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

Optimal. Leaf size=209 \[ \frac{\left (c-\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e-\sqrt{e^2-4 d f}}}\right )}{\sqrt{2} \sqrt{f} \sqrt{e-\sqrt{e^2-4 d f}}}+\frac{\left (\frac{c e-2 a f}{\sqrt{e^2-4 d f}}+c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{\sqrt{e^2-4 d f}+e}}\right )}{\sqrt{2} \sqrt{f} \sqrt{\sqrt{e^2-4 d f}+e}}-\frac{b \tanh ^{-1}\left (\frac{e+2 f x^2}{\sqrt{e^2-4 d f}}\right )}{\sqrt{e^2-4 d f}} \]

[Out]

((c - (c*e - 2*a*f)/Sqrt[e^2 - 4*d*f])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e - Sqrt[
e^2 - 4*d*f]]])/(Sqrt[2]*Sqrt[f]*Sqrt[e - Sqrt[e^2 - 4*d*f]]) + ((c + (c*e - 2*a
*f)/Sqrt[e^2 - 4*d*f])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e + Sqrt[e^2 - 4*d*f]]])/
(Sqrt[2]*Sqrt[f]*Sqrt[e + Sqrt[e^2 - 4*d*f]]) - (b*ArcTanh[(e + 2*f*x^2)/Sqrt[e^
2 - 4*d*f]])/Sqrt[e^2 - 4*d*f]

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Rubi [A]  time = 0.869576, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ \frac{\left (c-\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e-\sqrt{e^2-4 d f}}}\right )}{\sqrt{2} \sqrt{f} \sqrt{e-\sqrt{e^2-4 d f}}}+\frac{\left (\frac{c e-2 a f}{\sqrt{e^2-4 d f}}+c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{\sqrt{e^2-4 d f}+e}}\right )}{\sqrt{2} \sqrt{f} \sqrt{\sqrt{e^2-4 d f}+e}}-\frac{b \tanh ^{-1}\left (\frac{e+2 f x^2}{\sqrt{e^2-4 d f}}\right )}{\sqrt{e^2-4 d f}} \]

Antiderivative was successfully verified.

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

[Out]

((c - (c*e - 2*a*f)/Sqrt[e^2 - 4*d*f])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e - Sqrt[
e^2 - 4*d*f]]])/(Sqrt[2]*Sqrt[f]*Sqrt[e - Sqrt[e^2 - 4*d*f]]) + ((c + (c*e - 2*a
*f)/Sqrt[e^2 - 4*d*f])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e + Sqrt[e^2 - 4*d*f]]])/
(Sqrt[2]*Sqrt[f]*Sqrt[e + Sqrt[e^2 - 4*d*f]]) - (b*ArcTanh[(e + 2*f*x^2)/Sqrt[e^
2 - 4*d*f]])/Sqrt[e^2 - 4*d*f]

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Rubi in Sympy [A]  time = 60.4013, size = 221, normalized size = 1.06 \[ - \frac{b \operatorname{atanh}{\left (\frac{e + 2 f x^{2}}{\sqrt{- 4 d f + e^{2}}} \right )}}{\sqrt{- 4 d f + e^{2}}} - \frac{\sqrt{2} \left (2 a f - c e - c \sqrt{- 4 d f + e^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e + \sqrt{- 4 d f + e^{2}}}} \right )}}{2 \sqrt{f} \sqrt{e + \sqrt{- 4 d f + e^{2}}} \sqrt{- 4 d f + e^{2}}} + \frac{\sqrt{2} \left (2 a f - c e + c \sqrt{- 4 d f + e^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e - \sqrt{- 4 d f + e^{2}}}} \right )}}{2 \sqrt{f} \sqrt{e - \sqrt{- 4 d f + e^{2}}} \sqrt{- 4 d f + e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*atanh((e + 2*f*x**2)/sqrt(-4*d*f + e**2))/sqrt(-4*d*f + e**2) - sqrt(2)*(2*a*
f - c*e - c*sqrt(-4*d*f + e**2))*atan(sqrt(2)*sqrt(f)*x/sqrt(e + sqrt(-4*d*f + e
**2)))/(2*sqrt(f)*sqrt(e + sqrt(-4*d*f + e**2))*sqrt(-4*d*f + e**2)) + sqrt(2)*(
2*a*f - c*e + c*sqrt(-4*d*f + e**2))*atan(sqrt(2)*sqrt(f)*x/sqrt(e - sqrt(-4*d*f
+ e**2)))/(2*sqrt(f)*sqrt(e - sqrt(-4*d*f + e**2))*sqrt(-4*d*f + e**2))

