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

Optimal. Leaf size=368 \[ \frac{x \left (A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (\frac{A \left (b^2-12 a c\right )+4 a b C}{\sqrt{b^2-4 a c}}-2 a C+A b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (-\frac{-12 a A c+4 a b C+A b^2}{\sqrt{b^2-4 a c}}-2 a C+A b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 B c \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{B \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

-(B*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (x*(A*(b^2 - 2*a*c) -
a*b*C + c*(A*b - 2*a*C)*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[c
]*(A*b - 2*a*C + (A*(b^2 - 12*a*c) + 4*a*b*C)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]
*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)*Sqrt[b - Sq
rt[b^2 - 4*a*c]]) + (Sqrt[c]*(A*b - 2*a*C - (A*b^2 - 12*a*A*c + 4*a*b*C)/Sqrt[b^
2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*
a*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*B*c*ArcTanh[(b + 2*c*x^2)/Sqrt
[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

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Rubi [A]  time = 1.95437, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ \frac{x \left (A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (\frac{A \left (b^2-12 a c\right )+4 a b C}{\sqrt{b^2-4 a c}}-2 a C+A b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (-\frac{-12 a A c+4 a b C+A b^2}{\sqrt{b^2-4 a c}}-2 a C+A b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 B c \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{B \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(B*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (x*(A*(b^2 - 2*a*c) -
a*b*C + c*(A*b - 2*a*C)*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[c
]*(A*b - 2*a*C + (A*(b^2 - 12*a*c) + 4*a*b*C)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]
*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)*Sqrt[b - Sq
rt[b^2 - 4*a*c]]) + (Sqrt[c]*(A*b - 2*a*C - (A*b^2 - 12*a*A*c + 4*a*b*C)/Sqrt[b^
2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*
a*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*B*c*ArcTanh[(b + 2*c*x^2)/Sqrt
[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 2.88028, size = 393, normalized size = 1.07 \[ \frac{1}{4} \left (\frac{4 a c x (A+x (B+C x))+2 a b (B+C x)-2 A b x \left (b+c x^2\right )}{a \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \sqrt{c} \left (A \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right )-2 a C \left (\sqrt{b^2-4 a c}-2 b\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} \left (A \left (-b \sqrt{b^2-4 a c}-12 a c+b^2\right )+2 a C \left (\sqrt{b^2-4 a c}+2 b\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 B c \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{4 B c \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*a*b*(B + C*x) - 2*A*b*x*(b + c*x^2) + 4*a*c*x*(A + x*(B + C*x)))/(a*(-b^2 +
4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(A*(b^2 - 12*a*c + b*Sqrt[b^2 - 4
*a*c]) - 2*a*(-2*b + Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - S
qrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[
2]*Sqrt[c]*(A*(b^2 - 12*a*c - b*Sqrt[b^2 - 4*a*c]) + 2*a*(2*b + Sqrt[b^2 - 4*a*c
])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^
(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (4*B*c*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2
])/(b^2 - 4*a*c)^(3/2) + (4*B*c*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a
*c)^(3/2))/4

