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

Optimal. Leaf size=66 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt{d} e^{5/2}}-\frac{x (c d-b e)}{e^2}+\frac{c x^3}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.0961722, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt{d} e^{5/2}}-\frac{x (c d-b e)}{e^2}+\frac{c x^3}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.9889, size = 58, normalized size = 0.88 \[ \frac{c x^{3}}{3 e} + \frac{x \left (b e - c d\right )}{e^{2}} + \frac{\left (a e^{2} - b d e + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{\sqrt{d} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.104986, size = 65, normalized size = 0.98 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt{d} e^{5/2}}+\frac{x (b e-c d)}{e^2}+\frac{c x^3}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 84, normalized size = 1.3 \[{\frac{c{x}^{3}}{3\,e}}+{\frac{bx}{e}}-{\frac{cdx}{{e}^{2}}}+{a\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{bd}{e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{c{d}^{2}}{{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285536, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (c d^{2} - b d e + a e^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) + 2 \,{\left (c e x^{3} - 3 \,{\left (c d - b e\right )} x\right )} \sqrt{-d e}}{6 \, \sqrt{-d e} e^{2}}, \frac{3 \,{\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (c e x^{3} - 3 \,{\left (c d - b e\right )} x\right )} \sqrt{d e}}{3 \, \sqrt{d e} e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*(c*d^2 - b*d*e + a*e^2)*log((2*d*e*x + (e*x^2 - d)*sqrt(-d*e))/(e*x^2 +
d)) + 2*(c*e*x^3 - 3*(c*d - b*e)*x)*sqrt(-d*e))/(sqrt(-d*e)*e^2), 1/3*(3*(c*d^2
- b*d*e + a*e^2)*arctan(sqrt(d*e)*x/d) + (c*e*x^3 - 3*(c*d - b*e)*x)*sqrt(d*e))/
(sqrt(d*e)*e^2)]

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Sympy [A]  time = 2.16484, size = 117, normalized size = 1.77 \[ \frac{c x^{3}}{3 e} - \frac{\sqrt{- \frac{1}{d e^{5}}} \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (- d e^{2} \sqrt{- \frac{1}{d e^{5}}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{d e^{5}}} \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (d e^{2} \sqrt{- \frac{1}{d e^{5}}} + x \right )}}{2} + \frac{x \left (b e - c d\right )}{e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.267936, size = 76, normalized size = 1.15 \[ \frac{{\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{\sqrt{d}} + \frac{1}{3} \,{\left (c x^{3} e^{2} - 3 \, c d x e + 3 \, b x e^{2}\right )} e^{\left (-3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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