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

Optimal. Leaf size=150 \[ -\frac{x \left (7 c d^2-e (5 a e+b d)\right )}{24 d^2 e^2 \left (d+e x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e (5 a e+b d)+c d^2\right )}{16 d^{7/2} e^{5/2}}+\frac{x \left (e (5 a e+b d)+c d^2\right )}{16 d^3 e^2 \left (d+e x^2\right )}+\frac{x \left (a+\frac{d (c d-b e)}{e^2}\right )}{6 d \left (d+e x^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.308372, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{x \left (7 c d^2-e (5 a e+b d)\right )}{24 d^2 e^2 \left (d+e x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e (5 a e+b d)+c d^2\right )}{16 d^{7/2} e^{5/2}}+\frac{x \left (e (5 a e+b d)+c d^2\right )}{16 d^3 e^2 \left (d+e x^2\right )}+\frac{x \left (a+\frac{d (c d-b e)}{e^2}\right )}{6 d \left (d+e x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 38.8595, size = 144, normalized size = 0.96 \[ \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{6 d e^{2} \left (d + e x^{2}\right )^{3}} + \frac{x \left (5 a e^{2} + b d e - 7 c d^{2}\right )}{24 d^{2} e^{2} \left (d + e x^{2}\right )^{2}} + \frac{x \left (5 a e^{2} + b d e + c d^{2}\right )}{16 d^{3} e^{2} \left (d + e x^{2}\right )} + \frac{\left (5 a e^{2} + b d e + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{16 d^{\frac{7}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a*e**2 - b*d*e + c*d**2)/(6*d*e**2*(d + e*x**2)**3) + x*(5*a*e**2 + b*d*e - 7
*c*d**2)/(24*d**2*e**2*(d + e*x**2)**2) + x*(5*a*e**2 + b*d*e + c*d**2)/(16*d**3
*e**2*(d + e*x**2)) + (5*a*e**2 + b*d*e + c*d**2)*atan(sqrt(e)*x/sqrt(d))/(16*d*
*(7/2)*e**(5/2))

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Mathematica [A]  time = 0.250498, size = 142, normalized size = 0.95 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e (5 a e+b d)+c d^2\right )}{16 d^{7/2} e^{5/2}}+\frac{x \left (e \left (a e \left (33 d^2+40 d e x^2+15 e^2 x^4\right )+b d \left (-3 d^2+8 d e x^2+3 e^2 x^4\right )\right )+c d^2 \left (-3 d^2-8 d e x^2+3 e^2 x^4\right )\right )}{48 d^3 e^2 \left (d+e x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*d^2*(-3*d^2 - 8*d*e*x^2 + 3*e^2*x^4) + e*(b*d*(-3*d^2 + 8*d*e*x^2 + 3*e^2*
x^4) + a*e*(33*d^2 + 40*d*e*x^2 + 15*e^2*x^4))))/(48*d^3*e^2*(d + e*x^2)^3) + ((
c*d^2 + e*(b*d + 5*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(16*d^(7/2)*e^(5/2))

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Maple [A]  time = 0.013, size = 158, normalized size = 1.1 \[{\frac{1}{ \left ( e{x}^{2}+d \right ) ^{3}} \left ({\frac{ \left ( 5\,a{e}^{2}+bde+c{d}^{2} \right ){x}^{5}}{16\,{d}^{3}}}+{\frac{ \left ( 5\,a{e}^{2}+bde-c{d}^{2} \right ){x}^{3}}{6\,{d}^{2}e}}+{\frac{ \left ( 11\,a{e}^{2}-bde-c{d}^{2} \right ) x}{16\,{e}^{2}d}} \right ) }+{\frac{5\,a}{16\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{b}{16\,{d}^{2}e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{c}{16\,{e}^{2}d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/16*(5*a*e^2+b*d*e+c*d^2)/d^3*x^5+1/6*(5*a*e^2+b*d*e-c*d^2)/d^2/e*x^3+1/16*(11
*a*e^2-b*d*e-c*d^2)/e^2/d*x)/(e*x^2+d)^3+5/16/d^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(
1/2))*a+1/16/d^2/e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b+1/16/d/e^2/(d*e)^(1/2)*
arctan(x*e/(d*e)^(1/2))*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)/(e*x^2 + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281113, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (c d^{2} e^{3} + b d e^{4} + 5 \, a e^{5}\right )} x^{6} + c d^{5} + b d^{4} e + 5 \, a d^{3} e^{2} + 3 \,{\left (c d^{3} e^{2} + b d^{2} e^{3} + 5 \, a d e^{4}\right )} x^{4} + 3 \,{\left (c d^{4} e + b d^{3} e^{2} + 5 \, a d^{2} e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) + 2 \,{\left (3 \,{\left (c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{5} - 8 \,{\left (c d^{3} e - b d^{2} e^{2} - 5 \, a d e^{3}\right )} x^{3} - 3 \,{\left (c d^{4} + b d^{3} e - 11 \, a d^{2} e^{2}\right )} x\right )} \sqrt{-d e}}{96 \,{\left (d^{3} e^{5} x^{6} + 3 \, d^{4} e^{4} x^{4} + 3 \, d^{5} e^{3} x^{2} + d^{6} e^{2}\right )} \sqrt{-d e}}, \frac{3 \,{\left ({\left (c d^{2} e^{3} + b d e^{4} + 5 \, a e^{5}\right )} x^{6} + c d^{5} + b d^{4} e + 5 \, a d^{3} e^{2} + 3 \,{\left (c d^{3} e^{2} + b d^{2} e^{3} + 5 \, a d e^{4}\right )} x^{4} + 3 \,{\left (c d^{4} e + b d^{3} e^{2} + 5 \, a d^{2} e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (3 \,{\left (c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{5} - 8 \,{\left (c d^{3} e - b d^{2} e^{2} - 5 \, a d e^{3}\right )} x^{3} - 3 \,{\left (c d^{4} + b d^{3} e - 11 \, a d^{2} e^{2}\right )} x\right )} \sqrt{d e}}{48 \,{\left (d^{3} e^{5} x^{6} + 3 \, d^{4} e^{4} x^{4} + 3 \, d^{5} e^{3} x^{2} + d^{6} e^{2}\right )} \sqrt{d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.4936, size = 241, normalized size = 1.61 \[ - \frac{\sqrt{- \frac{1}{d^{7} e^{5}}} \left (5 a e^{2} + b d e + c d^{2}\right ) \log{\left (- d^{4} e^{2} \sqrt{- \frac{1}{d^{7} e^{5}}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{d^{7} e^{5}}} \left (5 a e^{2} + b d e + c d^{2}\right ) \log{\left (d^{4} e^{2} \sqrt{- \frac{1}{d^{7} e^{5}}} + x \right )}}{32} + \frac{x^{5} \left (15 a e^{4} + 3 b d e^{3} + 3 c d^{2} e^{2}\right ) + x^{3} \left (40 a d e^{3} + 8 b d^{2} e^{2} - 8 c d^{3} e\right ) + x \left (33 a d^{2} e^{2} - 3 b d^{3} e - 3 c d^{4}\right )}{48 d^{6} e^{2} + 144 d^{5} e^{3} x^{2} + 144 d^{4} e^{4} x^{4} + 48 d^{3} e^{5} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(d**7*e**5))*(5*a*e**2 + b*d*e + c*d**2)*log(-d**4*e**2*sqrt(-1/(d**7*e
**5)) + x)/32 + sqrt(-1/(d**7*e**5))*(5*a*e**2 + b*d*e + c*d**2)*log(d**4*e**2*s
qrt(-1/(d**7*e**5)) + x)/32 + (x**5*(15*a*e**4 + 3*b*d*e**3 + 3*c*d**2*e**2) + x
**3*(40*a*d*e**3 + 8*b*d**2*e**2 - 8*c*d**3*e) + x*(33*a*d**2*e**2 - 3*b*d**3*e
- 3*c*d**4))/(48*d**6*e**2 + 144*d**5*e**3*x**2 + 144*d**4*e**4*x**4 + 48*d**3*e
**5*x**6)

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GIAC/XCAS [A]  time = 0.269919, size = 181, normalized size = 1.21 \[ \frac{{\left (c d^{2} + b d e + 5 \, a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{16 \, d^{\frac{7}{2}}} + \frac{{\left (3 \, c d^{2} x^{5} e^{2} + 3 \, b d x^{5} e^{3} - 8 \, c d^{3} x^{3} e + 15 \, a x^{5} e^{4} + 8 \, b d^{2} x^{3} e^{2} - 3 \, c d^{4} x + 40 \, a d x^{3} e^{3} - 3 \, b d^{3} x e + 33 \, a d^{2} x e^{2}\right )} e^{\left (-2\right )}}{48 \,{\left (x^{2} e + d\right )}^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(c*d^2 + b*d*e + 5*a*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-5/2)/d^(7/2) + 1/48
*(3*c*d^2*x^5*e^2 + 3*b*d*x^5*e^3 - 8*c*d^3*x^3*e + 15*a*x^5*e^4 + 8*b*d^2*x^3*e
^2 - 3*c*d^4*x + 40*a*d*x^3*e^3 - 3*b*d^3*x*e + 33*a*d^2*x*e^2)*e^(-2)/((x^2*e +
d)^3*d^3)