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

Optimal. Leaf size=252 \[ -\frac{A \log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac{A \log (x)}{a^3}+\frac{2 c x^2 \left (6 a^2 B c+A \left (b^3-7 a b c\right )\right )+A \left (16 a^2 c^2-15 a b^2 c+2 b^4\right )+6 a^2 b B c}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\left (12 a^3 B c^2-A \left (30 a^2 b c^2-10 a b^3 c+b^5\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2}}-\frac{-A \left (b^2-2 a c\right )+c x^2 (-(A b-2 a B))+a b B}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

-(a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x^2)/(4*a*(b^2 - 4*a*c)*(a + b*x^2 +
 c*x^4)^2) + (6*a^2*b*B*c + A*(2*b^4 - 15*a*b^2*c + 16*a^2*c^2) + 2*c*(6*a^2*B*c
 + A*(b^3 - 7*a*b*c))*x^2)/(4*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - ((12*a^
3*B*c^2 - A*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 -
4*a*c]])/(2*a^3*(b^2 - 4*a*c)^(5/2)) + (A*Log[x])/a^3 - (A*Log[a + b*x^2 + c*x^4
])/(4*a^3)

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Rubi [A]  time = 1.06121, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{A \log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac{A \log (x)}{a^3}+\frac{2 c x^2 \left (6 a^2 B c+A \left (b^3-7 a b c\right )\right )+A \left (16 a^2 c^2-15 a b^2 c+2 b^4\right )+6 a^2 b B c}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\left (12 a^3 B c^2-A \left (30 a^2 b c^2-10 a b^3 c+b^5\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2}}+\frac{c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x^2)/(4*a*(b^2 - 4*a*c)*(a + b*x^2 +
c*x^4)^2) + (6*a^2*b*B*c + A*(2*b^4 - 15*a*b^2*c + 16*a^2*c^2) + 2*c*(6*a^2*B*c
+ A*(b^3 - 7*a*b*c))*x^2)/(4*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - ((12*a^3
*B*c^2 - A*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4
*a*c]])/(2*a^3*(b^2 - 4*a*c)^(5/2)) + (A*Log[x])/a^3 - (A*Log[a + b*x^2 + c*x^4]
)/(4*a^3)

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Rubi in Sympy [A]  time = 137.877, size = 257, normalized size = 1.02 \[ \frac{A \log{\left (x^{2} \right )}}{2 a^{3}} - \frac{A \log{\left (a + b x^{2} + c x^{4} \right )}}{4 a^{3}} + \frac{- 2 A a c + A b^{2} - B a b + c x^{2} \left (A b - 2 B a\right )}{4 a \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{2}} + \frac{16 A a^{2} c^{2} - 15 A a b^{2} c + 2 A b^{4} + 6 B a^{2} b c + 2 c x^{2} \left (- 7 A a b c + A b^{3} + 6 B a^{2} c\right )}{4 a^{2} \left (- 4 a c + b^{2}\right )^{2} \left (a + b x^{2} + c x^{4}\right )} + \frac{\left (30 A a^{2} b c^{2} - 10 A a b^{3} c + A b^{5} - 12 B a^{3} c^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{3} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*log(x**2)/(2*a**3) - A*log(a + b*x**2 + c*x**4)/(4*a**3) + (-2*A*a*c + A*b**2
- B*a*b + c*x**2*(A*b - 2*B*a))/(4*a*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)**2) +
 (16*A*a**2*c**2 - 15*A*a*b**2*c + 2*A*b**4 + 6*B*a**2*b*c + 2*c*x**2*(-7*A*a*b*
c + A*b**3 + 6*B*a**2*c))/(4*a**2*(-4*a*c + b**2)**2*(a + b*x**2 + c*x**4)) + (3
0*A*a**2*b*c**2 - 10*A*a*b**3*c + A*b**5 - 12*B*a**3*c**2)*atanh((b + 2*c*x**2)/
sqrt(-4*a*c + b**2))/(2*a**3*(-4*a*c + b**2)**(5/2))

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Mathematica [A]  time = 1.16134, size = 396, normalized size = 1.57 \[ \frac{\frac{a^2 \left (A \left (-2 a c+b^2+b c x^2\right )-a B \left (b+2 c x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{a \left (2 a^2 c \left (8 A c+3 b B+6 B c x^2\right )-a A b c \left (15 b+14 c x^2\right )+2 A b^3 \left (b+c x^2\right )\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\left (A \left (16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c-8 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}+b^5\right )-12 a^3 B c^2\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{\left (12 a^3 B c^2+A \left (16 a^2 c^2 \sqrt{b^2-4 a c}-30 a^2 b c^2+10 a b^3 c-8 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}-b^5\right )\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+4 A \log (x)}{4 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((a^2*(-(a*B*(b + 2*c*x^2)) + A*(b^2 - 2*a*c + b*c*x^2)))/((b^2 - 4*a*c)*(a + b*
x^2 + c*x^4)^2) + (a*(2*A*b^3*(b + c*x^2) - a*A*b*c*(15*b + 14*c*x^2) + 2*a^2*c*
(3*b*B + 8*A*c + 6*B*c*x^2)))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + 4*A*Log[x]
 - ((-12*a^3*B*c^2 + A*(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) - ((12*a^3*B*c^2 + A*(-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 = 0.033, size = 1645, normalized size = 6.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x/(c*x^4+b*x^2+a)^3,x)