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Maple [B]  time = 0.141, size = 2851, normalized size = 7.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c^2/(4*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
anh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^3-8*c^3/(4*a*c-b^2)^2*2^(
1/2)/(4*a*c+3*b^2)*a^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))*C+3/2*c/(4*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^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))*C*b^4-2*c^2/(4*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^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^3+12*c^3/(4*a*c-b
^2)^2*2^(1/2)/(4*a*c+3*b^2)*a/((-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*(-4*a*c+b^2)^(1/2)+8*c^2/(4*a*c-b^2)^2*
2^(1/2)/(4*a*c+3*b^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*(-4*a*c+b^2)^(1/2)*b^2+4*c^3/(4*a*c-b^2)^2*2^(1
/2)/(4*a*c+3*b^2)*a/((-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-3/4*c/(4*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^2)/a/((-
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^5-3*c/(4*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^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))*C*(-4*a*c+b^2)^(1/2
)*b^3-4*c^2/(4*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^2)*a/((-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))*C*b^2+12*c^3/(4*a*c-b
^2)^2*2^(1/2)/(4*a*c+3*b^2)*a/((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*(-4*a*c+b^2)^(1/2)+8*c^2/(4*a*c-b^2)^2*2^(
1/2)/(4*a*c+3*b^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*(-4*a*c+b^2)^(1/2)*b^2-4*c^3/(4*a*c-b^2)^2*2^(1/2)/(4
*a*c+3*b^2)*a/((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+3/4*c/(4*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^2)/a/((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^5-
3*c/(4*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^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))*C*(-4*a*c+b^2)^(1/2)*b^3+4*c^2/(4*
a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^2)*a/((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))*C*b^2-4*c^2/(4*a*c-b^2)^2*2^(1/2)/(4*a*
c+3*b^2)*a/((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))*C*(-4*a*c+b^2)^(1/2)*b-3/4*c/(4*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^2
)/a/((-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*(-4*a*c+b^2)^(1/2)*b^4-4*c^2/(4*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^2)*
a/((-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))*C*(-4*a*c+b^2)^(1/2)*b-3/4*c/(4*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^2)/a/((
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*(-4*a*c+b^2)^(1/2)*b^4+1/4/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(
1/2))/a*x*A*(-4*a*c+b^2)^(1/2)*b^2-1/4/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/
2)+1/2*b/c)/a*x*A*(-4*a*c+b^2)^(1/2)*b^2-4*c^2/(4*a*c-b^2)^2/(4*a*c+3*b^2)*a*ln(
(4*a*c+3*b^2)*a*(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b))*B*(-4*a*c+b^2)^(1/2)+3*c/(4*a*c
-b^2)^2/(4*a*c+3*b^2)*ln((4*a*c+3*b^2)*a*(2*c*x^2+(-4*a*c+b^2)^(1/2)+b))*B*(-4*a
*c+b^2)^(1/2)*b^2-3*c/(4*a*c-b^2)^2/(4*a*c+3*b^2)*ln((4*a*c+3*b^2)*a*(-2*c*x^2+(
-4*a*c+b^2)^(1/2)-b))*B*(-4*a*c+b^2)^(1/2)*b^2+4*c^2/(4*a*c-b^2)^2/(4*a*c+3*b^2)
*a*ln((4*a*c+3*b^2)*a*(2*c*x^2+(-4*a*c+b^2)^(1/2)+b))*B*(-4*a*c+b^2)^(1/2)-1/2/(
4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*B*b^2-1/2/(4*a*c-b^2)^2/(x^2
+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*B*b^2+2*c/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-
4*a*c+b^2)^(1/2))*B*a-1/2/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*x
*C*b^2-1/2/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*x*C*b^2+8*c^3/(4
*a*c-b^2)^2*2^(1/2)/(4*a*c+3*b^2)*a^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))*C-3/2*c/(4*a*c-b^2)^2*2^(1/2)/(4*a*c
+3*b^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))*C*b^4+2*c/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*B*a
-c/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*x*A*(-4*a*c+b^2)^(1/2)+1
/4/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))/a*x*A*b^3+c/(4*a*c-b^2)^
2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*x*A*(-4*a*c+b^2)^(1/2)-c/(4*a*c-b^2)^2/
(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*A*x*b+2*c/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/
c*(-4*a*c+b^2)^(1/2))*C*x*a-c/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/
c)*A*x*b+1/4/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)/a*x*A*b^3+2*c/
(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*C*x*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{2 \, B a c x^{2} +{\left (2 \, C a - A b\right )} c x^{3} + B a b +{\left (C a b - A b^{2} + 2 \, A a c\right )} x}{2 \,{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )}} + \frac{-\int \frac{4 \, B a c x +{\left (2 \, C a - A b\right )} c x^{2} - C a b - A b^{2} + 6 \, A a c}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a b^{2} - 4 \, a^{2} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*B*a*c*x^2 + (2*C*a - A*b)*c*x^3 + B*a*b + (C*a*b - A*b^2 + 2*A*a*c)*x)/(
(a*b^2*c - 4*a^2*c^2)*x^4 + a^2*b^2 - 4*a^3*c + (a*b^3 - 4*a^2*b*c)*x^2) + 1/2*i
ntegrate(-(4*B*a*c*x + (2*C*a - A*b)*c*x^2 - C*a*b - A*b^2 + 6*A*a*c)/(c*x^4 + b
*x^2 + a), x)/(a*b^2 - 4*a^2*c)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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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((C*x**2+B*x+A)/(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((C*x^2 + B*x + A)/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError