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

Optimal. Leaf size=189 \[ -\frac{\sqrt{c} \left (\frac{A b-2 a B}{\sqrt{b^2-4 a c}}+A\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (A-\frac{A b-2 a B}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{A}{a x} \]

[Out]

-(A/(a*x)) - (Sqrt[c]*(A + (A*b - 2*a*B)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt
[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (
Sqrt[c]*(A - (A*b - 2*a*B)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b
+ Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 0.761438, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{\sqrt{c} \left (\frac{A b-2 a B}{\sqrt{b^2-4 a c}}+A\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (A-\frac{A b-2 a B}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{A}{a x} \]

Antiderivative was successfully verified.

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

[Out]

-(A/(a*x)) - (Sqrt[c]*(A + (A*b - 2*a*B)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt
[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (
Sqrt[c]*(A - (A*b - 2*a*B)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b
+ Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi in Sympy [A]  time = 64.0168, size = 194, normalized size = 1.03 \[ - \frac{A}{a x} + \frac{\sqrt{2} \sqrt{c} \left (A b - A \sqrt{- 4 a c + b^{2}} - 2 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 a \sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt{2} \sqrt{c} \left (A b + A \sqrt{- 4 a c + b^{2}} - 2 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 a \sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(a*x) + sqrt(2)*sqrt(c)*(A*b - A*sqrt(-4*a*c + b**2) - 2*B*a)*atan(sqrt(2)*sq
rt(c)*x/sqrt(b + sqrt(-4*a*c + b**2)))/(2*a*sqrt(b + sqrt(-4*a*c + b**2))*sqrt(-
4*a*c + b**2)) - sqrt(2)*sqrt(c)*(A*b + A*sqrt(-4*a*c + b**2) - 2*B*a)*atan(sqrt
(2)*sqrt(c)*x/sqrt(b - sqrt(-4*a*c + b**2)))/(2*a*sqrt(b - sqrt(-4*a*c + b**2))*
sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.521094, size = 206, normalized size = 1.09 \[ -\frac{\frac{\sqrt{2} \sqrt{c} \left (A \left (\sqrt{b^2-4 a c}+b\right )-2 a B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (A \left (\sqrt{b^2-4 a c}-b\right )+2 a B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 A}{x}}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

-((2*A)/x + (Sqrt[2]*Sqrt[c]*(-2*a*B + A*(b + Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2
]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2
- 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(2*a*B + A*(-b + Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[
2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2
- 4*a*c]]))/(2*a)

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Maple [B]  time = 0.028, size = 353, normalized size = 1.9 \[ -{\frac{c\sqrt{2}A}{2\,a}\arctan \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}+{\frac{c\sqrt{2}Ab}{2\,a}\arctan \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}-{c\sqrt{2}B\arctan \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}+{\frac{c\sqrt{2}A}{2\,a}{\it Artanh} \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}+{\frac{c\sqrt{2}Ab}{2\,a}{\it Artanh} \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}-{c\sqrt{2}B{\it Artanh} \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}-{\frac{A}{ax}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c/a*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c
+b^2)^(1/2))*c)^(1/2))*A+1/2*c/a/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/
2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b-c/(-4*a*c+
b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a
*c+b^2)^(1/2))*c)^(1/2))*B+1/2*c/a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arc
tanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A+1/2*c/a/(-4*a*c+b^2)^(1/2)
*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)
^(1/2))*c)^(1/2))*A*b-c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(
1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B-A/a/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{\int \frac{A c x^{2} - B a + A b}{c x^{4} + b x^{2} + a}\,{d x}}{a} - \frac{A}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate((A*c*x^2 - B*a + A*b)/(c*x^4 + b*x^2 + a), x)/a - A/(a*x)

