3.66 \(\int \frac{d+e x^2+f x^4}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=234 \[ \frac{(2 b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac{\log (x) (2 b d-a e)}{a^3}-\frac{2 a^2 c e+c x^2 \left (-a b e-2 a (c d-a f)+b^2 d\right )-a b^2 e-a b (3 c d-a f)+b^3 d}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{d}{2 a^2 x^2}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (6 a^2 b c e+4 a^2 c (3 c d-a f)-a b^3 e-12 a b^2 c d+2 b^4 d\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}} \]

[Out]

-d/(2*a^2*x^2) - (b^3*d - a*b^2*e + 2*a^2*c*e - a*b*(3*c*d - a*f) + c*(b^2*d - a
*b*e - 2*a*(c*d - a*f))*x^2)/(2*a^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4
*d - 12*a*b^2*c*d - a*b^3*e + 6*a^2*b*c*e + 4*a^2*c*(3*c*d - a*f))*ArcTanh[(b +
2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*(b^2 - 4*a*c)^(3/2)) - ((2*b*d - a*e)*Log[x]
)/a^3 + ((2*b*d - a*e)*Log[a + b*x^2 + c*x^4])/(4*a^3)

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Rubi [A]  time = 1.4444, antiderivative size = 234, 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 \[ \frac{(2 b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac{\log (x) (2 b d-a e)}{a^3}-\frac{2 a^2 c e+c x^2 \left (-a b e-2 a (c d-a f)+b^2 d\right )-a b^2 e-a b (3 c d-a f)+b^3 d}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{d}{2 a^2 x^2}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (6 a^2 b c e+4 a^2 c (3 c d-a f)-a b^3 e-12 a b^2 c d+2 b^4 d\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2 + f*x^4)/(x^3*(a + b*x^2 + c*x^4)^2),x]

[Out]

-d/(2*a^2*x^2) - (b^3*d - a*b^2*e + 2*a^2*c*e - a*b*(3*c*d - a*f) + c*(b^2*d - a
*b*e - 2*a*(c*d - a*f))*x^2)/(2*a^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4
*d - 12*a*b^2*c*d - a*b^3*e + 6*a^2*b*c*e + 4*a^2*c*(3*c*d - a*f))*ArcTanh[(b +
2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*(b^2 - 4*a*c)^(3/2)) - ((2*b*d - a*e)*Log[x]
)/a^3 + ((2*b*d - a*e)*Log[a + b*x^2 + c*x^4])/(4*a^3)

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Rubi in Sympy [A]  time = 59.2522, size = 265, normalized size = 1.13 \[ \frac{c \left (2 a^{2} f - a b e - 2 a c d + b^{2} d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{2} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{d}{2 a^{2} x^{2}} - \frac{a^{2} b f + 2 a^{2} c e - a b^{2} e - 3 a b c d + b^{3} d + c x^{2} \left (2 a^{2} f - a b e - 2 a c d + b^{2} d\right )}{2 a^{2} \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} - \frac{\left (\frac{a e}{2} - b d\right ) \log{\left (a + b x^{2} + c x^{4} \right )}}{2 a^{3}} + \frac{\left (a e - 2 b d\right ) \log{\left (x^{2} \right )}}{2 a^{3}} - \frac{\left (- a b e - 2 a c d + 2 b^{2} d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{3} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**4+e*x**2+d)/x**3/(c*x**4+b*x**2+a)**2,x)

