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

Optimal. Leaf size=189 \[ \frac{\sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]

[Out]

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

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Rubi [A]  time = 0.459179, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{\sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.9693, size = 177, normalized size = 0.94 \[ - \frac{\sqrt{2} \sqrt{c} d \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{2} \sqrt{c} d \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{e \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.705602, size = 194, normalized size = 1.03 \[ \frac{\frac{2 \sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b-\sqrt{b^2-4 a c}}}-\frac{2 \sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{\sqrt{b^2-4 a c}+b}}+e \left (\log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )-\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )\right )}{2 \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.039, size = 231, normalized size = 1.2 \[{\frac{e}{8\,ac-2\,{b}^{2}}\sqrt{-4\,ac+{b}^{2}}\ln \left ( 2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}+b \right ) }+2\,{\frac{\sqrt{-4\,ac+{b}^{2}}cd\sqrt{2}}{ \left ( 8\,ac-2\,{b}^{2} \right ) \sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}\arctan \left ({\frac{cx\sqrt{2}}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}} \right ) }-{\frac{e}{8\,ac-2\,{b}^{2}}\sqrt{-4\,ac+{b}^{2}}\ln \left ( -2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}-b \right ) }+2\,{\frac{\sqrt{-4\,ac+{b}^{2}}cd\sqrt{2}}{ \left ( 8\,ac-2\,{b}^{2} \right ) \sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}{\it Artanh} \left ({\frac{cx\sqrt{2}}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^4+b*x^2+a),x)

[Out]

(-4*a*c+b^2)^(1/2)/(8*a*c-2*b^2)*e*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)+2*c*(-4*a*c+
b^2)^(1/2)/(8*a*c-2*b^2)*d*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2
^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))-(-4*a*c+b^2)^(1/2)/(8*a*c-2*b^2)*e*ln(-
2*c*x^2+(-4*a*c+b^2)^(1/2)-b)+2*c*(-4*a*c+b^2)^(1/2)/(8*a*c-2*b^2)*d*2^(1/2)/((-
b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(
1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.647762, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done