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

Optimal. Leaf size=115 \[ -\frac{x \left (5 c d^2-e (3 a e+b d)\right )}{8 d^2 e^2 \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e (3 a e+b d)+3 c d^2\right )}{8 d^{5/2} e^{5/2}}+\frac{x \left (a+\frac{d (c d-b e)}{e^2}\right )}{4 d \left (d+e x^2\right )^2} \]

[Out]

((a + (d*(c*d - b*e))/e^2)*x)/(4*d*(d + e*x^2)^2) - ((5*c*d^2 - e*(b*d + 3*a*e))
*x)/(8*d^2*e^2*(d + e*x^2)) + ((3*c*d^2 + e*(b*d + 3*a*e))*ArcTan[(Sqrt[e]*x)/Sq
rt[d]])/(8*d^(5/2)*e^(5/2))

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Rubi [A]  time = 0.2063, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{x \left (5 c d^2-e (3 a e+b d)\right )}{8 d^2 e^2 \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e (3 a e+b d)+3 c d^2\right )}{8 d^{5/2} e^{5/2}}+\frac{x \left (a+\frac{d (c d-b e)}{e^2}\right )}{4 d \left (d+e x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + (d*(c*d - b*e))/e^2)*x)/(4*d*(d + e*x^2)^2) - ((5*c*d^2 - e*(b*d + 3*a*e))
*x)/(8*d^2*e^2*(d + e*x^2)) + ((3*c*d^2 + e*(b*d + 3*a*e))*ArcTan[(Sqrt[e]*x)/Sq
rt[d]])/(8*d^(5/2)*e^(5/2))

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Rubi in Sympy [A]  time = 34.0753, size = 110, normalized size = 0.96 \[ \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{4 d e^{2} \left (d + e x^{2}\right )^{2}} + \frac{x \left (3 a e^{2} + b d e - 5 c d^{2}\right )}{8 d^{2} e^{2} \left (d + e x^{2}\right )} + \frac{\left (3 a e^{2} + b d e + 3 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 d^{\frac{5}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a*e**2 - b*d*e + c*d**2)/(4*d*e**2*(d + e*x**2)**2) + x*(3*a*e**2 + b*d*e - 5
*c*d**2)/(8*d**2*e**2*(d + e*x**2)) + (3*a*e**2 + b*d*e + 3*c*d**2)*atan(sqrt(e)
*x/sqrt(d))/(8*d**(5/2)*e**(5/2))

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Mathematica [A]  time = 0.16758, size = 110, normalized size = 0.96 \[ \frac{x \left (e \left (a e \left (5 d+3 e x^2\right )+b d \left (e x^2-d\right )\right )-c d^2 \left (3 d+5 e x^2\right )\right )}{8 d^2 e^2 \left (d+e x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e (3 a e+b d)+3 c d^2\right )}{8 d^{5/2} e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 131, normalized size = 1.1 \[{\frac{1}{ \left ( e{x}^{2}+d \right ) ^{2}} \left ({\frac{ \left ( 3\,a{e}^{2}+bde-5\,c{d}^{2} \right ){x}^{3}}{8\,{d}^{2}e}}+{\frac{ \left ( 5\,a{e}^{2}-bde-3\,c{d}^{2} \right ) x}{8\,{e}^{2}d}} \right ) }+{\frac{3\,a}{8\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{b}{8\,de}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,c}{8\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/8*(3*a*e^2+b*d*e-5*c*d^2)/d^2/e*x^3+1/8*(5*a*e^2-b*d*e-3*c*d^2)/e^2/d*x)/(e*x
^2+d)^2+3/8/d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a+1/8/d/e/(d*e)^(1/2)*arctan
(x*e/(d*e)^(1/2))*b+3/8/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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277568, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (3 \, c d^{4} + b d^{3} e + 3 \, a d^{2} e^{2} +{\left (3 \, c d^{2} e^{2} + b d e^{3} + 3 \, a e^{4}\right )} x^{4} + 2 \,{\left (3 \, c d^{3} e + b d^{2} e^{2} + 3 \, a d e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) - 2 \,{\left ({\left (5 \, c d^{2} e - b d e^{2} - 3 \, a e^{3}\right )} x^{3} +{\left (3 \, c d^{3} + b d^{2} e - 5 \, a d e^{2}\right )} x\right )} \sqrt{-d e}}{16 \,{\left (d^{2} e^{4} x^{4} + 2 \, d^{3} e^{3} x^{2} + d^{4} e^{2}\right )} \sqrt{-d e}}, \frac{{\left (3 \, c d^{4} + b d^{3} e + 3 \, a d^{2} e^{2} +{\left (3 \, c d^{2} e^{2} + b d e^{3} + 3 \, a e^{4}\right )} x^{4} + 2 \,{\left (3 \, c d^{3} e + b d^{2} e^{2} + 3 \, a d e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) -{\left ({\left (5 \, c d^{2} e - b d e^{2} - 3 \, a e^{3}\right )} x^{3} +{\left (3 \, c d^{3} + b d^{2} e - 5 \, a d e^{2}\right )} x\right )} \sqrt{d e}}{8 \,{\left (d^{2} e^{4} x^{4} + 2 \, d^{3} e^{3} x^{2} + d^{4} e^{2}\right )} \sqrt{d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*((3*c*d^4 + b*d^3*e + 3*a*d^2*e^2 + (3*c*d^2*e^2 + b*d*e^3 + 3*a*e^4)*x^4
+ 2*(3*c*d^3*e + b*d^2*e^2 + 3*a*d*e^3)*x^2)*log((2*d*e*x + (e*x^2 - d)*sqrt(-d*
e))/(e*x^2 + d)) - 2*((5*c*d^2*e - b*d*e^2 - 3*a*e^3)*x^3 + (3*c*d^3 + b*d^2*e -
5*a*d*e^2)*x)*sqrt(-d*e))/((d^2*e^4*x^4 + 2*d^3*e^3*x^2 + d^4*e^2)*sqrt(-d*e)),
1/8*((3*c*d^4 + b*d^3*e + 3*a*d^2*e^2 + (3*c*d^2*e^2 + b*d*e^3 + 3*a*e^4)*x^4 +
2*(3*c*d^3*e + b*d^2*e^2 + 3*a*d*e^3)*x^2)*arctan(sqrt(d*e)*x/d) - ((5*c*d^2*e
- b*d*e^2 - 3*a*e^3)*x^3 + (3*c*d^3 + b*d^2*e - 5*a*d*e^2)*x)*sqrt(d*e))/((d^2*e
^4*x^4 + 2*d^3*e^3*x^2 + d^4*e^2)*sqrt(d*e))]

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Sympy [A]  time = 6.63788, size = 196, normalized size = 1.7 \[ - \frac{\sqrt{- \frac{1}{d^{5} e^{5}}} \left (3 a e^{2} + b d e + 3 c d^{2}\right ) \log{\left (- d^{3} e^{2} \sqrt{- \frac{1}{d^{5} e^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{d^{5} e^{5}}} \left (3 a e^{2} + b d e + 3 c d^{2}\right ) \log{\left (d^{3} e^{2} \sqrt{- \frac{1}{d^{5} e^{5}}} + x \right )}}{16} + \frac{x^{3} \left (3 a e^{3} + b d e^{2} - 5 c d^{2} e\right ) + x \left (5 a d e^{2} - b d^{2} e - 3 c d^{3}\right )}{8 d^{4} e^{2} + 16 d^{3} e^{3} x^{2} + 8 d^{2} e^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(d**5*e**5))*(3*a*e**2 + b*d*e + 3*c*d**2)*log(-d**3*e**2*sqrt(-1/(d**5
*e**5)) + x)/16 + sqrt(-1/(d**5*e**5))*(3*a*e**2 + b*d*e + 3*c*d**2)*log(d**3*e*
*2*sqrt(-1/(d**5*e**5)) + x)/16 + (x**3*(3*a*e**3 + b*d*e**2 - 5*c*d**2*e) + x*(
5*a*d*e**2 - b*d**2*e - 3*c*d**3))/(8*d**4*e**2 + 16*d**3*e**3*x**2 + 8*d**2*e**
4*x**4)

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GIAC/XCAS [A]  time = 0.271979, size = 136, normalized size = 1.18 \[ \frac{{\left (3 \, c d^{2} + b d e + 3 \, a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{8 \, d^{\frac{5}{2}}} - \frac{{\left (5 \, c d^{2} x^{3} e - b d x^{3} e^{2} + 3 \, c d^{3} x - 3 \, a x^{3} e^{3} + b d^{2} x e - 5 \, a d x e^{2}\right )} e^{\left (-2\right )}}{8 \,{\left (x^{2} e + d\right )}^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(3*c*d^2 + b*d*e + 3*a*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-5/2)/d^(5/2) - 1/8
*(5*c*d^2*x^3*e - b*d*x^3*e^2 + 3*c*d^3*x - 3*a*x^3*e^3 + b*d^2*x*e - 5*a*d*x*e^
2)*e^(-2)/((x^2*e + d)^2*d^2)