3.53 \(\int \frac{d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=174 \[ -\frac{\log \left (a+b x^2+c x^4\right ) \left (-a b e-a (c d-a f)+b^2 d\right )}{4 a^3}+\frac{\log (x) \left (-a b e-a (c d-a f)+b^2 d\right )}{a^3}+\frac{b d-a e}{2 a^2 x^2}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )}{2 a^3 \sqrt{b^2-4 a c}}-\frac{d}{4 a x^4} \]

[Out]

-d/(4*a*x^4) + (b*d - a*e)/(2*a^2*x^2) + ((b^3*d - a*b^2*e + 2*a^2*c*e - a*b*(3*
c*d - a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*Sqrt[b^2 - 4*a*c])
+ ((b^2*d - a*b*e - a*(c*d - a*f))*Log[x])/a^3 - ((b^2*d - a*b*e - a*(c*d - a*f)
)*Log[a + b*x^2 + c*x^4])/(4*a^3)

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Rubi [A]  time = 0.78159, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\log \left (a+b x^2+c x^4\right ) \left (-a b e-a (c d-a f)+b^2 d\right )}{4 a^3}+\frac{\log (x) \left (-a b e-a (c d-a f)+b^2 d\right )}{a^3}+\frac{b d-a e}{2 a^2 x^2}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )}{2 a^3 \sqrt{b^2-4 a c}}-\frac{d}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)),x]

[Out]

-d/(4*a*x^4) + (b*d - a*e)/(2*a^2*x^2) + ((b^3*d - a*b^2*e + 2*a^2*c*e - a*b*(3*
c*d - a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*Sqrt[b^2 - 4*a*c])
+ ((b^2*d - a*b*e - a*(c*d - a*f))*Log[x])/a^3 - ((b^2*d - a*b*e - a*(c*d - a*f)
)*Log[a + b*x^2 + c*x^4])/(4*a^3)

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Rubi in Sympy [A]  time = 136.097, size = 167, normalized size = 0.96 \[ - \frac{d}{4 a x^{4}} - \frac{a e - b d}{2 a^{2} x^{2}} + \frac{\left (a^{2} f - a b e - a c d + b^{2} d\right ) \log{\left (x^{2} \right )}}{2 a^{3}} - \frac{\left (a^{2} f - a b e - a c d + b^{2} d\right ) \log{\left (a + b x^{2} + c x^{4} \right )}}{4 a^{3}} + \frac{\left (a^{2} b f + 2 a^{2} c e - a b^{2} e - 3 a b c d + b^{3} d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{3} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**4+e*x**2+d)/x**5/(c*x**4+b*x**2+a),x)

[Out]

-d/(4*a*x**4) - (a*e - b*d)/(2*a**2*x**2) + (a**2*f - a*b*e - a*c*d + b**2*d)*lo
g(x**2)/(2*a**3) - (a**2*f - a*b*e - a*c*d + b**2*d)*log(a + b*x**2 + c*x**4)/(4
*a**3) + (a**2*b*f + 2*a**2*c*e - a*b**2*e - 3*a*b*c*d + b**3*d)*atanh((b + 2*c*
x**2)/sqrt(-4*a*c + b**2))/(2*a**3*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.680589, size = 314, normalized size = 1.8 \[ -\frac{\frac{a^2 d}{x^4}-4 \log (x) \left (-a b e+a (a f-c d)+b^2 d\right )+\frac{\log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (a b \left (-e \sqrt{b^2-4 a c}+a f-3 c d\right )+a \left (-c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+2 a c e\right )+b^2 \left (d \sqrt{b^2-4 a c}-a e\right )+b^3 d\right )}{\sqrt{b^2-4 a c}}+\frac{\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (-a b \left (e \sqrt{b^2-4 a c}+a f-3 c d\right )+a \left (a f \sqrt{b^2-4 a c}-c \left (d \sqrt{b^2-4 a c}+2 a e\right )\right )+b^2 \left (d \sqrt{b^2-4 a c}+a e\right )+b^3 (-d)\right )}{\sqrt{b^2-4 a c}}+\frac{2 a (a e-b d)}{x^2}}{4 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)),x]

[Out]

