3.1333 \(\int \frac{x^4}{\left (a+b x^6\right )^2} \, dx\)

Optimal. Leaf size=234 \[ \frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{x^5}{6 a \left (a+b x^6\right )} \]

[Out]

x^5/(6*a*(a + b*x^6)) + ArcTan[(b^(1/6)*x)/a^(1/6)]/(18*a^(7/6)*b^(5/6)) - ArcTa
n[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)]/(36*a^(7/6)*b^(5/6)) + ArcTan[(Sqrt[3
]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)]/(36*a^(7/6)*b^(5/6)) + Log[a^(1/3) - Sqrt[3]*a
^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(7/6)*b^(5/6)) - Log[a^(1/3) + Sqr
t[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(7/6)*b^(5/6))

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Rubi [A]  time = 1.08368, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{x^5}{6 a \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^6)^2,x]

[Out]

x^5/(6*a*(a + b*x^6)) + ArcTan[(b^(1/6)*x)/a^(1/6)]/(18*a^(7/6)*b^(5/6)) - ArcTa
n[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)]/(36*a^(7/6)*b^(5/6)) + ArcTan[(Sqrt[3
]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)]/(36*a^(7/6)*b^(5/6)) + Log[a^(1/3) - Sqrt[3]*a
^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(7/6)*b^(5/6)) - Log[a^(1/3) + Sqr
t[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(7/6)*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)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.300508, size = 193, normalized size = 0.82 \[ \frac{\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{5/6}}-\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{5/6}}+\frac{4 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}-\frac{2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{b^{5/6}}+\frac{12 \sqrt [6]{a} x^5}{a+b x^6}}{72 a^{7/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^6)^2,x]

[Out]

((12*a^(1/6)*x^5)/(a + b*x^6) + (4*ArcTan[(b^(1/6)*x)/a^(1/6)])/b^(5/6) - (2*Arc
Tan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)])/b^(5/6) + (2*ArcTan[Sqrt[3] + (2*b^(1/6)*x
)/a^(1/6)])/b^(5/6) + (Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)
*x^2])/b^(5/6) - (Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]
)/b^(5/6))/(72*a^(7/6))

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Maple [B]  time = 0.354, size = 349, normalized size = 1.5 \[{\frac{x}{18\,ab} \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{1}{18\,ab}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{x}{18\,ab} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{\sqrt{3}}{72\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( \sqrt{3}\sqrt [6]{{\frac{a}{b}}}x-{x}^{2}-\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{36\,ab}\arctan \left ( -\sqrt{3}+2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{x}{18\,ab} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{\sqrt{3}}{72\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{36\,ab}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^6+a)^2,x)

[Out]

1/18/b/a*x/(x^2+(a/b)^(1/3))+1/18/b/a/(a/b)^(1/6)*arctan(x/(a/b)^(1/6))+1/18/b/a
/(x^2-3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))*x-1/36/b/a/(x^2-3^(1/2)*(a/b)^(1/6)*x+(
a/b)^(1/3))*3^(1/2)*(a/b)^(1/6)+1/72/a^2*3^(1/2)*(a/b)^(5/6)*ln(3^(1/2)*(a/b)^(1
/6)*x-x^2-(a/b)^(1/3))+1/36/b/a/(a/b)^(1/6)*arctan(-3^(1/2)+2*x/(a/b)^(1/6))+1/1
8/b/a/(x^2+3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))*x+1/36/b/a/(x^2+3^(1/2)*(a/b)^(1/6
)*x+(a/b)^(1/3))*3^(1/2)*(a/b)^(1/6)-1/72/a^2*3^(1/2)*(a/b)^(5/6)*ln(x^2+3^(1/2)
*(a/b)^(1/6)*x+(a/b)^(1/3))+1/36/b/a/(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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244848, size = 620, normalized size = 2.65 \[ \frac{12 \, x^{5} + 4 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \arctan \left (\frac{\sqrt{3} a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}}}{a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} + 2 \, x + 2 \, \sqrt{a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}}}\right ) + 4 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \arctan \left (-\frac{\sqrt{3} a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}}}{a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - 2 \, x - 2 \, \sqrt{-a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}}}\right ) +{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}\right ) -{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (-a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}\right ) + 2 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} + x\right ) - 2 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (-a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} + x\right )}{72 \,{\left (a b x^{6} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^6 + a)^2,x, algorithm="fricas")

[Out]

1/72*(12*x^5 + 4*sqrt(3)*(a*b*x^6 + a^2)*(-1/(a^7*b^5))^(1/6)*arctan(sqrt(3)*a^6
*b^4*(-1/(a^7*b^5))^(5/6)/(a^6*b^4*(-1/(a^7*b^5))^(5/6) + 2*x + 2*sqrt(a^6*b^4*x
*(-1/(a^7*b^5))^(5/6) - a^5*b^3*(-1/(a^7*b^5))^(2/3) + x^2))) + 4*sqrt(3)*(a*b*x
^6 + a^2)*(-1/(a^7*b^5))^(1/6)*arctan(-sqrt(3)*a^6*b^4*(-1/(a^7*b^5))^(5/6)/(a^6
*b^4*(-1/(a^7*b^5))^(5/6) - 2*x - 2*sqrt(-a^6*b^4*x*(-1/(a^7*b^5))^(5/6) - a^5*b
^3*(-1/(a^7*b^5))^(2/3) + x^2))) + (a*b*x^6 + a^2)*(-1/(a^7*b^5))^(1/6)*log(a^6*
b^4*x*(-1/(a^7*b^5))^(5/6) - a^5*b^3*(-1/(a^7*b^5))^(2/3) + x^2) - (a*b*x^6 + a^
2)*(-1/(a^7*b^5))^(1/6)*log(-a^6*b^4*x*(-1/(a^7*b^5))^(5/6) - a^5*b^3*(-1/(a^7*b
^5))^(2/3) + x^2) + 2*(a*b*x^6 + a^2)*(-1/(a^7*b^5))^(1/6)*log(a^6*b^4*(-1/(a^7*
b^5))^(5/6) + x) - 2*(a*b*x^6 + a^2)*(-1/(a^7*b^5))^(1/6)*log(-a^6*b^4*(-1/(a^7*
b^5))^(5/6) + x))/(a*b*x^6 + a^2)

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Sympy [A]  time = 4.13771, size = 46, normalized size = 0.2 \[ \frac{x^{5}}{6 a^{2} + 6 a b x^{6}} + \operatorname{RootSum}{\left (2176782336 t^{6} a^{7} b^{5} + 1, \left ( t \mapsto t \log{\left (60466176 t^{5} a^{6} b^{4} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**6+a)**2,x)

[Out]

x**5/(6*a**2 + 6*a*b*x**6) + RootSum(2176782336*_t**6*a**7*b**5 + 1, Lambda(_t,
_t*log(60466176*_t**5*a**6*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)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError