3.666 \(\int \frac{x^2}{\left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=204 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}+\frac{x^3}{4 a \left (a+c x^4\right )} \]

[Out]

x^3/(4*a*(a + c*x^4)) - ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(8*Sqrt[2]*a^(5/
4)*c^(3/4)) + ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(8*Sqrt[2]*a^(5/4)*c^(3/4)
) + Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(16*Sqrt[2]*a^(5/4)*c
^(3/4)) - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(16*Sqrt[2]*a^(
5/4)*c^(3/4))

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Rubi [A]  time = 0.242339, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{5/4} c^{3/4}}+\frac{x^3}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + c*x^4)^2,x]

[Out]

x^3/(4*a*(a + c*x^4)) - ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(8*Sqrt[2]*a^(5/
4)*c^(3/4)) + ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(8*Sqrt[2]*a^(5/4)*c^(3/4)
) + Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(16*Sqrt[2]*a^(5/4)*c
^(3/4)) - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(16*Sqrt[2]*a^(
5/4)*c^(3/4))

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Rubi in Sympy [A]  time = 53.3747, size = 185, normalized size = 0.91 \[ \frac{x^{3}}{4 a \left (a + c x^{4}\right )} + \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 a^{\frac{5}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 a^{\frac{5}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{5}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{5}{4}} c^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/(4*a*(a + c*x**4)) + sqrt(2)*log(-sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + s
qrt(c)*x**2)/(32*a**(5/4)*c**(3/4)) - sqrt(2)*log(sqrt(2)*a**(1/4)*c**(1/4)*x +
sqrt(a) + sqrt(c)*x**2)/(32*a**(5/4)*c**(3/4)) - sqrt(2)*atan(1 - sqrt(2)*c**(1/
4)*x/a**(1/4))/(16*a**(5/4)*c**(3/4)) + sqrt(2)*atan(1 + sqrt(2)*c**(1/4)*x/a**(
1/4))/(16*a**(5/4)*c**(3/4))

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Mathematica [A]  time = 0.272686, size = 184, normalized size = 0.9 \[ \frac{\frac{\sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}-\frac{\sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}-\frac{2 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac{2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{c^{3/4}}+\frac{8 \sqrt [4]{a} x^3}{a+c x^4}}{32 a^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + c*x^4)^2,x]

[Out]

((8*a^(1/4)*x^3)/(a + c*x^4) - (2*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)
])/c^(3/4) + (2*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(3/4) + (Sqrt
[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(3/4) - (Sqrt[2]*L
og[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(3/4))/(32*a^(5/4))

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Maple [A]  time = 0.009, size = 154, normalized size = 0.8 \[{\frac{{x}^{3}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{\sqrt{2}}{32\,ac}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{\sqrt{2}}{16\,ac}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{\sqrt{2}}{16\,ac}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(c*x^4+a)^2,x)

[Out]

1/4*x^3/a/(c*x^4+a)+1/32/a/c/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(
a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+1/16/a/c/(a/c)^(1/4)*2^(1/2
)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+1/16/a/c/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a
/c)^(1/4)*x-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245625, size = 230, normalized size = 1.13 \[ \frac{4 \, x^{3} + 4 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{4} c^{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{4}}}{x + \sqrt{-a^{3} c \sqrt{-\frac{1}{a^{5} c^{3}}} + x^{2}}}\right ) +{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{4}} \log \left (a^{4} c^{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{4}} + x\right ) -{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{1}{4}} \log \left (-a^{4} c^{2} \left (-\frac{1}{a^{5} c^{3}}\right )^{\frac{3}{4}} + x\right )}{16 \,{\left (a c x^{4} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(4*x^3 + 4*(a*c*x^4 + a^2)*(-1/(a^5*c^3))^(1/4)*arctan(a^4*c^2*(-1/(a^5*c^3
))^(3/4)/(x + sqrt(-a^3*c*sqrt(-1/(a^5*c^3)) + x^2))) + (a*c*x^4 + a^2)*(-1/(a^5
*c^3))^(1/4)*log(a^4*c^2*(-1/(a^5*c^3))^(3/4) + x) - (a*c*x^4 + a^2)*(-1/(a^5*c^
3))^(1/4)*log(-a^4*c^2*(-1/(a^5*c^3))^(3/4) + x))/(a*c*x^4 + a^2)

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Sympy [A]  time = 1.84723, size = 46, normalized size = 0.23 \[ \frac{x^{3}}{4 a^{2} + 4 a c x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a^{5} c^{3} + 1, \left ( t \mapsto t \log{\left (4096 t^{3} a^{4} c^{2} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(c*x**4+a)**2,x)

[Out]

x**3/(4*a**2 + 4*a*c*x**4) + RootSum(65536*_t**4*a**5*c**3 + 1, Lambda(_t, _t*lo
g(4096*_t**3*a**4*c**2 + x)))

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GIAC/XCAS [A]  time = 0.22397, size = 265, normalized size = 1.3 \[ \frac{x^{3}}{4 \,{\left (c x^{4} + a\right )} a} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} c^{3}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c^{3}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^3/((c*x^4 + a)*a) + 1/16*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + s
qrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^3) + 1/16*sqrt(2)*(a*c^3)^(3/4)*arctan(1
/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^3) - 1/32*sqrt(2)*(a*
c^3)^(3/4)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^3) + 1/32*sqrt(2)*
(a*c^3)^(3/4)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^3)