3.24 \(\int \frac{x \left (A+B x+C x^2\right )}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=223 \[ -\frac{(2 A c-b C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}-\frac{B \sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{B \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{C \log \left (a+b x^2+c x^4\right )}{4 c} \]

[Out]

-((B*Sqrt[b - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 -
4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])) + (B*Sqrt[b + Sqrt[b^2 - 4*a*c]]*
ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b
^2 - 4*a*c]) - ((2*A*c - b*C)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c*Sqr
t[b^2 - 4*a*c]) + (C*Log[a + b*x^2 + c*x^4])/(4*c)

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Rubi [A]  time = 0.520649, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346 \[ -\frac{(2 A c-b C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}-\frac{B \sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{B \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{C \log \left (a+b x^2+c x^4\right )}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[(x*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4),x]

[Out]

-((B*Sqrt[b - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 -
4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])) + (B*Sqrt[b + Sqrt[b^2 - 4*a*c]]*
ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b
^2 - 4*a*c]) - ((2*A*c - b*C)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c*Sqr
t[b^2 - 4*a*c]) + (C*Log[a + b*x^2 + c*x^4])/(4*c)

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Rubi in Sympy [A]  time = 64.4513, size = 209, normalized size = 0.94 \[ - \frac{\sqrt{2} B \sqrt{b - \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{c} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{2} B \sqrt{b + \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{c} \sqrt{- 4 a c + b^{2}}} + \frac{C \log{\left (a + b x^{2} + c x^{4} \right )}}{4 c} - \frac{\left (2 A c - C b\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(C*x**2+B*x+A)/(c*x**4+b*x**2+a),x)

[Out]

-sqrt(2)*B*sqrt(b - sqrt(-4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*x/sqrt(b - sqrt(-4
*a*c + b**2)))/(2*sqrt(c)*sqrt(-4*a*c + b**2)) + sqrt(2)*B*sqrt(b + sqrt(-4*a*c
+ b**2))*atan(sqrt(2)*sqrt(c)*x/sqrt(b + sqrt(-4*a*c + b**2)))/(2*sqrt(c)*sqrt(-
4*a*c + b**2)) + C*log(a + b*x**2 + c*x**4)/(4*c) - (2*A*c - C*b)*atanh((b + 2*c
*x**2)/sqrt(-4*a*c + b**2))/(2*c*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.790396, size = 240, normalized size = 1.08 \[ \frac{\left (C \left (\sqrt{b^2-4 a c}-b\right )+2 A c\right ) \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )-\left (2 A c-C \left (\sqrt{b^2-4 a c}+b\right )\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )-2 \sqrt{2} B \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )+2 \sqrt{2} B \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 c \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4),x]

[Out]

(-2*Sqrt[2]*B*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqr
t[b - Sqrt[b^2 - 4*a*c]]] + 2*Sqrt[2]*B*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcT
an[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]] + (2*A*c + (-b + Sqrt[b^2 -
4*a*c])*C)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2] - (2*A*c - (b + Sqrt[b^2 - 4*a*
c])*C)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(4*c*Sqrt[b^2 - 4*a*c])

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Maple [B]  time = 0.035, size = 728, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(C*x^2+B*x+A)/(c*x^4+b*x^2+a),x)

[Out]

1/2/(4*a*c-b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*A*(-4*a*c+b^2)^(1/2)-1/4/c/(4*a
*c-b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*C*(-4*a*c+b^2)^(1/2)*b+1/(4*a*c-b^2)*ln
(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*a*C-1/4/c/(4*a*c-b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/
2)+b)*b^2*C-1/2/(4*a*c-b^2)*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))*B*(-4*a*c+b^2)^(1/2)*b+2*c/(4*a*c-b^2)
*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))*a*B-1/2/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
an(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2*B-1/2/(4*a*c-b^2)*ln(-2*c*x
^2+(-4*a*c+b^2)^(1/2)-b)*A*(-4*a*c+b^2)^(1/2)+1/4/c/(4*a*c-b^2)*ln(-2*c*x^2+(-4*
a*c+b^2)^(1/2)-b)*C*(-4*a*c+b^2)^(1/2)*b+1/(4*a*c-b^2)*ln(-2*c*x^2+(-4*a*c+b^2)^
(1/2)-b)*a*C-1/4/c/(4*a*c-b^2)*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*b^2*C-1/2/(4*a*
c-b^2)*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))*B*(-4*a*c+b^2)^(1/2)*b-2*c/(4*a*c-b^2)*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))*
a*B+1/2/(4*a*c-b^2)*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))*b^2*B

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} x}{c x^{4} + b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((C*x^2 + B*x + A)*x/(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((C*x^2 + B*x + A)*x/(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(x*(C*x**2+B*x+A)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.2154, 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)*x/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

Done