[Out]

6/(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*c^2-1/4/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*B*b^3+1/2/a^2/(c*
x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b^5+5*a/(c*x^4+b*x^2+a)^2/(16*a^
2*c^2-8*a*b^2*c+b^4)*x^2*B*c^2+5/2*a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4
)*b*B*c+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))*A*b^2-15/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)+(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))*A*b*c^2+5/a^2/(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))*A*b^3*c+9/2/(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*B-1/2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+
b^4)*x^2*A*b*c^2+1/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*B*b^2*c+A*ln
(x)/a^3+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*A+6*a/(c*x^4+b*x^
2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*A*c^2+3/4/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*
b^2*c+b^4)*A*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))*A-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))*A*b^4-1/2/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))*A*b^5-21/4/(c*x^4+b*x^2+a)^2/(16*a^2*
c^2-8*a*b^2*c+b^4)*A*b^2*c+3/(c*x^4+b*x^2+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^
6*B-3/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b^3*c+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*A*b^4-7/2/a/(c*x^4+b*x^2+a)^2*c^3/(16*
a^2*c^2-8*a*b^2*c+b^4)*x^6*A*b+1/2/a^2/(c*x^4+b*x^2+a)^2*c^2/(16*a^2*c^2-8*a*b^2
*c+b^4)*x^6*A*b^3-29/4/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*A*
b^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((B*x^2 + A)/((c*x^4 + b*x^2 + a)^3*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.71744, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(((A*b^5*c^2 - 10*A*a*b^3*c^3 - 6*(2*B*a^3 - 5*A*a^2*b)*c^4)*x^8 + A*a^2*b^
5 - 10*A*a^3*b^3*c + 2*(A*b^6*c - 10*A*a*b^4*c^2 - 6*(2*B*a^3*b - 5*A*a^2*b^2)*c
^3)*x^6 + (A*b^7 - 8*A*a*b^5*c - 12*(2*B*a^4 - 5*A*a^3*b)*c^3 - 2*(6*B*a^3*b^2 -
 5*A*a^2*b^3)*c^2)*x^4 - 6*(2*B*a^5 - 5*A*a^4*b)*c^2 + 2*(A*a*b^6 - 10*A*a^2*b^4
*c - 6*(2*B*a^4*b - 5*A*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)) - (B*a^3*b^3 - 3*A*a^2*b^4 - 24*A*a^4*c^2 - 2*(A*a*b^3*c^2 + (6*B*a^3 -
7*A*a^2*b)*c^3)*x^6 - (4*A*a*b^4*c + 16*A*a^3*c^3 + (18*B*a^3*b - 29*A*a^2*b^2)*
c^2)*x^4 - 2*(A*a*b^5 + (10*B*a^4 - A*a^3*b)*c^2 + 2*(B*a^3*b^2 - 3*A*a^2*b^3)*c
)*x^2 - (10*B*a^4*b - 21*A*a^3*b^2)*c + ((A*b^4*c^2 - 8*A*a*b^2*c^3 + 16*A*a^2*c
^4)*x^8 + A*a^2*b^4 - 8*A*a^3*b^2*c + 16*A*a^4*c^2 + 2*(A*b^5*c - 8*A*a*b^3*c^2
+ 16*A*a^2*b*c^3)*x^6 + (A*b^6 - 6*A*a*b^4*c + 32*A*a^3*c^3)*x^4 + 2*(A*a*b^5 -
8*A*a^2*b^3*c + 16*A*a^3*b*c^2)*x^2)*log(c*x^4 + b*x^2 + a) - 4*((A*b^4*c^2 - 8*
A*a*b^2*c^3 + 16*A*a^2*c^4)*x^8 + A*a^2*b^4 - 8*A*a^3*b^2*c + 16*A*a^4*c^2 + 2*(
A*b^5*c - 8*A*a*b^3*c^2 + 16*A*a^2*b*c^3)*x^6 + (A*b^6 - 6*A*a*b^4*c + 32*A*a^3*
c^3)*x^4 + 2*(A*a*b^5 - 8*A*a^2*b^3*c + 16*A*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*((A*b^5*c^2 - 10*A*a*b^3*c^3 - 6*(2*B*a^3 - 5*A*a^2*b)*c
^4)*x^8 + A*a^2*b^5 - 10*A*a^3*b^3*c + 2*(A*b^6*c - 10*A*a*b^4*c^2 - 6*(2*B*a^3*
b - 5*A*a^2*b^2)*c^3)*x^6 + (A*b^7 - 8*A*a*b^5*c - 12*(2*B*a^4 - 5*A*a^3*b)*c^3
- 2*(6*B*a^3*b^2 - 5*A*a^2*b^3)*c^2)*x^4 - 6*(2*B*a^5 - 5*A*a^4*b)*c^2 + 2*(A*a*
b^6 - 10*A*a^2*b^4*c - 6*(2*B*a^4*b - 5*A*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)) + (B*a^3*b^3 - 3*A*a^2*b^4 - 24*A*a^4*c^2 -
 2*(A*a*b^3*c^2 + (6*B*a^3 - 7*A*a^2*b)*c^3)*x^6 - (4*A*a*b^4*c + 16*A*a^3*c^3 +
 (18*B*a^3*b - 29*A*a^2*b^2)*c^2)*x^4 - 2*(A*a*b^5 + (10*B*a^4 - A*a^3*b)*c^2 +
2*(B*a^3*b^2 - 3*A*a^2*b^3)*c)*x^2 - (10*B*a^4*b - 21*A*a^3*b^2)*c + ((A*b^4*c^2
 - 8*A*a*b^2*c^3 + 16*A*a^2*c^4)*x^8 + A*a^2*b^4 - 8*A*a^3*b^2*c + 16*A*a^4*c^2
+ 2*(A*b^5*c - 8*A*a*b^3*c^2 + 16*A*a^2*b*c^3)*x^6 + (A*b^6 - 6*A*a*b^4*c + 32*A
*a^3*c^3)*x^4 + 2*(A*a*b^5 - 8*A*a^2*b^3*c + 16*A*a^3*b*c^2)*x^2)*log(c*x^4 + b*
x^2 + a) - 4*((A*b^4*c^2 - 8*A*a*b^2*c^3 + 16*A*a^2*c^4)*x^8 + A*a^2*b^4 - 8*A*a
^3*b^2*c + 16*A*a^4*c^2 + 2*(A*b^5*c - 8*A*a*b^3*c^2 + 16*A*a^2*b*c^3)*x^6 + (A*
b^6 - 6*A*a*b^4*c + 32*A*a^3*c^3)*x^4 + 2*(A*a*b^5 - 8*A*a^2*b^3*c + 16*A*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))]

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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((B*x**2+A)/x/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 15.9649, size = 568, normalized size = 2.25 \[ -\frac{{\left (A b^{5} - 10 \, A a b^{3} c - 12 \, B a^{3} c^{2} + 30 \, A a^{2} b c^{2}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{A{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac{A{\rm ln}\left (x^{2}\right )}{2 \, a^{3}} + \frac{3 \, A b^{4} c^{2} x^{8} - 24 \, A a b^{2} c^{3} x^{8} + 48 \, A a^{2} c^{4} x^{8} + 6 \, A b^{5} c x^{6} - 44 \, A a b^{3} c^{2} x^{6} + 24 \, B a^{3} c^{3} x^{6} + 68 \, A a^{2} b c^{3} x^{6} + 3 \, A b^{6} x^{4} - 10 \, A a b^{4} c x^{4} + 36 \, B a^{3} b c^{2} x^{4} - 58 \, A a^{2} b^{2} c^{2} x^{4} + 128 \, A a^{3} c^{3} x^{4} + 10 \, A a b^{5} x^{2} + 8 \, B a^{3} b^{2} c x^{2} - 72 \, A a^{2} b^{3} c x^{2} + 40 \, B a^{4} c^{2} x^{2} + 92 \, A a^{3} b c^{2} x^{2} - 2 \, B a^{3} b^{3} + 9 \, A a^{2} b^{4} + 20 \, B a^{4} b c - 66 \, A a^{3} b^{2} c + 96 \, A a^{4} c^{2}}{8 \,{\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(A*b^5 - 10*A*a*b^3*c - 12*B*a^3*c^2 + 30*A*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)) -
 1/4*A*ln(c*x^4 + b*x^2 + a)/a^3 + 1/2*A*ln(x^2)/a^3 + 1/8*(3*A*b^4*c^2*x^8 - 24
*A*a*b^2*c^3*x^8 + 48*A*a^2*c^4*x^8 + 6*A*b^5*c*x^6 - 44*A*a*b^3*c^2*x^6 + 24*B*
a^3*c^3*x^6 + 68*A*a^2*b*c^3*x^6 + 3*A*b^6*x^4 - 10*A*a*b^4*c*x^4 + 36*B*a^3*b*c
^2*x^4 - 58*A*a^2*b^2*c^2*x^4 + 128*A*a^3*c^3*x^4 + 10*A*a*b^5*x^2 + 8*B*a^3*b^2
*c*x^2 - 72*A*a^2*b^3*c*x^2 + 40*B*a^4*c^2*x^2 + 92*A*a^3*b*c^2*x^2 - 2*B*a^3*b^
3 + 9*A*a^2*b^4 + 20*B*a^4*b*c - 66*A*a^3*b^2*c + 96*A*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)

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