3.879 \(\int \frac{1}{x \left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=200 \[ -\frac{\log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac{\log (x)}{a^3}+\frac{16 a^2 c^2+2 b c x^2 \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

(b^2 - 2*a*c + b*c*x^2)/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*b^4 - 15*
a*b^2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c)*x^2)/(4*a^2*(b^2 - 4*a*c)^2*(a + b*x^
2 + c*x^4)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2
- 4*a*c]])/(2*a^3*(b^2 - 4*a*c)^(5/2)) + Log[x]/a^3 - Log[a + b*x^2 + c*x^4]/(4*
a^3)

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Rubi [A]  time = 0.674616, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ -\frac{\log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac{\log (x)}{a^3}+\frac{16 a^2 c^2+2 b c x^2 \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^2 + c*x^4)^3),x]

[Out]

(b^2 - 2*a*c + b*c*x^2)/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*b^4 - 15*
a*b^2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c)*x^2)/(4*a^2*(b^2 - 4*a*c)^2*(a + b*x^
2 + c*x^4)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2
- 4*a*c]])/(2*a^3*(b^2 - 4*a*c)^(5/2)) + Log[x]/a^3 - Log[a + b*x^2 + c*x^4]/(4*
a^3)

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Rubi in Sympy [A]  time = 79.0637, size = 196, normalized size = 0.98 \[ \frac{- 2 a c + b^{2} + b c x^{2}}{4 a \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{2}} + \frac{16 a^{2} c^{2} - 15 a b^{2} c + 2 b^{4} + 2 b c x^{2} \left (- 7 a c + b^{2}\right )}{4 a^{2} \left (- 4 a c + b^{2}\right )^{2} \left (a + b x^{2} + c x^{4}\right )} + \frac{b \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{3} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{\log{\left (x^{2} \right )}}{2 a^{3}} - \frac{\log{\left (a + b x^{2} + c x^{4} \right )}}{4 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(c*x**4+b*x**2+a)**3,x)

[Out]

(-2*a*c + b**2 + b*c*x**2)/(4*a*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)**2) + (16*
a**2*c**2 - 15*a*b**2*c + 2*b**4 + 2*b*c*x**2*(-7*a*c + b**2))/(4*a**2*(-4*a*c +
b**2)**2*(a + b*x**2 + c*x**4)) + b*(30*a**2*c**2 - 10*a*b**2*c + b**4)*atanh((
b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(2*a**3*(-4*a*c + b**2)**(5/2)) + log(x**2)/(
2*a**3) - log(a + b*x**2 + c*x**4)/(4*a**3)

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Mathematica [A]  time = 0.927753, size = 342, normalized size = 1.71 \[ \frac{\frac{a^2 \left (-2 a c+b^2+b c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{a \left (16 a^2 c^2-15 a b^2 c-14 a b c^2 x^2+2 b^4+2 b^3 c x^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\left (16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c-8 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}+b^5\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{\left (-16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c+8 a b^2 c \sqrt{b^2-4 a c}-b^4 \sqrt{b^2-4 a c}+b^5\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+4 \log (x)}{4 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^2 + c*x^4)^3),x]

[Out]

((a^2*(b^2 - 2*a*c + b*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (a*(2*b^4
- 15*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*x^2 - 14*a*b*c^2*x^2))/((b^2 - 4*a*c)^2*(a
+ b*x^2 + c*x^4)) + 4*Log[x] - ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 + b^4*Sqrt[b^2
- 4*a*c] - 8*a*b^2*c*Sqrt[b^2 - 4*a*c] + 16*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b - S
qrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2) + ((b^5 - 10*a*b^3*c + 30*a^2*b
*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c*Sqrt[b^2 - 4*a*c] - 16*a^2*c^2*Sqrt[b^2
- 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/(4*a^3)

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Maple [B]  time = 0.044, size = 1200, normalized size = 6. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^4+b*x^2+a)^3,x)

[Out]

-7/2/a/(c*x^4+b*x^2+a)^2*b*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2/a^2/(c*x^4+b*x
^2+a)^2*b^3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+4/(c*x^4+b*x^2+a)^2*c^3/(16*a^2*c
^2-8*a*b^2*c+b^4)*x^4-29/4/a/(c*x^4+b*x^2+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^
4*b^2+1/a^2/(c*x^4+b*x^2+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*b^4-1/2/(c*x^4+b*
x^2+a)^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*c^2-3/a/(c*x^4+b*x^2+a)^2*b^3/(16*a^2*
c^2-8*a*b^2*c+b^4)*x^2*c+1/2/a^2/(c*x^4+b*x^2+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4
)*x^2+6*a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2-21/4/(c*x^4+b*x^2+a)^
2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^2*c+3/4/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c
+b^4)*b^4-4/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^
4+b*x^2+a))+2/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*
x^4+b*x^2+a))*b^2-1/4/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^
4)*(c*x^4+b*x^2+a))*b^4-15/a/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*
a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*x^2*(16*a^2*c^2-8*a*b^2*c+b^4)+(1
6*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a
^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*b*c^2+5/a^2/(1024*a^5*c^5-1280*a^4*b^2*c^4+64
0*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*x^2*(16*a^2*c^2
-8*a*b^2*c+b^4)+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640
*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*b^3*c-1/2/a^3/(1024*a^5*c^5
-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan(
(2*c*x^2*(16*a^2*c^2-8*a*b^2*c+b^4)+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-
1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*b^5+ln(
x)/a^3

