3.313 \(\int \frac{x^5}{a+b x^4+c x^8} \, dx\)

Optimal. Leaf size=159 \[ \frac{\sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}} \]

[Out]

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

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Rubi [A]  time = 0.273555, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^4 + c*x^8),x]

[Out]

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

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Rubi in Sympy [A]  time = 38.7665, size = 144, normalized size = 0.91 \[ - \frac{\sqrt{2} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x^{2}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 \sqrt{c} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{2} \sqrt{b + \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x^{2}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 \sqrt{c} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(c*x**8+b*x**4+a),x)

[Out]

-sqrt(2)*sqrt(b - sqrt(-4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*x**2/sqrt(b - sqrt(-
4*a*c + b**2)))/(4*sqrt(c)*sqrt(-4*a*c + b**2)) + sqrt(2)*sqrt(b + sqrt(-4*a*c +
b**2))*atan(sqrt(2)*sqrt(c)*x**2/sqrt(b + sqrt(-4*a*c + b**2)))/(4*sqrt(c)*sqrt
(-4*a*c + b**2))

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Mathematica [A]  time = 0.148516, size = 171, normalized size = 1.08 \[ \frac{\left (\sqrt{b^2-4 a c}-b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )+\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} \sqrt{c} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^4 + c*x^8),x]

[Out]

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

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Maple [A]  time = 0.021, size = 216, normalized size = 1.4 \[{\frac{\sqrt{2}}{4}\arctan \left ({c{x}^{2}\sqrt{2}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}+{\frac{b\sqrt{2}}{4}\arctan \left ({c{x}^{2}\sqrt{2}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}-{\frac{\sqrt{2}}{4}{\it Artanh} \left ({c{x}^{2}\sqrt{2}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}+{\frac{b\sqrt{2}}{4}{\it Artanh} \left ({c{x}^{2}\sqrt{2}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(c*x^8+b*x^4+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{c x^{8} + b x^{4} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^8 + b*x^4 + a),x, algorithm="maxima")

[Out]

integrate(x^5/(c*x^8 + b*x^4 + a), x)

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Fricas [A]  time = 0.274801, size = 765, normalized size = 4.81 \[ \frac{1}{4} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{b + \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (x^{2} + \frac{\sqrt{\frac{1}{2}}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{-\frac{b + \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}\right ) - \frac{1}{4} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{b + \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (x^{2} - \frac{\sqrt{\frac{1}{2}}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{-\frac{b + \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}\right ) - \frac{1}{4} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{b - \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (x^{2} + \frac{\sqrt{\frac{1}{2}}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{-\frac{b - \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}\right ) + \frac{1}{4} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{b - \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (x^{2} - \frac{\sqrt{\frac{1}{2}}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{-\frac{b - \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^8 + b*x^4 + a),x, algorithm="fricas")

[Out]

1/4*sqrt(1/2)*sqrt(-(b + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a
*c^2))*log(x^2 + sqrt(1/2)*(b^2*c - 4*a*c^2)*sqrt(-(b + (b^2*c - 4*a*c^2)/sqrt(b
^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))/sqrt(b^2*c^2 - 4*a*c^3)) - 1/4*sqrt(1/2)*s
qrt(-(b + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))*log(x^2
- sqrt(1/2)*(b^2*c - 4*a*c^2)*sqrt(-(b + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^
3))/(b^2*c - 4*a*c^2))/sqrt(b^2*c^2 - 4*a*c^3)) - 1/4*sqrt(1/2)*sqrt(-(b - (b^2*
c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))*log(x^2 + sqrt(1/2)*(b^
2*c - 4*a*c^2)*sqrt(-(b - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*
a*c^2))/sqrt(b^2*c^2 - 4*a*c^3)) + 1/4*sqrt(1/2)*sqrt(-(b - (b^2*c - 4*a*c^2)/sq
rt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))*log(x^2 - sqrt(1/2)*(b^2*c - 4*a*c^2)*
sqrt(-(b - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))/sqrt(b^
2*c^2 - 4*a*c^3))

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Sympy [A]  time = 7.94046, size = 76, normalized size = 0.48 \[ \operatorname{RootSum}{\left (t^{4} \left (4096 a^{2} c^{3} - 2048 a b^{2} c^{2} + 256 b^{4} c\right ) + t^{2} \left (- 64 a b c + 16 b^{3}\right ) + a, \left ( t \mapsto t \log{\left (512 t^{3} a c^{2} - 128 t^{3} b^{2} c - 4 t b + x^{2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(c*x**8+b*x**4+a),x)

[Out]

RootSum(_t**4*(4096*a**2*c**3 - 2048*a*b**2*c**2 + 256*b**4*c) + _t**2*(-64*a*b*
c + 16*b**3) + a, Lambda(_t, _t*log(512*_t**3*a*c**2 - 128*_t**3*b**2*c - 4*_t*b
+ x**2)))

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GIAC/XCAS [A]  time = 0.395299, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^8 + b*x^4 + a),x, algorithm="giac")

[Out]

Done