[Out]

c*(2*a**2*f - a*b*e - 2*a*c*d + b**2*d)*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2)
)/(a**2*(-4*a*c + b**2)**(3/2)) - d/(2*a**2*x**2) - (a**2*b*f + 2*a**2*c*e - a*b
**2*e - 3*a*b*c*d + b**3*d + c*x**2*(2*a**2*f - a*b*e - 2*a*c*d + b**2*d))/(2*a*
*2*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)) - (a*e/2 - b*d)*log(a + b*x**2 + c*x**
4)/(2*a**3) + (a*e - 2*b*d)*log(x**2)/(2*a**3) - (-a*b*e - 2*a*c*d + 2*b**2*d)*a
tanh((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 = 1.43176, size = 403, normalized size = 1.72 \[ \frac{\frac{\log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (4 a^2 c \left (e \sqrt{b^2-4 a c}-a f+3 c d\right )-a b^2 \left (e \sqrt{b^2-4 a c}+12 c d\right )+2 a b c \left (3 a e-4 d \sqrt{b^2-4 a c}\right )+b^3 \left (2 d \sqrt{b^2-4 a c}-a e\right )+2 b^4 d\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (4 a^2 c \left (e \sqrt{b^2-4 a c}+a f-3 c d\right )+a b^2 \left (12 c d-e \sqrt{b^2-4 a c}\right )-2 a b c \left (4 d \sqrt{b^2-4 a c}+3 a e\right )+b^3 \left (2 d \sqrt{b^2-4 a c}+a e\right )-2 b^4 d\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{2 a \left (b^2 \left (c d x^2-a e\right )+a b \left (a f-c \left (3 d+e x^2\right )\right )+2 a c \left (a \left (e+f x^2\right )-c d x^2\right )+b^3 d\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+4 \log (x) (a e-2 b d)-\frac{2 a d}{x^2}}{4 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2 + f*x^4)/(x^3*(a + b*x^2 + c*x^4)^2),x]

[Out]

((-2*a*d)/x^2 - (2*a*(b^3*d + b^2*(-(a*e) + c*d*x^2) + a*b*(a*f - c*(3*d + e*x^2
)) + 2*a*c*(-(c*d*x^2) + a*(e + f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) +
4*(-2*b*d + a*e)*Log[x] + ((2*b^4*d + b^3*(2*Sqrt[b^2 - 4*a*c]*d - a*e) + 2*a*b*
c*(-4*Sqrt[b^2 - 4*a*c]*d + 3*a*e) - a*b^2*(12*c*d + Sqrt[b^2 - 4*a*c]*e) + 4*a^
2*c*(3*c*d + Sqrt[b^2 - 4*a*c]*e - a*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(
b^2 - 4*a*c)^(3/2) + ((-2*b^4*d + b^3*(2*Sqrt[b^2 - 4*a*c]*d + a*e) - 2*a*b*c*(4
*Sqrt[b^2 - 4*a*c]*d + 3*a*e) + a*b^2*(12*c*d - Sqrt[b^2 - 4*a*c]*e) + 4*a^2*c*(
-3*c*d + Sqrt[b^2 - 4*a*c]*e + a*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2
- 4*a*c)^(3/2))/(4*a^3)

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Maple [B]  time = 0.03, size = 1156, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^4+e*x^2+d)/x^3/(c*x^4+b*x^2+a)^2,x)

[Out]

1/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^2*f-1/2/a/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^2*b*
e-1/a/(c*x^4+b*x^2+a)*c^2/(4*a*c-b^2)*x^2*d+1/2/a^2/(c*x^4+b*x^2+a)*c/(4*a*c-b^2
)*x^2*b^2*d+1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b*f+1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*c*
e-1/2/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^2*e-3/2/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b*c*
d+1/2/a^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^3*d-1/a/(4*a*c-b^2)*c*ln((4*a*c-b^2)*(c*
x^4+b*x^2+a))*e+1/4/a^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*b^2*e+2/a^2/
(4*a*c-b^2)*c*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*b*d-1/2/a^3/(4*a*c-b^2)*ln((4*a*c-
b^2)*(c*x^4+b*x^2+a))*b^3*d+2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*a
rctan((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*f-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*e-6/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))*c^2*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)*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^3*e+6/a^
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))*b^2*c*d-1/a^3/(6
4*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^4*d-1/2*d/a^2/x^2+
1/a^2*ln(x)*e-2/a^3*ln(x)*b*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)^2*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.09619, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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^3),x, algorithm="fricas")