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Mathematica [A]  time = 0.425308, size = 234, normalized size = 1.12 \[ \frac{\frac{\sqrt{2} \left (2 a f+c \left (\sqrt{e^2-4 d f}-e\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e-\sqrt{e^2-4 d f}}}\right )}{\sqrt{f} \sqrt{e-\sqrt{e^2-4 d f}}}+\frac{\sqrt{2} \left (c \left (\sqrt{e^2-4 d f}+e\right )-2 a f\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{\sqrt{e^2-4 d f}+e}}\right )}{\sqrt{f} \sqrt{\sqrt{e^2-4 d f}+e}}+b \log \left (\sqrt{e^2-4 d f}-e-2 f x^2\right )-b \log \left (\sqrt{e^2-4 d f}+e+2 f x^2\right )}{2 \sqrt{e^2-4 d f}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[2]*(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f]))*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e
- Sqrt[e^2 - 4*d*f]]])/(Sqrt[f]*Sqrt[e - Sqrt[e^2 - 4*d*f]]) + (Sqrt[2]*(-2*a*f
+ c*(e + Sqrt[e^2 - 4*d*f]))*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e + Sqrt[e^2 - 4*d
*f]]])/(Sqrt[f]*Sqrt[e + Sqrt[e^2 - 4*d*f]]) + b*Log[-e + Sqrt[e^2 - 4*d*f] - 2*
f*x^2] - b*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x^2])/(2*Sqrt[e^2 - 4*d*f])

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Maple [B]  time = 0.073, size = 616, normalized size = 3. \[ -{\frac{b}{8\,df-2\,{e}^{2}}\sqrt{-4\,df+{e}^{2}}\ln \left ( -2\,f{x}^{2}+\sqrt{-4\,df+{e}^{2}}-e \right ) }-2\,{\frac{f\sqrt{2}cd}{ \left ( 4\,df-{e}^{2} \right ) \sqrt{ \left ( -e+\sqrt{-4\,df+{e}^{2}} \right ) f}}{\it Artanh} \left ({\frac{fx\sqrt{2}}{\sqrt{ \left ( -e+\sqrt{-4\,df+{e}^{2}} \right ) f}}} \right ) }+{\frac{c\sqrt{2}{e}^{2}}{8\,df-2\,{e}^{2}}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( -e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( -e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}}+{\frac{f\sqrt{2}a}{4\,df-{e}^{2}}\sqrt{-4\,df+{e}^{2}}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( -e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( -e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}}-{\frac{c\sqrt{2}e}{8\,df-2\,{e}^{2}}\sqrt{-4\,df+{e}^{2}}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( -e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( -e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}}+{\frac{b}{8\,df-2\,{e}^{2}}\sqrt{-4\,df+{e}^{2}}\ln \left ( 2\,f{x}^{2}+\sqrt{-4\,df+{e}^{2}}+e \right ) }+2\,{\frac{f\sqrt{2}cd}{ \left ( 4\,df-{e}^{2} \right ) \sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}\arctan \left ({\frac{fx\sqrt{2}}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}} \right ) }-{\frac{c\sqrt{2}{e}^{2}}{8\,df-2\,{e}^{2}}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}}+{\frac{f\sqrt{2}a}{4\,df-{e}^{2}}\sqrt{-4\,df+{e}^{2}}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}}-{\frac{c\sqrt{2}e}{8\,df-2\,{e}^{2}}\sqrt{-4\,df+{e}^{2}}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-4*d*f+e^2)^(1/2)/(4*d*f-e^2)*b*ln(-2*f*x^2+(-4*d*f+e^2)^(1/2)-e)-2*f/(4*d
*f-e^2)*2^(1/2)/((-e+(-4*d*f+e^2)^(1/2))*f)^(1/2)*arctanh(x*f*2^(1/2)/((-e+(-4*d
*f+e^2)^(1/2))*f)^(1/2))*c*d+1/2/(4*d*f-e^2)*2^(1/2)/((-e+(-4*d*f+e^2)^(1/2))*f)
^(1/2)*arctanh(x*f*2^(1/2)/((-e+(-4*d*f+e^2)^(1/2))*f)^(1/2))*c*e^2+f*(-4*d*f+e^
2)^(1/2)/(4*d*f-e^2)*2^(1/2)/((-e+(-4*d*f+e^2)^(1/2))*f)^(1/2)*arctanh(x*f*2^(1/
2)/((-e+(-4*d*f+e^2)^(1/2))*f)^(1/2))*a-1/2*(-4*d*f+e^2)^(1/2)/(4*d*f-e^2)*2^(1/
2)/((-e+(-4*d*f+e^2)^(1/2))*f)^(1/2)*arctanh(x*f*2^(1/2)/((-e+(-4*d*f+e^2)^(1/2)
)*f)^(1/2))*c*e+1/2*(-4*d*f+e^2)^(1/2)/(4*d*f-e^2)*b*ln(2*f*x^2+(-4*d*f+e^2)^(1/
2)+e)+2*f/(4*d*f-e^2)*2^(1/2)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2)*arctan(x*f*2^(1/2
)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2))*c*d-1/2/(4*d*f-e^2)*2^(1/2)/((e+(-4*d*f+e^2)
^(1/2))*f)^(1/2)*arctan(x*f*2^(1/2)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2))*c*e^2+f*(-
4*d*f+e^2)^(1/2)/(4*d*f-e^2)*2^(1/2)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2)*arctan(x*f
*2^(1/2)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2))*a-1/2*(-4*d*f+e^2)^(1/2)/(4*d*f-e^2)*
2^(1/2)/((e+(-4*d*f+e^2)^(1/2))*f)^(1/2)*arctan(x*f*2^(1/2)/((e+(-4*d*f+e^2)^(1/
2))*f)^(1/2))*c*e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{c x^{2} + b x + a}{f x^{4} + e x^{2} + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x + a)/(f*x^4 + e*x^2 + d), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.0255, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done