16\, \left ( e{x}^{2}+d \right ) ^{3}d}}-{\frac{{b}^{2}{x}^{3}}{6\,e \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{11\,{a}^{2}x}{16\, \left ( e{x}^{2}+d \right ) ^{3}d}}+{\frac{5\,{a}^{2}}{16\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,{e}^{2}{x}^{5}{a}^{2}}{16\, \left ( e{x}^{2}+d \right ) ^{3}{d}^{3}}}+{\frac{ac{x}^{5}}{8\, \left ( e{x}^{2}+d \right ) ^{3}d}}+{\frac{ab{x}^{3}}{3\, \left ( e{x}^{2}+d \right ) ^{3}d}}-{\frac{adxc}{8\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{29\,d{x}^{5}{c}^{2}}{16\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{5\,{a}^{2}e{x}^{3}}{6\, \left ( e{x}^{2}+d \right ) ^{3}{d}^{2}}}+{\frac{ac}{8\,{e}^{2}d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{35\,{c}^{2}d}{16\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{{x}^{3}ac}{3\,e \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{17\,{d}^{2}{x}^{3}{c}^{2}}{6\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{19\,{d}^{3}x{c}^{2}}{16\,{e}^{4} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{e{x}^{5}ab}{8\, \left ( e{x}^{2}+d \right ) ^{3}{d}^{2}}}-{\frac{5\,b{x}^{3}cd}{3\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{3}}}-{\frac{5\,bc{d}^{2}x}{8\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{ab}{8\,{d}^{2}e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{11\,{x}^{5}bc}{8\,e \left ( e{x}^{2}+d \right ) ^{3}}}-{\frac{abx}{8\,e \left ( e{x}^{2}+d \right ) ^{3}}}-{\frac{{b}^{2}dx}{16\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{{b}^{2}}{16\,{e}^{2}d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,bc}{8\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^2*x/e^4+1/16/(e*x^2+d)^3/d*x^5*b^2-1/6/e/(e*x^2+d)^3*x^3*b^2+11/16/(e*x^2+d)^3
/d*x*a^2+5/16/d^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a^2+5/16*e^2/(e*x^2+d)^3/d
^3*x^5*a^2+1/8/(e*x^2+d)^3/d*x^5*a*c+1/3/(e*x^2+d)^3/d*x^3*a*b-1/8/e^2/(e*x^2+d)
^3*d*x*a*c+29/16/e^2/(e*x^2+d)^3*d*x^5*c^2+5/6*e/(e*x^2+d)^3/d^2*x^3*a^2+1/8/e^2
/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a*c-35/16/e^4*d/(d*e)^(1/2)*arctan(x*e/(d
*e)^(1/2))*c^2-1/3/e/(e*x^2+d)^3*x^3*a*c+17/6/e^3/(e*x^2+d)^3*d^2*x^3*c^2+19/16/
e^4/(e*x^2+d)^3*d^3*x*c^2+1/8*e/(e*x^2+d)^3/d^2*x^5*a*b-5/3/e^2/(e*x^2+d)^3*x^3*
b*c*d-5/8/e^3/(e*x^2+d)^3*b*c*d^2*x+1/8/e/d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2)
)*a*b-11/8/e/(e*x^2+d)^3*x^5*b*c-1/8/e/(e*x^2+d)^3*a*b*x-1/16/e^2/(e*x^2+d)^3*b^
2*d*x+1/16/e^2/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b^2+5/8/e^3/(d*e)^(1/2)*arc
tan(x*e/(d*e)^(1/2))*b*c

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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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299297, 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)^4,x, algorithm="fricas")

[Out]