[Out]

[-1/4*(((4*a^3*c^2*f - 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d + (a*b^3*c - 6*a^2*
b*c^2)*e)*x^6 + (4*a^3*b*c*f - 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d + (a*b^4 - 6*
a^2*b^2*c)*e)*x^4 + (4*a^4*c*f - 2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d + (a^2*b^
3 - 6*a^3*b*c)*e)*x^2)*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
*c*e - 2*a^3*c*f - 2*(a*b^2*c - 3*a^2*c^2)*d)*x^4 - 2*(a^3*b*f + (2*a*b^3 - 7*a^
2*b*c)*d - (a^2*b^2 - 2*a^3*c)*e)*x^2 - 2*(a^2*b^2 - 4*a^3*c)*d + ((2*(b^3*c - 4
*a*b*c^2)*d - (a*b^2*c - 4*a^2*c^2)*e)*x^6 + (2*(b^4 - 4*a*b^2*c)*d - (a*b^3 - 4
*a^2*b*c)*e)*x^4 + (2*(a*b^3 - 4*a^2*b*c)*d - (a^2*b^2 - 4*a^3*c)*e)*x^2)*log(c*
x^4 + b*x^2 + a) - 4*((2*(b^3*c - 4*a*b*c^2)*d - (a*b^2*c - 4*a^2*c^2)*e)*x^6 +
(2*(b^4 - 4*a*b^2*c)*d - (a*b^3 - 4*a^2*b*c)*e)*x^4 + (2*(a*b^3 - 4*a^2*b*c)*d -
(a^2*b^2 - 4*a^3*c)*e)*x^2)*log(x))*sqrt(b^2 - 4*a*c))/(((a^3*b^2*c - 4*a^4*c^2
)*x^6 + (a^3*b^3 - 4*a^4*b*c)*x^4 + (a^4*b^2 - 4*a^5*c)*x^2)*sqrt(b^2 - 4*a*c)),
-1/4*(2*((4*a^3*c^2*f - 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d + (a*b^3*c - 6*a^
2*b*c^2)*e)*x^6 + (4*a^3*b*c*f - 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d + (a*b^4 -
6*a^2*b^2*c)*e)*x^4 + (4*a^4*c*f - 2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d + (a^2*
b^3 - 6*a^3*b*c)*e)*x^2)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c))
- (2*(a^2*b*c*e - 2*a^3*c*f - 2*(a*b^2*c - 3*a^2*c^2)*d)*x^4 - 2*(a^3*b*f + (2*
a*b^3 - 7*a^2*b*c)*d - (a^2*b^2 - 2*a^3*c)*e)*x^2 - 2*(a^2*b^2 - 4*a^3*c)*d + ((
2*(b^3*c - 4*a*b*c^2)*d - (a*b^2*c - 4*a^2*c^2)*e)*x^6 + (2*(b^4 - 4*a*b^2*c)*d
- (a*b^3 - 4*a^2*b*c)*e)*x^4 + (2*(a*b^3 - 4*a^2*b*c)*d - (a^2*b^2 - 4*a^3*c)*e)
*x^2)*log(c*x^4 + b*x^2 + a) - 4*((2*(b^3*c - 4*a*b*c^2)*d - (a*b^2*c - 4*a^2*c^
2)*e)*x^6 + (2*(b^4 - 4*a*b^2*c)*d - (a*b^3 - 4*a^2*b*c)*e)*x^4 + (2*(a*b^3 - 4*
a^2*b*c)*d - (a^2*b^2 - 4*a^3*c)*e)*x^2)*log(x))*sqrt(-b^2 + 4*a*c))/(((a^3*b^2*
c - 4*a^4*c^2)*x^6 + (a^3*b^3 - 4*a^4*b*c)*x^4 + (a^4*b^2 - 4*a^5*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**3/(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^3),x, algorithm="giac")

[Out]

Exception raised: TypeError