-((a^2*d)/x^4 + (2*a*(-(b*d) + a*e))/x^2 - 4*(b^2*d - a*b*e + a*(-(c*d) + a*f))*
Log[x] + ((b^3*d + b^2*(Sqrt[b^2 - 4*a*c]*d - a*e) + a*b*(-3*c*d - Sqrt[b^2 - 4*
a*c]*e + a*f) + a*(-(c*Sqrt[b^2 - 4*a*c]*d) + 2*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*
Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c] + ((-(b^3*d) + b^2*(Sqrt
[b^2 - 4*a*c]*d + a*e) - a*b*(-3*c*d + Sqrt[b^2 - 4*a*c]*e + a*f) + a*(-(c*(Sqrt
[b^2 - 4*a*c]*d + 2*a*e)) + a*Sqrt[b^2 - 4*a*c]*f))*Log[b + Sqrt[b^2 - 4*a*c] +
2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*a^3)

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Maple [B]  time = 0.016, size = 356, normalized size = 2.1 \[ -{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) f}{4\,a}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) be}{4\,{a}^{2}}}+{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) d}{4\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}d}{4\,{a}^{3}}}-{\frac{bf}{2\,a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{ce}{a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}e}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{3\,bcd}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}d}{2\,{a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{d}{4\,a{x}^{4}}}-{\frac{e}{2\,a{x}^{2}}}+{\frac{bd}{2\,{a}^{2}{x}^{2}}}+{\frac{\ln \left ( x \right ) f}{a}}-{\frac{\ln \left ( x \right ) be}{{a}^{2}}}-{\frac{\ln \left ( x \right ) cd}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}d}{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a),x)

[Out]

-1/4/a*ln(c*x^4+b*x^2+a)*f+1/4/a^2*ln(c*x^4+b*x^2+a)*b*e+1/4/a^2*c*ln(c*x^4+b*x^
2+a)*d-1/4/a^3*ln(c*x^4+b*x^2+a)*b^2*d-1/2/a/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b
)/(4*a*c-b^2)^(1/2))*b*f-1/a/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1
/2))*c*e+1/2/a^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^2*e+3
/2/a^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*c*d-1/2/a^3/(4*
a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^3*d-1/4*d/a/x^4-1/2/a/x^2
*e+1/2/a^2/x^2*b*d+1/a*ln(x)*f-1/a^2*ln(x)*b*e-1/a^2*ln(x)*c*d+1/a^3*ln(x)*b^2*d

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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)/((c*x^4 + b*x^2 + a)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.926894, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (a^{2} b f +{\left (b^{3} - 3 \, a b c\right )} d -{\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} x^{4} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left ({\left (a b e - a^{2} f -{\left (b^{2} - a c\right )} d\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \,{\left (a b e - a^{2} f -{\left (b^{2} - a c\right )} d\right )} x^{4} \log \left (x\right ) - a^{2} d + 2 \,{\left (a b d - a^{2} e\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}{4 \, \sqrt{b^{2} - 4 \, a c} a^{3} x^{4}}, -\frac{2 \,{\left (a^{2} b f +{\left (b^{3} - 3 \, a b c\right )} d -{\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} x^{4} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left ({\left (a b e - a^{2} f -{\left (b^{2} - a c\right )} d\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \,{\left (a b e - a^{2} f -{\left (b^{2} - a c\right )} d\right )} x^{4} \log \left (x\right ) - a^{2} d + 2 \,{\left (a b d - a^{2} e\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \, \sqrt{-b^{2} + 4 \, a c} a^{3} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^5),x, algorithm="fricas")

[Out]

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

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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((f*x**4+e*x**2+d)/x**5/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.295183, size = 286, normalized size = 1.64 \[ -\frac{{\left (b^{2} d - a c d + a^{2} f - a b e\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac{{\left (b^{2} d - a c d + a^{2} f - a b e\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{3}} - \frac{{\left (b^{3} d - 3 \, a b c d + a^{2} b f - a b^{2} e + 2 \, a^{2} c e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{3}} - \frac{3 \, b^{2} d x^{4} - 3 \, a c d x^{4} + 3 \, a^{2} f x^{4} - 3 \, a b x^{4} e - 2 \, a b d x^{2} + 2 \, a^{2} x^{2} e + a^{2} d}{4 \, a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^5),x, algorithm="giac")

[Out]

-1/4*(b^2*d - a*c*d + a^2*f - a*b*e)*ln(c*x^4 + b*x^2 + a)/a^3 + 1/2*(b^2*d - a*
c*d + a^2*f - a*b*e)*ln(x^2)/a^3 - 1/2*(b^3*d - 3*a*b*c*d + a^2*b*f - a*b^2*e +
2*a^2*c*e)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^3) - 1
/4*(3*b^2*d*x^4 - 3*a*c*d*x^4 + 3*a^2*f*x^4 - 3*a*b*x^4*e - 2*a*b*d*x^2 + 2*a^2*
x^2*e + a^2*d)/(a^3*x^4)