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

Optimal. Leaf size=74 \[ \frac{x \left (a+\frac{c d^2}{e^2}\right )}{2 d \left (d+e x^2\right )}-\frac{\left (3 c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} e^{5/2}}+\frac{c x}{e^2} \]

[Out]

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

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Rubi [A]  time = 0.113144, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x \left (a+\frac{c d^2}{e^2}\right )}{2 d \left (d+e x^2\right )}-\frac{\left (3 c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} e^{5/2}}+\frac{c x}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.8383, size = 65, normalized size = 0.88 \[ \frac{c x}{e^{2}} + \frac{x \left (\frac{a}{2 d} + \frac{c d}{2 e^{2}}\right )}{d + e x^{2}} + \frac{\left (a e^{2} - 3 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 d^{\frac{3}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x/e**2 + x*(a/(2*d) + c*d/(2*e**2))/(d + e*x**2) + (a*e**2 - 3*c*d**2)*atan(sq
rt(e)*x/sqrt(d))/(2*d**(3/2)*e**(5/2))

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Mathematica [A]  time = 0.0833572, size = 78, normalized size = 1.05 \[ \frac{x \left (a e^2+c d^2\right )}{2 d e^2 \left (d+e x^2\right )}-\frac{\left (3 c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} e^{5/2}}+\frac{c x}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

(c*x)/e^2 + ((c*d^2 + a*e^2)*x)/(2*d*e^2*(d + e*x^2)) - ((3*c*d^2 - a*e^2)*ArcTa
n[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*e^(5/2))

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Maple [A]  time = 0.012, size = 82, normalized size = 1.1 \[{\frac{cx}{{e}^{2}}}+{\frac{ax}{2\,d \left ( e{x}^{2}+d \right ) }}+{\frac{cdx}{2\,{e}^{2} \left ( e{x}^{2}+d \right ) }}+{\frac{a}{2\,d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{3\,cd}{2\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)/(e*x^2+d)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29123, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, c d^{3} - a d e^{2} +{\left (3 \, c d^{2} e - a e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) - 2 \,{\left (2 \, c d e x^{3} +{\left (3 \, c d^{2} + a e^{2}\right )} x\right )} \sqrt{-d e}}{4 \,{\left (d e^{3} x^{2} + d^{2} e^{2}\right )} \sqrt{-d e}}, -\frac{{\left (3 \, c d^{3} - a d e^{2} +{\left (3 \, c d^{2} e - a e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) -{\left (2 \, c d e x^{3} +{\left (3 \, c d^{2} + a e^{2}\right )} x\right )} \sqrt{d e}}{2 \,{\left (d e^{3} x^{2} + d^{2} e^{2}\right )} \sqrt{d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.29111, size = 138, normalized size = 1.86 \[ \frac{c x}{e^{2}} + \frac{x \left (a e^{2} + c d^{2}\right )}{2 d^{2} e^{2} + 2 d e^{3} x^{2}} - \frac{\sqrt{- \frac{1}{d^{3} e^{5}}} \left (a e^{2} - 3 c d^{2}\right ) \log{\left (- d^{2} e^{2} \sqrt{- \frac{1}{d^{3} e^{5}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{d^{3} e^{5}}} \left (a e^{2} - 3 c d^{2}\right ) \log{\left (d^{2} e^{2} \sqrt{- \frac{1}{d^{3} e^{5}}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270064, size = 84, normalized size = 1.14 \[ c x e^{\left (-2\right )} - \frac{{\left (3 \, c d^{2} - a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{2 \, d^{\frac{3}{2}}} + \frac{{\left (c d^{2} x + a x e^{2}\right )} e^{\left (-2\right )}}{2 \,{\left (x^{2} e + d\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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