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

Optimal. Leaf size=185 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} c^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.210221, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} c^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + c*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 47.7867, size = 172, normalized size = 0.93 \[ \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 \sqrt [4]{a} c^{\frac{3}{4}}} - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 \sqrt [4]{a} c^{\frac{3}{4}}} - \frac{\sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 \sqrt [4]{a} c^{\frac{3}{4}}} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 \sqrt [4]{a} c^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*log(-sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(8*a**(1/4)*c
**(3/4)) - sqrt(2)*log(sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(8*
a**(1/4)*c**(3/4)) - sqrt(2)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(4*a**(1/4)*c
**(3/4)) + sqrt(2)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(4*a**(1/4)*c**(3/4))

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Mathematica [A]  time = 0.0507221, size = 134, normalized size = 0.72 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} c^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + c*x^4),x]

[Out]

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

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Maple [A]  time = 0.004, size = 128, normalized size = 0.7 \[{\frac{\sqrt{2}}{8\,c}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{\sqrt{2}}{4\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{\sqrt{2}}{4\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(c*x^4+a),x)

[Out]

1/8/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)))+1/4/c/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/
4)*x+1)+1/4/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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238958, size = 153, normalized size = 0.83 \[ \left (-\frac{1}{a c^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{a c^{2} \left (-\frac{1}{a c^{3}}\right )^{\frac{3}{4}}}{x + \sqrt{-a c \sqrt{-\frac{1}{a c^{3}}} + x^{2}}}\right ) + \frac{1}{4} \, \left (-\frac{1}{a c^{3}}\right )^{\frac{1}{4}} \log \left (a c^{2} \left (-\frac{1}{a c^{3}}\right )^{\frac{3}{4}} + x\right ) - \frac{1}{4} \, \left (-\frac{1}{a c^{3}}\right )^{\frac{1}{4}} \log \left (-a c^{2} \left (-\frac{1}{a c^{3}}\right )^{\frac{3}{4}} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.374871, size = 26, normalized size = 0.14 \[ \operatorname{RootSum}{\left (256 t^{4} a c^{3} + 1, \left ( t \mapsto t \log{\left (64 t^{3} a c^{2} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(c*x**4+a),x)

[Out]

RootSum(256*_t**4*a*c**3 + 1, Lambda(_t, _t*log(64*_t**3*a*c**2 + x)))

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GIAC/XCAS [A]  time = 0.225741, size = 242, normalized size = 1.31 \[ \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{3}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{3}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{3}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*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*c^3) + 1/4*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*c^3) - 1/8*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 + sqrt(2)*x*(a/
c)^(1/4) + sqrt(a/c))/(a*c^3) + 1/8*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 - sqrt(2)*x*(a/
c)^(1/4) + sqrt(a/c))/(a*c^3)