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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(1/((c*x^4 + b*x^2 + a)^3*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.712955, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^3*x),x, algorithm="fricas")

[Out]

[1/4*(((b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*x^8 + a^2*b^5 - 10*a^3*b^3*c + 30
*a^4*b*c^2 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*x^6 + (b^7 - 8*a*b^5*c +
10*a^2*b^3*c^2 + 60*a^3*b*c^3)*x^4 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*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*b^3*c^2 - 7*a^2*b*c^3)*
x^6 + 3*a^2*b^4 - 21*a^3*b^2*c + 24*a^4*c^2 + (4*a*b^4*c - 29*a^2*b^2*c^2 + 16*a
^3*c^3)*x^4 + 2*(a*b^5 - 6*a^2*b^3*c - a^3*b*c^2)*x^2 - ((b^4*c^2 - 8*a*b^2*c^3
+ 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3
*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*
c + 16*a^3*b*c^2)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((b^4*c^2 - 8*a*b^2*c^3 + 16*a
^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c
+ 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16
*a^3*b*c^2)*x^2)*log(x))*sqrt(b^2 - 4*a*c))/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2
+ (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^8 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2
+ 16*a^5*b*c^3)*x^6 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^4 + 2*(a^4*b^5 - 8
*a^5*b^3*c + 16*a^6*b*c^2)*x^2)*sqrt(b^2 - 4*a*c)), -1/4*(2*((b^5*c^2 - 10*a*b^3
*c^3 + 30*a^2*b*c^4)*x^8 + a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2 + 2*(b^6*c - 10
*a*b^4*c^2 + 30*a^2*b^2*c^3)*x^6 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*
c^3)*x^4 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*x^2)*arctan(-(2*c*x^2 + b)*
sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (2*(a*b^3*c^2 - 7*a^2*b*c^3)*x^6 + 3*a^2*b^4
- 21*a^3*b^2*c + 24*a^4*c^2 + (4*a*b^4*c - 29*a^2*b^2*c^2 + 16*a^3*c^3)*x^4 + 2
*(a*b^5 - 6*a^2*b^3*c - a^3*b*c^2)*x^2 - ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x
^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4
*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^
2)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2
*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 +
(b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2
)*log(x))*sqrt(-b^2 + 4*a*c))/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2 + (a^3*b^4*c^
2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^8 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^
3)*x^6 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^4 + 2*(a^4*b^5 - 8*a^5*b^3*c + 1
6*a^6*b*c^2)*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(1/x/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 15.6035, size = 436, normalized size = 2.18 \[ -\frac{{\left (b^{5} - 10 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{3 \, b^{4} c^{2} x^{8} - 24 \, a b^{2} c^{3} x^{8} + 48 \, a^{2} c^{4} x^{8} + 6 \, b^{5} c x^{6} - 44 \, a b^{3} c^{2} x^{6} + 68 \, a^{2} b c^{3} x^{6} + 3 \, b^{6} x^{4} - 10 \, a b^{4} c x^{4} - 58 \, a^{2} b^{2} c^{2} x^{4} + 128 \, a^{3} c^{3} x^{4} + 10 \, a b^{5} x^{2} - 72 \, a^{2} b^{3} c x^{2} + 92 \, a^{3} b c^{2} x^{2} + 9 \, a^{2} b^{4} - 66 \, a^{3} b^{2} c + 96 \, a^{4} c^{2}}{8 \,{\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} - \frac{{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac{{\rm ln}\left (x^{2}\right )}{2 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^3*x),x, algorithm="giac")

[Out]

-1/2*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/
((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*sqrt(-b^2 + 4*a*c)) + 1/8*(3*b^4*c^2*x^8 -
24*a*b^2*c^3*x^8 + 48*a^2*c^4*x^8 + 6*b^5*c*x^6 - 44*a*b^3*c^2*x^6 + 68*a^2*b*c
^3*x^6 + 3*b^6*x^4 - 10*a*b^4*c*x^4 - 58*a^2*b^2*c^2*x^4 + 128*a^3*c^3*x^4 + 10*
a*b^5*x^2 - 72*a^2*b^3*c*x^2 + 92*a^3*b*c^2*x^2 + 9*a^2*b^4 - 66*a^3*b^2*c + 96*
a^4*c^2)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*(c*x^4 + b*x^2 + a)^2) - 1/4*ln(c
*x^4 + b*x^2 + a)/a^3 + 1/2*ln(x^2)/a^3