3.323 \(\int \frac{x^2}{a+b x^4+c x^8} \, dx\)
Optimal. Leaf size=315 \[ -\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2^{3/4} \sqrt{b^2-4 a c} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2^{3/4} \sqrt{b^2-4 a c} \sqrt [4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2^{3/4} \sqrt{b^2-4 a c} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2^{3/4} \sqrt{b^2-4 a c} \sqrt [4]{\sqrt{b^2-4 a c}-b}} \]
[Out]
-((c^(1/4)*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2^(3/4)*
Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(1/4))) + (c^(1/4)*ArcTan[(2^(1/4)*c^
(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2^(3/4)*Sqrt[b^2 - 4*a*c]*(-b + Sqrt[
b^2 - 4*a*c])^(1/4)) + (c^(1/4)*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a
*c])^(1/4)])/(2^(3/4)*Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - (c^(1/
4)*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2^(3/4)*Sqrt[b^
2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(1/4))
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Rubi [A] time = 0.600244, antiderivative size = 315, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222
\[ -\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2^{3/4} \sqrt{b^2-4 a c} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2^{3/4} \sqrt{b^2-4 a c} \sqrt [4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2^{3/4} \sqrt{b^2-4 a c} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2^{3/4} \sqrt{b^2-4 a c} \sqrt [4]{\sqrt{b^2-4 a c}-b}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a + b*x^4 + c*x^8),x]
[Out]
-((c^(1/4)*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2^(3/4)*
Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(1/4))) + (c^(1/4)*ArcTan[(2^(1/4)*c^
(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2^(3/4)*Sqrt[b^2 - 4*a*c]*(-b + Sqrt[
b^2 - 4*a*c])^(1/4)) + (c^(1/4)*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a
*c])^(1/4)])/(2^(3/4)*Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - (c^(1/
4)*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2^(3/4)*Sqrt[b^
2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(1/4))
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Rubi in Sympy [A] time = 90.8588, size = 291, normalized size = 0.92 \[ \frac{\sqrt [4]{2} \sqrt [4]{c} \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt [4]{2} \sqrt [4]{c} \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt [4]{2} \sqrt [4]{c} \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt [4]{2} \sqrt [4]{c} \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(c*x**8+b*x**4+a),x)
[Out]
2**(1/4)*c**(1/4)*atan(2**(1/4)*c**(1/4)*x/(-b + sqrt(-4*a*c + b**2))**(1/4))/(2
*(-b + sqrt(-4*a*c + b**2))**(1/4)*sqrt(-4*a*c + b**2)) - 2**(1/4)*c**(1/4)*atan
h(2**(1/4)*c**(1/4)*x/(-b + sqrt(-4*a*c + b**2))**(1/4))/(2*(-b + sqrt(-4*a*c +
b**2))**(1/4)*sqrt(-4*a*c + b**2)) - 2**(1/4)*c**(1/4)*atan(2**(1/4)*c**(1/4)*x/
(-b - sqrt(-4*a*c + b**2))**(1/4))/(2*(-b - sqrt(-4*a*c + b**2))**(1/4)*sqrt(-4*
a*c + b**2)) + 2**(1/4)*c**(1/4)*atanh(2**(1/4)*c**(1/4)*x/(-b - sqrt(-4*a*c + b
**2))**(1/4))/(2*(-b - sqrt(-4*a*c + b**2))**(1/4)*sqrt(-4*a*c + b**2))
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Mathematica [C] time = 0.0320882, size = 43, normalized size = 0.14 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right ] \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a + b*x^4 + c*x^8),x]
[Out]
RootSum[a + b*#1^4 + c*#1^8 & , Log[x - #1]/(b*#1 + 2*c*#1^5) & ]/4
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Maple [C] time = 0.001, size = 43, normalized size = 0.1 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+{{\it \_Z}}^{4}b+a \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(c*x^8+b*x^4+a),x)
[Out]
1/4*sum(_R^2/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{c x^{8} + b x^{4} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(c*x^8 + b*x^4 + a),x, algorithm="maxima")
[Out]
integrate(x^2/(c*x^8 + b*x^4 + a), x)
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Fricas [A] time = 0.336273, size = 3767, normalized size = 11.96 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(c*x^8 + b*x^4 + a),x, algorithm="fricas")
[Out]
-sqrt(sqrt(1/2)*sqrt(-(b + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*
a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))*a
rctan(1/2*sqrt(1/2)*(b^4 - 8*a*b^2*c + 16*a^2*c^2 - (a*b^7 - 12*a^2*b^5*c + 48*a
^3*b^3*c^2 - 64*a^4*b*c^3)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5
*c^3))*sqrt(sqrt(1/2)*sqrt(-(b + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6
- 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^
2)))*sqrt(-(b + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c +
48*a^4*b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2))/(c*x + sqrt(1
/2)*c*sqrt((2*c*x^2 - sqrt(1/2)*(b^3 - 4*a*b*c - (a*b^6 - 12*a^2*b^4*c + 48*a^3*
b^2*c^2 - 64*a^4*c^3)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)
)*sqrt(-(b + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48
*a^4*b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))/c))) + sqrt(sqr
t(1/2)*sqrt(-(b - (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c
+ 48*a^4*b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))*arctan(-1/
2*sqrt(1/2)*(b^4 - 8*a*b^2*c + 16*a^2*c^2 + (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c
^2 - 64*a^4*b*c^3)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3))*s
qrt(sqrt(1/2)*sqrt(-(b - (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^
3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))*sqr
t(-(b - (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*
b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2))/(c*x + sqrt(1/2)*c*sq
rt((2*c*x^2 - sqrt(1/2)*(b^3 - 4*a*b*c + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2
- 64*a^4*c^3)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3))*sqrt(-
(b - (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2
*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))/c))) - 1/4*sqrt(sqrt(1/
2)*sqrt(-(b + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 4
8*a^4*b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))*log(1/2*sqrt(1
/2)*(b^4 - 8*a*b^2*c + 16*a^2*c^2 - (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*
a^4*b*c^3)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3))*sqrt(sqrt
(1/2)*sqrt(-(b + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c
+ 48*a^4*b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))*sqrt(-(b +
(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2
- 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)) + c*x) + 1/4*sqrt(sqrt(1/2)*s
qrt(-(b + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^
4*b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))*log(-1/2*sqrt(1/2)
*(b^4 - 8*a*b^2*c + 16*a^2*c^2 - (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4
*b*c^3)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3))*sqrt(sqrt(1/
2)*sqrt(-(b + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 4
8*a^4*b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))*sqrt(-(b + (a*
b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 6
4*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)) + c*x) - 1/4*sqrt(sqrt(1/2)*sqrt
(-(b - (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b
^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))*log(1/2*sqrt(1/2)*(b^
4 - 8*a*b^2*c + 16*a^2*c^2 + (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c
^3)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3))*sqrt(sqrt(1/2)*s
qrt(-(b - (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^
4*b^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))*sqrt(-(b - (a*b^4
- 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^
5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)) + c*x) + 1/4*sqrt(sqrt(1/2)*sqrt(-(b
- (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c
^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))*log(-1/2*sqrt(1/2)*(b^4 -
8*a*b^2*c + 16*a^2*c^2 + (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)
/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3))*sqrt(sqrt(1/2)*sqrt
(-(b - (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b
^2*c^2 - 64*a^5*c^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)))*sqrt(-(b - (a*b^4 - 8
*a^2*b^2*c + 16*a^3*c^2)/sqrt(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c
^3))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)) + c*x)
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Sympy [A] time = 12.0883, size = 172, normalized size = 0.55 \[ \operatorname{RootSum}{\left (t^{8} \left (16777216 a^{5} c^{4} - 16777216 a^{4} b^{2} c^{3} + 6291456 a^{3} b^{4} c^{2} - 1048576 a^{2} b^{6} c + 65536 a b^{8}\right ) + t^{4} \left (4096 a^{2} b c^{2} - 2048 a b^{3} c + 256 b^{5}\right ) + c, \left ( t \mapsto t \log{\left (x + \frac{1048576 t^{7} a^{4} b c^{3} - 786432 t^{7} a^{3} b^{3} c^{2} + 196608 t^{7} a^{2} b^{5} c - 16384 t^{7} a b^{7} - 512 t^{3} a^{2} c^{2} + 384 t^{3} a b^{2} c - 64 t^{3} b^{4}}{c} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(c*x**8+b*x**4+a),x)
[Out]
RootSum(_t**8*(16777216*a**5*c**4 - 16777216*a**4*b**2*c**3 + 6291456*a**3*b**4*
c**2 - 1048576*a**2*b**6*c + 65536*a*b**8) + _t**4*(4096*a**2*b*c**2 - 2048*a*b*
*3*c + 256*b**5) + c, Lambda(_t, _t*log(x + (1048576*_t**7*a**4*b*c**3 - 786432*
_t**7*a**3*b**3*c**2 + 196608*_t**7*a**2*b**5*c - 16384*_t**7*a*b**7 - 512*_t**3
*a**2*c**2 + 384*_t**3*a*b**2*c - 64*_t**3*b**4)/c)))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{c x^{8} + b x^{4} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(c*x^8 + b*x^4 + a),x, algorithm="giac")
[Out]
integrate(x^2/(c*x^8 + b*x^4 + a), x)