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Fricas [A]  time = 0.50651, size = 3934, normalized size = 20.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(sqrt(1/2)*a*x*sqrt(-(B^2*a^2*b - 2*A*B*a*b^2 + A^2*b^3 + (4*A*B*a^2 - 3*A^2
*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2
- 4*A^3*B*a*b^3 + A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a^2*b + A^4*a
*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(2*(A^4*a*c^3 + (A^3*B*a*
b - A^4*b^2)*c^2 - (B^4*a^3 - 3*A*B^3*a^2*b + 3*A^2*B^2*a*b^2 - A^3*B*b^3)*c)*x
+ sqrt(1/2)*(B^3*a^3*b^2 - 3*A*B^2*a^2*b^3 + 3*A^2*B*a*b^4 - A^3*b^5 + 4*(A^2*B*
a^3 - A^3*a^2*b)*c^2 - (4*B^3*a^4 - 12*A*B^2*a^3*b + 13*A^2*B*a^2*b^2 - 5*A^3*a*
b^3)*c - (B*a^4*b^3 - A*a^3*b^4 - 8*A*a^5*c^2 - 2*(2*B*a^5*b - 3*A*a^4*b^2)*c)*s
qrt((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4 + A^4
*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a^2*b + A^4*a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))*
sqrt(-(B^2*a^2*b - 2*A*B*a*b^2 + A^2*b^3 + (4*A*B*a^2 - 3*A^2*a*b)*c + (a^3*b^2
- 4*a^4*c)*sqrt((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A
^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a^2*b + A^4*a*b^2)*c)/(a^6*b^2 -
4*a^7*c)))/(a^3*b^2 - 4*a^4*c))) - sqrt(1/2)*a*x*sqrt(-(B^2*a^2*b - 2*A*B*a*b^2
+ A^2*b^3 + (4*A*B*a^2 - 3*A^2*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt((B^4*a^4 - 4*A
*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*
B^2*a^3 - 2*A^3*B*a^2*b + A^4*a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c
))*log(2*(A^4*a*c^3 + (A^3*B*a*b - A^4*b^2)*c^2 - (B^4*a^3 - 3*A*B^3*a^2*b + 3*A
^2*B^2*a*b^2 - A^3*B*b^3)*c)*x - sqrt(1/2)*(B^3*a^3*b^2 - 3*A*B^2*a^2*b^3 + 3*A^
2*B*a*b^4 - A^3*b^5 + 4*(A^2*B*a^3 - A^3*a^2*b)*c^2 - (4*B^3*a^4 - 12*A*B^2*a^3*
b + 13*A^2*B*a^2*b^2 - 5*A^3*a*b^3)*c - (B*a^4*b^3 - A*a^3*b^4 - 8*A*a^5*c^2 - 2
*(2*B*a^5*b - 3*A*a^4*b^2)*c)*sqrt((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2
- 4*A^3*B*a*b^3 + A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a^2*b + A^4*a
*b^2)*c)/(a^6*b^2 - 4*a^7*c)))*sqrt(-(B^2*a^2*b - 2*A*B*a*b^2 + A^2*b^3 + (4*A*B
*a^2 - 3*A^2*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*
B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a
^2*b + A^4*a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))) + sqrt(1/2)*a*x
*sqrt(-(B^2*a^2*b - 2*A*B*a*b^2 + A^2*b^3 + (4*A*B*a^2 - 3*A^2*a*b)*c - (a^3*b^2
- 4*a^4*c)*sqrt((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 +
A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a^2*b + A^4*a*b^2)*c)/(a^6*b^2
- 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(2*(A^4*a*c^3 + (A^3*B*a*b - A^4*b^2)*c^2 -
(B^4*a^3 - 3*A*B^3*a^2*b + 3*A^2*B^2*a*b^2 - A^3*B*b^3)*c)*x + sqrt(1/2)*(B^3*a
^3*b^2 - 3*A*B^2*a^2*b^3 + 3*A^2*B*a*b^4 - A^3*b^5 + 4*(A^2*B*a^3 - A^3*a^2*b)*c
^2 - (4*B^3*a^4 - 12*A*B^2*a^3*b + 13*A^2*B*a^2*b^2 - 5*A^3*a*b^3)*c + (B*a^4*b^
3 - A*a^3*b^4 - 8*A*a^5*c^2 - 2*(2*B*a^5*b - 3*A*a^4*b^2)*c)*sqrt((B^4*a^4 - 4*A
*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*
B^2*a^3 - 2*A^3*B*a^2*b + A^4*a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))*sqrt(-(B^2*a^2*b -
2*A*B*a*b^2 + A^2*b^3 + (4*A*B*a^2 - 3*A^2*a*b)*c - (a^3*b^2 - 4*a^4*c)*sqrt((B
^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4 + A^4*a^2*c
^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a^2*b + A^4*a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b
^2 - 4*a^4*c))) - sqrt(1/2)*a*x*sqrt(-(B^2*a^2*b - 2*A*B*a*b^2 + A^2*b^3 + (4*A*
B*a^2 - 3*A^2*a*b)*c - (a^3*b^2 - 4*a^4*c)*sqrt((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2
*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*
a^2*b + A^4*a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(2*(A^4*a*c^
3 + (A^3*B*a*b - A^4*b^2)*c^2 - (B^4*a^3 - 3*A*B^3*a^2*b + 3*A^2*B^2*a*b^2 - A^3
*B*b^3)*c)*x - sqrt(1/2)*(B^3*a^3*b^2 - 3*A*B^2*a^2*b^3 + 3*A^2*B*a*b^4 - A^3*b^
5 + 4*(A^2*B*a^3 - A^3*a^2*b)*c^2 - (4*B^3*a^4 - 12*A*B^2*a^3*b + 13*A^2*B*a^2*b
^2 - 5*A^3*a*b^3)*c + (B*a^4*b^3 - A*a^3*b^4 - 8*A*a^5*c^2 - 2*(2*B*a^5*b - 3*A*
a^4*b^2)*c)*sqrt((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 +
A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a^2*b + A^4*a*b^2)*c)/(a^6*b^2
- 4*a^7*c)))*sqrt(-(B^2*a^2*b - 2*A*B*a*b^2 + A^2*b^3 + (4*A*B*a^2 - 3*A^2*a*b)*
c - (a^3*b^2 - 4*a^4*c)*sqrt((B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^
3*B*a*b^3 + A^4*b^4 + A^4*a^2*c^2 - 2*(A^2*B^2*a^3 - 2*A^3*B*a^2*b + A^4*a*b^2)*
c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))) - 2*A)/(a*x)

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Sympy [A]  time = 36.2761, size = 490, normalized size = 2.59 \[ - \frac{A}{a x} + \operatorname{RootSum}{\left (t^{4} \left (256 a^{5} c^{2} - 128 a^{4} b^{2} c + 16 a^{3} b^{4}\right ) + t^{2} \left (48 A^{2} a^{2} b c^{2} - 28 A^{2} a b^{3} c + 4 A^{2} b^{5} - 64 A B a^{3} c^{2} + 48 A B a^{2} b^{2} c - 8 A B a b^{4} - 16 B^{2} a^{3} b c + 4 B^{2} a^{2} b^{3}\right ) + A^{4} c^{3} - 2 A^{3} B b c^{2} + 2 A^{2} B^{2} a c^{2} + A^{2} B^{2} b^{2} c - 2 A B^{3} a b c + B^{4} a^{2} c, \left ( t \mapsto t \log{\left (x + \frac{64 t^{3} A a^{5} c^{2} - 48 t^{3} A a^{4} b^{2} c + 8 t^{3} A a^{3} b^{4} + 32 t^{3} B a^{5} b c - 8 t^{3} B a^{4} b^{3} + 10 t A^{3} a^{2} b c^{2} - 10 t A^{3} a b^{3} c + 2 t A^{3} b^{5} - 12 t A^{2} B a^{3} c^{2} + 24 t A^{2} B a^{2} b^{2} c - 6 t A^{2} B a b^{4} - 18 t A B^{2} a^{3} b c + 6 t A B^{2} a^{2} b^{3} + 4 t B^{3} a^{4} c - 2 t B^{3} a^{3} b^{2}}{- A^{4} a c^{3} + A^{4} b^{2} c^{2} - A^{3} B a b c^{2} - A^{3} B b^{3} c + 3 A^{2} B^{2} a b^{2} c - 3 A B^{3} a^{2} b c + B^{4} a^{3} c} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(a*x) + RootSum(_t**4*(256*a**5*c**2 - 128*a**4*b**2*c + 16*a**3*b**4) + _t**
2*(48*A**2*a**2*b*c**2 - 28*A**2*a*b**3*c + 4*A**2*b**5 - 64*A*B*a**3*c**2 + 48*
A*B*a**2*b**2*c - 8*A*B*a*b**4 - 16*B**2*a**3*b*c + 4*B**2*a**2*b**3) + A**4*c**
3 - 2*A**3*B*b*c**2 + 2*A**2*B**2*a*c**2 + A**2*B**2*b**2*c - 2*A*B**3*a*b*c + B
**4*a**2*c, Lambda(_t, _t*log(x + (64*_t**3*A*a**5*c**2 - 48*_t**3*A*a**4*b**2*c
+ 8*_t**3*A*a**3*b**4 + 32*_t**3*B*a**5*b*c - 8*_t**3*B*a**4*b**3 + 10*_t*A**3*
a**2*b*c**2 - 10*_t*A**3*a*b**3*c + 2*_t*A**3*b**5 - 12*_t*A**2*B*a**3*c**2 + 24
*_t*A**2*B*a**2*b**2*c - 6*_t*A**2*B*a*b**4 - 18*_t*A*B**2*a**3*b*c + 6*_t*A*B**
2*a**2*b**3 + 4*_t*B**3*a**4*c - 2*_t*B**3*a**3*b**2)/(-A**4*a*c**3 + A**4*b**2*
c**2 - A**3*B*a*b*c**2 - A**3*B*b**3*c + 3*A**2*B**2*a*b**2*c - 3*A*B**3*a**2*b*
c + B**4*a**3*c))))

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GIAC/XCAS [A]  time = 0.883985, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done