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

Optimal. Leaf size=103 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )}{2 c^2 \sqrt{b^2-4 a c}}+\frac{(c e-b f) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{f x^2}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.346625, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )}{2 c^2 \sqrt{b^2-4 a c}}+\frac{(c e-b f) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{f x^2}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{2}} f\, dx}{2 c} - \frac{\left (b f - c e\right ) \log{\left (a + b x^{2} + c x^{4} \right )}}{4 c^{2}} - \frac{\left (- 2 a c f + b^{2} f - b c e + 2 c^{2} d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c^{2} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.118854, size = 100, normalized size = 0.97 \[ \frac{\frac{2 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) \left (-c (2 a f+b e)+b^2 f+2 c^2 d\right )}{\sqrt{4 a c-b^2}}+(c e-b f) \log \left (a+b x^2+c x^4\right )+2 c f x^2}{4 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.005, size = 211, normalized size = 2.1 \[{\frac{f{x}^{2}}{2\,c}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bf}{4\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) e}{4\,c}}-{\frac{fa}{c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{d\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}f}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{be}{2\,c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28979, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, c^{2} d - b c e +{\left (b^{2} - 2 \, a c\right )} f\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left (2 \, c f x^{2} +{\left (c e - b f\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{4 \, \sqrt{b^{2} - 4 \, a c} c^{2}}, \frac{2 \,{\left (2 \, c^{2} d - b c e +{\left (b^{2} - 2 \, a c\right )} f\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, c f x^{2} +{\left (c e - b f\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \, \sqrt{-b^{2} + 4 \, a c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*((2*c^2*d - b*c*e + (b^2 - 2*a*c)*f)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c
^2)*x^2 + (2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^
2 + a)) - (2*c*f*x^2 + (c*e - b*f)*log(c*x^4 + b*x^2 + a))*sqrt(b^2 - 4*a*c))/(s
qrt(b^2 - 4*a*c)*c^2), 1/4*(2*(2*c^2*d - b*c*e + (b^2 - 2*a*c)*f)*arctan(-(2*c*x
^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (2*c*f*x^2 + (c*e - b*f)*log(c*x^4 +
b*x^2 + a))*sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^2)]

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Sympy [A]  time = 36.9497, size = 498, normalized size = 4.83 \[ \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) \log{\left (x^{2} + \frac{- a b f - 8 a c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) + 2 a c e + 2 b^{2} c \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) - b c d}{2 a c f - b^{2} f + b c e - 2 c^{2} d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) \log{\left (x^{2} + \frac{- a b f - 8 a c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) + 2 a c e + 2 b^{2} c \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) - b c d}{2 a c f - b^{2} f + b c e - 2 c^{2} d} \right )} + \frac{f x^{2}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-sqrt(-4*a*c + b**2)*(2*a*c*f - b**2*f + b*c*e - 2*c**2*d)/(4*c**2*(4*a*c - b**
2)) - (b*f - c*e)/(4*c**2))*log(x**2 + (-a*b*f - 8*a*c**2*(-sqrt(-4*a*c + b**2)*
(2*a*c*f - b**2*f + b*c*e - 2*c**2*d)/(4*c**2*(4*a*c - b**2)) - (b*f - c*e)/(4*c
**2)) + 2*a*c*e + 2*b**2*c*(-sqrt(-4*a*c + b**2)*(2*a*c*f - b**2*f + b*c*e - 2*c
**2*d)/(4*c**2*(4*a*c - b**2)) - (b*f - c*e)/(4*c**2)) - b*c*d)/(2*a*c*f - b**2*
f + b*c*e - 2*c**2*d)) + (sqrt(-4*a*c + b**2)*(2*a*c*f - b**2*f + b*c*e - 2*c**2
*d)/(4*c**2*(4*a*c - b**2)) - (b*f - c*e)/(4*c**2))*log(x**2 + (-a*b*f - 8*a*c**
2*(sqrt(-4*a*c + b**2)*(2*a*c*f - b**2*f + b*c*e - 2*c**2*d)/(4*c**2*(4*a*c - b*
*2)) - (b*f - c*e)/(4*c**2)) + 2*a*c*e + 2*b**2*c*(sqrt(-4*a*c + b**2)*(2*a*c*f
- b**2*f + b*c*e - 2*c**2*d)/(4*c**2*(4*a*c - b**2)) - (b*f - c*e)/(4*c**2)) - b
*c*d)/(2*a*c*f - b**2*f + b*c*e - 2*c**2*d)) + f*x**2/(2*c)

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GIAC/XCAS [A]  time = 0.317974, size = 134, normalized size = 1.3 \[ \frac{f x^{2}}{2 \, c} - \frac{{\left (b f - c e\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, c^{2}} + \frac{{\left (2 \, c^{2} d + b^{2} f - 2 \, a c f - b c e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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