[-1/96*(3*(35*c^2*d^7 - 10*b*c*d^6*e - 2*a*b*d^4*e^3 - 5*a^2*d^3*e^4 - (b^2 + 2*
a*c)*d^5*e^2 + (35*c^2*d^4*e^3 - 10*b*c*d^3*e^4 - 2*a*b*d*e^6 - 5*a^2*e^7 - (b^2
+ 2*a*c)*d^2*e^5)*x^6 + 3*(35*c^2*d^5*e^2 - 10*b*c*d^4*e^3 - 2*a*b*d^2*e^5 - 5*
a^2*d*e^6 - (b^2 + 2*a*c)*d^3*e^4)*x^4 + 3*(35*c^2*d^6*e - 10*b*c*d^5*e^2 - 2*a*
b*d^3*e^4 - 5*a^2*d^2*e^5 - (b^2 + 2*a*c)*d^4*e^3)*x^2)*log((2*d*e*x + (e*x^2 -
d)*sqrt(-d*e))/(e*x^2 + d)) - 2*(48*c^2*d^3*e^3*x^7 + 3*(77*c^2*d^4*e^2 - 22*b*c
*d^3*e^3 + 2*a*b*d*e^5 + 5*a^2*e^6 + (b^2 + 2*a*c)*d^2*e^4)*x^5 + 8*(35*c^2*d^5*
e - 10*b*c*d^4*e^2 + 2*a*b*d^2*e^4 + 5*a^2*d*e^5 - (b^2 + 2*a*c)*d^3*e^3)*x^3 +
3*(35*c^2*d^6 - 10*b*c*d^5*e - 2*a*b*d^3*e^3 + 11*a^2*d^2*e^4 - (b^2 + 2*a*c)*d^
4*e^2)*x)*sqrt(-d*e))/((d^3*e^7*x^6 + 3*d^4*e^6*x^4 + 3*d^5*e^5*x^2 + d^6*e^4)*s
qrt(-d*e)), -1/48*(3*(35*c^2*d^7 - 10*b*c*d^6*e - 2*a*b*d^4*e^3 - 5*a^2*d^3*e^4
- (b^2 + 2*a*c)*d^5*e^2 + (35*c^2*d^4*e^3 - 10*b*c*d^3*e^4 - 2*a*b*d*e^6 - 5*a^2
*e^7 - (b^2 + 2*a*c)*d^2*e^5)*x^6 + 3*(35*c^2*d^5*e^2 - 10*b*c*d^4*e^3 - 2*a*b*d
^2*e^5 - 5*a^2*d*e^6 - (b^2 + 2*a*c)*d^3*e^4)*x^4 + 3*(35*c^2*d^6*e - 10*b*c*d^5
*e^2 - 2*a*b*d^3*e^4 - 5*a^2*d^2*e^5 - (b^2 + 2*a*c)*d^4*e^3)*x^2)*arctan(sqrt(d
*e)*x/d) - (48*c^2*d^3*e^3*x^7 + 3*(77*c^2*d^4*e^2 - 22*b*c*d^3*e^3 + 2*a*b*d*e^
5 + 5*a^2*e^6 + (b^2 + 2*a*c)*d^2*e^4)*x^5 + 8*(35*c^2*d^5*e - 10*b*c*d^4*e^2 +
2*a*b*d^2*e^4 + 5*a^2*d*e^5 - (b^2 + 2*a*c)*d^3*e^3)*x^3 + 3*(35*c^2*d^6 - 10*b*
c*d^5*e - 2*a*b*d^3*e^3 + 11*a^2*d^2*e^4 - (b^2 + 2*a*c)*d^4*e^2)*x)*sqrt(d*e))/
((d^3*e^7*x^6 + 3*d^4*e^6*x^4 + 3*d^5*e^5*x^2 + d^6*e^4)*sqrt(d*e))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265931, size = 400, normalized size = 1.6 \[ c^{2} x e^{\left (-4\right )} - \frac{{\left (35 \, c^{2} d^{4} - 10 \, b c d^{3} e - b^{2} d^{2} e^{2} - 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} - 5 \, a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{16 \, d^{\frac{7}{2}}} + \frac{{\left (87 \, c^{2} d^{4} x^{5} e^{2} - 66 \, b c d^{3} x^{5} e^{3} + 136 \, c^{2} d^{5} x^{3} e + 3 \, b^{2} d^{2} x^{5} e^{4} + 6 \, a c d^{2} x^{5} e^{4} - 80 \, b c d^{4} x^{3} e^{2} + 57 \, c^{2} d^{6} x + 6 \, a b d x^{5} e^{5} - 8 \, b^{2} d^{3} x^{3} e^{3} - 16 \, a c d^{3} x^{3} e^{3} - 30 \, b c d^{5} x e + 15 \, a^{2} x^{5} e^{6} + 16 \, a b d^{2} x^{3} e^{4} - 3 \, b^{2} d^{4} x e^{2} - 6 \, a c d^{4} x e^{2} + 40 \, a^{2} d x^{3} e^{5} - 6 \, a b d^{3} x e^{3} + 33 \, a^{2} d^{2} x e^{4}\right )} e^{\left (-4\right )}}{48 \,{\left (x^{2} e + d\right )}^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^2*x*e^(-4) - 1/16*(35*c^2*d^4 - 10*b*c*d^3*e - b^2*d^2*e^2 - 2*a*c*d^2*e^2 - 2
*a*b*d*e^3 - 5*a^2*e^4)*arctan(x*e^(1/2)/sqrt(d))*e^(-9/2)/d^(7/2) + 1/48*(87*c^
2*d^4*x^5*e^2 - 66*b*c*d^3*x^5*e^3 + 136*c^2*d^5*x^3*e + 3*b^2*d^2*x^5*e^4 + 6*a
*c*d^2*x^5*e^4 - 80*b*c*d^4*x^3*e^2 + 57*c^2*d^6*x + 6*a*b*d*x^5*e^5 - 8*b^2*d^3
*x^3*e^3 - 16*a*c*d^3*x^3*e^3 - 30*b*c*d^5*x*e + 15*a^2*x^5*e^6 + 16*a*b*d^2*x^3
*e^4 - 3*b^2*d^4*x*e^2 - 6*a*c*d^4*x*e^2 + 40*a^2*d*x^3*e^5 - 6*a*b*d^3*x*e^3 +
33*a^2*d^2*x*e^4)*e^(-4)/((x^2*e + d)^3*d^3)