3.37 \(\int \frac{d+e x+f x^2}{\left (a+b x^2+c x^4\right )^2} \, dx\)
Optimal. Leaf size=368 \[ \frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (-\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 c e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
[Out]
-(e*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (x*(b^2*d - 2*a*c*d -
a*b*f + c*(b*d - 2*a*f)*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[c
]*(b*d - 2*a*f + (b^2*d - 12*a*c*d + 4*a*b*f)/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]*(b*d - 2*a*f - (b^2*d - 12*a*c*d + 4*a*b*f)/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*c*e*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 = 1.92523, 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
\[ \frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (-\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 c e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x + f*x^2)/(a + b*x^2 + c*x^4)^2,x]
[Out]
-(e*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (x*(b^2*d - 2*a*c*d -
a*b*f + c*(b*d - 2*a*f)*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[c
]*(b*d - 2*a*f + (b^2*d - 12*a*c*d + 4*a*b*f)/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]*(b*d - 2*a*f - (b^2*d - 12*a*c*d + 4*a*b*f)/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*c*e*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((f*x**2+e*x+d)/(c*x**4+b*x**2+a)**2,x)
[Out]
Timed out
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Mathematica [A] time = 2.53327, size = 398, normalized size = 1.08 \[ \frac{1}{4} \left (\frac{2 a b (e+f x)+4 a c x (d+x (e+f x))-2 b d x \left (b+c x^2\right )}{a \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \sqrt{c} \left (b \left (d \sqrt{b^2-4 a c}+4 a f\right )-2 a \left (f \sqrt{b^2-4 a c}+6 c d\right )+b^2 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (b d \sqrt{b^2-4 a c}-2 a f \sqrt{b^2-4 a c}-4 a b f+12 a c d+b^2 (-d)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 c e \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{4 c e \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x + f*x^2)/(a + b*x^2 + c*x^4)^2,x]
[Out]
((2*a*b*(e + f*x) - 2*b*d*x*(b + c*x^2) + 4*a*c*x*(d + x*(e + f*x)))/(a*(-b^2 +
4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^2*d + b*(Sqrt[b^2 - 4*a*c]*d +
4*a*f) - 2*a*(6*c*d + Sqrt[b^2 - 4*a*c]*f))*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]]) + (Sqr
t[2]*Sqrt[c]*(-(b^2*d) + 12*a*c*d + b*Sqrt[b^2 - 4*a*c]*d - 4*a*b*f - 2*a*Sqrt[b
^2 - 4*a*c]*f)*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*c*e*Log[-b + Sqrt[b^2 - 4*a*c]
- 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + (4*c*e*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 = 0.163, size = 2851, normalized size = 7.8 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^2+e*x+d)/(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))*b^3*d-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))*(-4*a*c+b^2)^(1/2)*b^4*d+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))*b^2*f-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))*b*
d+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)*a
rctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^5*d+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))*(-4*a*c+b^2)^(1/2)*d-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))*(-4*a*c+b^2)^(1/2)*b^3*f+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))*b*d-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))*b^5*d+
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)*arcta
n(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b^2*d-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))*(-4*a*c+b^2)^(1/2)*b^3*f+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))*(-4*a*c+b^2)^(1/2)*b^2*d-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))*b^2*f+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))*(-4*a*c+b^2)^(1/2)*d-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*d*x*b^2*(-4*a*c+b^2)^(1/2)+c/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^
(1/2)+1/2*b/c)*d*x*(-4*a*c+b^2)^(1/2)-c/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1
/2)+1/2*b/c)*d*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*d*
x*b^3-c/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*d*x*(-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))*d*x*b+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*d*x*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))*(-4*a*c+b^2)^(1/2)*b*f-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))*(-4*a*c+b^2)^(1/2)*b^4*d-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))*(-4*a*c+b^2)^(1/2)*b*f+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*d*x*b^2*(-4*a*c+b^2)^(1/2)-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)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^
2)^(1/2))*c)^(1/2))*f+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))*b^4*f+2
*c/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*e*a-1/2/(4*a*c-b^2)^2/(x
^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*e*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))*e*b^2+2*c/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/
c)*e*a+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))*f-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))*b^4*f-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
))*b^3*d+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*f+2*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*f+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))*(-4*a*c+b^2)^(1/2)
*e+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))*(-4*a*c+b^2)^(1/2)*b^2*e-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))*(-4*a*c+b^2)^(1/2)*e-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))*(-4*a*c+b^2)^(1/
2)*b^2*e-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*b^2*f-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*b^2*f
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{2 \, a c e x^{2} -{\left (b c d - 2 \, a c f\right )} x^{3} + a b e +{\left (a b f -{\left (b^{2} - 2 \, a c\right )} d\right )} x}{2 \,{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )}} - \frac{\int \frac{4 \, a c e x - a b f -{\left (b c d - 2 \, a c f\right )} x^{2} -{\left (b^{2} - 6 \, a c\right )} d}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a b^{2} - 4 \, a^{2} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^2,x, algorithm="maxima")
[Out]
-1/2*(2*a*c*e*x^2 - (b*c*d - 2*a*c*f)*x^3 + a*b*e + (a*b*f - (b^2 - 2*a*c)*d)*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
*integrate((4*a*c*e*x - a*b*f - (b*c*d - 2*a*c*f)*x^2 - (b^2 - 6*a*c)*d)/(c*x^4
+ b*x^2 + a), x)/(a*b^2 - 4*a^2*c)
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^2,x, algorithm="fricas")
[Out]
Exception raised: NotImplementedError
_______________________________________________________________________________________
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**2+e*x+d)/(c*x**4+b*x**2+a)**2,x)
[Out]
Timed out
_______________________________________________________________________________________
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^2 + e*x + d)/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError