3.1320 \(\int \frac{x^4}{a+b x^6} \, dx\)

Optimal. Leaf size=215 \[ \frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} \sqrt [6]{a} b^{5/6}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} \sqrt [6]{a} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}} \]

[Out]

ArcTan[(b^(1/6)*x)/a^(1/6)]/(3*a^(1/6)*b^(5/6)) - ArcTan[(Sqrt[3]*a^(1/6) - 2*b^
(1/6)*x)/a^(1/6)]/(6*a^(1/6)*b^(5/6)) + ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a
^(1/6)]/(6*a^(1/6)*b^(5/6)) + Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*
x^2]/(4*Sqrt[3]*a^(1/6)*b^(5/6)) - Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(
1/3)*x^2]/(4*Sqrt[3]*a^(1/6)*b^(5/6))

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Rubi [A]  time = 1.02969, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ \frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} \sqrt [6]{a} b^{5/6}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} \sqrt [6]{a} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^6),x]

[Out]

ArcTan[(b^(1/6)*x)/a^(1/6)]/(3*a^(1/6)*b^(5/6)) - ArcTan[(Sqrt[3]*a^(1/6) - 2*b^
(1/6)*x)/a^(1/6)]/(6*a^(1/6)*b^(5/6)) + ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a
^(1/6)]/(6*a^(1/6)*b^(5/6)) + Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*
x^2]/(4*Sqrt[3]*a^(1/6)*b^(5/6)) - Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(
1/3)*x^2]/(4*Sqrt[3]*a^(1/6)*b^(5/6))

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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(x**4/(b*x**6+a),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0394209, size = 154, normalized size = 0.72 \[ \frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+4 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{12 \sqrt [6]{a} b^{5/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^6),x]

[Out]

(4*ArcTan[(b^(1/6)*x)/a^(1/6)] - 2*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*A
rcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] + Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b
^(1/6)*x + b^(1/3)*x^2] - Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1
/3)*x^2])/(12*a^(1/6)*b^(5/6))

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Maple [A]  time = 0.061, size = 162, normalized size = 0.8 \[{\frac{1}{3\,b}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}}{12\,a} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( \sqrt{3}\sqrt [6]{{\frac{a}{b}}}x-{x}^{2}-\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{6\,b}\arctan \left ( -\sqrt{3}+2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}}{12\,a} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{6\,b}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^6+a),x)

[Out]

1/3/b/(a/b)^(1/6)*arctan(x/(a/b)^(1/6))+1/12/a*3^(1/2)*(a/b)^(5/6)*ln(3^(1/2)*(a
/b)^(1/6)*x-x^2-(a/b)^(1/3))+1/6/b/(a/b)^(1/6)*arctan(-3^(1/2)+2*x/(a/b)^(1/6))-
1/12/a*3^(1/2)*(a/b)^(5/6)*ln(x^2+3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))+1/6/b/(a/b)
^(1/6)*arctan(2*x/(a/b)^(1/6)+3^(1/2))

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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^4/(b*x^6 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234888, size = 477, normalized size = 2.22 \[ \frac{1}{3} \, \sqrt{3} \left (-\frac{1}{a b^{5}}\right )^{\frac{1}{6}} \arctan \left (\frac{\sqrt{3} a b^{4} \left (-\frac{1}{a b^{5}}\right )^{\frac{5}{6}}}{a b^{4} \left (-\frac{1}{a b^{5}}\right )^{\frac{5}{6}} + 2 \, x + 2 \, \sqrt{a b^{4} x \left (-\frac{1}{a b^{5}}\right )^{\frac{5}{6}} - a b^{3} \left (-\frac{1}{a b^{5}}\right )^{\frac{2}{3}} + x^{2}}}\right ) + \frac{1}{3} \, \sqrt{3} \left (-\frac{1}{a b^{5}}\right )^{\frac{1}{6}} \arctan \left (-\frac{\sqrt{3} a b^{4} \left (-\frac{1}{a b^{5}}\right )^{\frac{5}{6}}}{a b^{4} \left (-\frac{1}{a b^{5}}\right )^{\frac{5}{6}} - 2 \, x - 2 \, \sqrt{-a b^{4} x \left (-\frac{1}{a b^{5}}\right )^{\frac{5}{6}} - a b^{3} \left (-\frac{1}{a b^{5}}\right )^{\frac{2}{3}} + x^{2}}}\right ) + \frac{1}{12} \, \left (-\frac{1}{a b^{5}}\right )^{\frac{1}{6}} \log \left (a b^{4} x \left (-\frac{1}{a b^{5}}\right )^{\frac{5}{6}} - a b^{3} \left (-\frac{1}{a b^{5}}\right )^{\frac{2}{3}} + x^{2}\right ) - \frac{1}{12} \, \left (-\frac{1}{a b^{5}}\right )^{\frac{1}{6}} \log \left (-a b^{4} x \left (-\frac{1}{a b^{5}}\right )^{\frac{5}{6}} - a b^{3} \left (-\frac{1}{a b^{5}}\right )^{\frac{2}{3}} + x^{2}\right ) + \frac{1}{6} \, \left (-\frac{1}{a b^{5}}\right )^{\frac{1}{6}} \log \left (a b^{4} \left (-\frac{1}{a b^{5}}\right )^{\frac{5}{6}} + x\right ) - \frac{1}{6} \, \left (-\frac{1}{a b^{5}}\right )^{\frac{1}{6}} \log \left (-a b^{4} \left (-\frac{1}{a b^{5}}\right )^{\frac{5}{6}} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^6 + a),x, algorithm="fricas")

[Out]

1/3*sqrt(3)*(-1/(a*b^5))^(1/6)*arctan(sqrt(3)*a*b^4*(-1/(a*b^5))^(5/6)/(a*b^4*(-
1/(a*b^5))^(5/6) + 2*x + 2*sqrt(a*b^4*x*(-1/(a*b^5))^(5/6) - a*b^3*(-1/(a*b^5))^
(2/3) + x^2))) + 1/3*sqrt(3)*(-1/(a*b^5))^(1/6)*arctan(-sqrt(3)*a*b^4*(-1/(a*b^5
))^(5/6)/(a*b^4*(-1/(a*b^5))^(5/6) - 2*x - 2*sqrt(-a*b^4*x*(-1/(a*b^5))^(5/6) -
a*b^3*(-1/(a*b^5))^(2/3) + x^2))) + 1/12*(-1/(a*b^5))^(1/6)*log(a*b^4*x*(-1/(a*b
^5))^(5/6) - a*b^3*(-1/(a*b^5))^(2/3) + x^2) - 1/12*(-1/(a*b^5))^(1/6)*log(-a*b^
4*x*(-1/(a*b^5))^(5/6) - a*b^3*(-1/(a*b^5))^(2/3) + x^2) + 1/6*(-1/(a*b^5))^(1/6
)*log(a*b^4*(-1/(a*b^5))^(5/6) + x) - 1/6*(-1/(a*b^5))^(1/6)*log(-a*b^4*(-1/(a*b
^5))^(5/6) + x)

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Sympy [A]  time = 0.442016, size = 26, normalized size = 0.12 \[ \operatorname{RootSum}{\left (46656 t^{6} a b^{5} + 1, \left ( t \mapsto t \log{\left (7776 t^{5} a b^{4} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**6+a),x)

[Out]

RootSum(46656*_t**6*a*b**5 + 1, Lambda(_t, _t*log(7776*_t**5*a*b**4 + x)))

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GIAC/XCAS [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(x^4/(b*x^6 + a),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError