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

Optimal. Leaf size=55 \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{5/2}}-\frac{c d x}{e^2}+\frac{c x^3}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.0758465, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{5/2}}-\frac{c d x}{e^2}+\frac{c x^3}{3 e} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)/(d + e*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.0536, size = 49, normalized size = 0.89 \[ - \frac{c d x}{e^{2}} + \frac{c x^{3}}{3 e} + \frac{\left (a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{\sqrt{d} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0570117, size = 55, normalized size = 1. \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{5/2}}-\frac{c d x}{e^2}+\frac{c x^3}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 57, normalized size = 1. \[{\frac{c{x}^{3}}{3\,e}}-{\frac{cdx}{{e}^{2}}}+{a\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{c{d}^{2}}{{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)/(e*x^2+d),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296494, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (c d^{2} + a e^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) + 2 \,{\left (c e x^{3} - 3 \, c d x\right )} \sqrt{-d e}}{6 \, \sqrt{-d e} e^{2}}, \frac{3 \,{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (c e x^{3} - 3 \, c d x\right )} \sqrt{d e}}{3 \, \sqrt{d e} e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.66288, size = 104, normalized size = 1.89 \[ - \frac{c d x}{e^{2}} + \frac{c x^{3}}{3 e} - \frac{\sqrt{- \frac{1}{d e^{5}}} \left (a e^{2} + c d^{2}\right ) \log{\left (- d e^{2} \sqrt{- \frac{1}{d e^{5}}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{d e^{5}}} \left (a e^{2} + c d^{2}\right ) \log{\left (d e^{2} \sqrt{- \frac{1}{d e^{5}}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.268124, size = 59, normalized size = 1.07 \[ \frac{{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{\sqrt{d}} + \frac{1}{3} \,{\left (c x^{3} e^{2} - 3 \, c d x e\right )} e^{\left (-3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*d^2 + 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)*e^(-3)