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

Optimal. Leaf size=166 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (4 a^2 c e-2 a b (a f+3 c d)+b^3 d\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{d \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{d \log (x)}{a^2}+\frac{x^2 (a b f-2 a c e+b c d)-a b e-2 a (c d-a f)+b^2 d}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.825503, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (4 a^2 c e-2 a b (a f+3 c d)+b^3 d\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{d \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{d \log (x)}{a^2}+\frac{x^2 (a b f-2 a c e+b c d)-a b e-2 a (c d-a f)+b^2 d}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.1041, size = 194, normalized size = 1.17 \[ \frac{2 a^{2} f - a b e - 2 a c d + b^{2} d + x^{2} \left (a b f - 2 a c e + b c d\right )}{2 a \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} - \frac{\left (a b f - 2 a c e + b c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{a \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{b d \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{2} \sqrt{- 4 a c + b^{2}}} + \frac{d \log{\left (x^{2} \right )}}{2 a^{2}} - \frac{d \log{\left (a + b x^{2} + c x^{4} \right )}}{4 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.927853, size = 268, normalized size = 1.61 \[ -\frac{-\frac{2 a \left (b \left (-a e+a f x^2+c d x^2\right )+2 a \left (a f-c \left (d+e x^2\right )\right )+b^2 d\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (4 a c \left (a e-d \sqrt{b^2-4 a c}\right )+b^2 d \sqrt{b^2-4 a c}-2 a b (a f+3 c d)+b^3 d\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (-4 a c \left (d \sqrt{b^2-4 a c}+a e\right )+b^2 d \sqrt{b^2-4 a c}+2 a b (a f+3 c d)+b^3 (-d)\right )}{\left (b^2-4 a c\right )^{3/2}}-4 d \log (x)}{4 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.024, size = 744, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*b*f+c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*e-1/2
/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*b*c*d-a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*f+1/2/(c*
x^4+b*x^2+a)/(4*a*c-b^2)*b*e+1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*c*d-1/2/a/(c*x^4+b*x^
2+a)/(4*a*c-b^2)*b^2*d-1/a/(4*a*c-b^2)*c*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*d+1/4/a
^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*b^2*d-1/(64*a^3*c^3-48*a^2*b^2*c^
2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-4
8*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b*f+2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c
-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^
2+12*a*b^4*c-b^6)^(1/2))*c*e-3/a/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2
)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4
*c-b^6)^(1/2))*b*c*d+1/2/a^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*ar
ctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b
^6)^(1/2))*b^3*d+d*ln(x)/a^2

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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)^2*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.34886, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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)^2*x),x, algorithm="fricas")

[Out]

[1/4*((4*a^3*c*e - 2*a^3*b*f + (4*a^2*c^2*e - 2*a^2*b*c*f + (b^3*c - 6*a*b*c^2)*
d)*x^4 + (4*a^2*b*c*e - 2*a^2*b^2*f + (b^4 - 6*a*b^2*c)*d)*x^2 + (a*b^3 - 6*a^2*
b*c)*d)*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*a^2*b*e - 4*a^3*f - 2*
(a*b*c*d - 2*a^2*c*e + a^2*b*f)*x^2 - 2*(a*b^2 - 2*a^2*c)*d + ((b^2*c - 4*a*c^2)
*d*x^4 + (b^3 - 4*a*b*c)*d*x^2 + (a*b^2 - 4*a^2*c)*d)*log(c*x^4 + b*x^2 + a) - 4
*((b^2*c - 4*a*c^2)*d*x^4 + (b^3 - 4*a*b*c)*d*x^2 + (a*b^2 - 4*a^2*c)*d)*log(x))
*sqrt(b^2 - 4*a*c))/((a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3
- 4*a^3*b*c)*x^2)*sqrt(b^2 - 4*a*c)), -1/4*(2*(4*a^3*c*e - 2*a^3*b*f + (4*a^2*c
^2*e - 2*a^2*b*c*f + (b^3*c - 6*a*b*c^2)*d)*x^4 + (4*a^2*b*c*e - 2*a^2*b^2*f + (
b^4 - 6*a*b^2*c)*d)*x^2 + (a*b^3 - 6*a^2*b*c)*d)*arctan(-(2*c*x^2 + b)*sqrt(-b^2
+ 4*a*c)/(b^2 - 4*a*c)) + (2*a^2*b*e - 4*a^3*f - 2*(a*b*c*d - 2*a^2*c*e + a^2*b
*f)*x^2 - 2*(a*b^2 - 2*a^2*c)*d + ((b^2*c - 4*a*c^2)*d*x^4 + (b^3 - 4*a*b*c)*d*x
^2 + (a*b^2 - 4*a^2*c)*d)*log(c*x^4 + b*x^2 + a) - 4*((b^2*c - 4*a*c^2)*d*x^4 +
(b^3 - 4*a*b*c)*d*x^2 + (a*b^2 - 4*a^2*c)*d)*log(x))*sqrt(-b^2 + 4*a*c))/((a^3*b
^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*sqrt(-b^
2 + 4*a*c))]

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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/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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)^2*x),x, algorithm="giac")

[Out]

Exception raised: TypeError