3.1334 \(\int \frac{x^3}{\left (a+b x^6\right )^2} \, dx\)

Optimal. Leaf size=142 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{4/3} b^{2/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{4/3} b^{2/3}}+\frac{x^4}{6 a \left (a+b x^6\right )} \]

[Out]

x^4/(6*a*(a + b*x^6)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))]/(6*S
qrt[3]*a^(4/3)*b^(2/3)) - Log[a^(1/3) + b^(1/3)*x^2]/(18*a^(4/3)*b^(2/3)) + Log[
a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(36*a^(4/3)*b^(2/3))

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Rubi [A]  time = 0.231423, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{4/3} b^{2/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{4/3} b^{2/3}}+\frac{x^4}{6 a \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^6)^2,x]

[Out]

x^4/(6*a*(a + b*x^6)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))]/(6*S
qrt[3]*a^(4/3)*b^(2/3)) - Log[a^(1/3) + b^(1/3)*x^2]/(18*a^(4/3)*b^(2/3)) + Log[
a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(36*a^(4/3)*b^(2/3))

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Rubi in Sympy [A]  time = 33.7226, size = 128, normalized size = 0.9 \[ \frac{x^{4}}{6 a \left (a + b x^{6}\right )} - \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} x^{2} \right )}}{18 a^{\frac{4}{3}} b^{\frac{2}{3}}} + \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{2} + b^{\frac{2}{3}} x^{4} \right )}}{36 a^{\frac{4}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x^{2}}{3}\right )}{\sqrt [3]{a}} \right )}}{18 a^{\frac{4}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**6+a)**2,x)

[Out]

x**4/(6*a*(a + b*x**6)) - log(a**(1/3) + b**(1/3)*x**2)/(18*a**(4/3)*b**(2/3)) +
log(a**(2/3) - a**(1/3)*b**(1/3)*x**2 + b**(2/3)*x**4)/(36*a**(4/3)*b**(2/3)) -
sqrt(3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x**2/3)/a**(1/3))/(18*a**(4/3)*b*
*(2/3))

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Mathematica [A]  time = 0.267345, size = 195, normalized size = 1.37 \[ \frac{-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{2/3}}+\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{2/3}}+\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{2/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{2/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{b^{2/3}}+\frac{6 \sqrt [3]{a} x^4}{a+b x^6}}{36 a^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x^6)^2,x]

[Out]

((6*a^(1/3)*x^4)/(a + b*x^6) - (2*Sqrt[3]*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)
])/b^(2/3) - (2*Sqrt[3]*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])/b^(2/3) - (2*Lo
g[a^(1/3) + b^(1/3)*x^2])/b^(2/3) + Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^
(1/3)*x^2]/b^(2/3) + Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/b^(2
/3))/(36*a^(4/3))

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Maple [A]  time = 0.006, size = 123, normalized size = 0.9 \[{\frac{{x}^{4}}{6\,a \left ( b{x}^{6}+a \right ) }}-{\frac{1}{18\,ab}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{36\,ab}\ln \left ({x}^{4}-{x}^{2}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}}{18\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{x}^{2}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^6+a)^2,x)

[Out]

1/6*x^4/a/(b*x^6+a)-1/18/a/b/(a/b)^(1/3)*ln(x^2+(a/b)^(1/3))+1/36/a/b/(a/b)^(1/3
)*ln(x^4-x^2*(a/b)^(1/3)+(a/b)^(2/3))+1/18/a*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^
(1/2)*(2/(a/b)^(1/3)*x^2-1))

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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^3/(b*x^6 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221748, size = 211, normalized size = 1.49 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} x^{4} - \sqrt{3}{\left (b x^{6} + a\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{4} + \left (-a b^{2}\right )^{\frac{2}{3}} x^{2} - a b\right ) + 2 \, \sqrt{3}{\left (b x^{6} + a\right )} \log \left (\left (-a b^{2}\right )^{\frac{2}{3}} x^{2} + a b\right ) - 6 \,{\left (b x^{6} + a\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x^{2} - \sqrt{3} a b}{3 \, a b}\right )\right )}}{108 \,{\left (a b x^{6} + a^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^6 + a)^2,x, algorithm="fricas")

[Out]

1/108*sqrt(3)*(6*sqrt(3)*(-a*b^2)^(1/3)*x^4 - sqrt(3)*(b*x^6 + a)*log((-a*b^2)^(
1/3)*b*x^4 + (-a*b^2)^(2/3)*x^2 - a*b) + 2*sqrt(3)*(b*x^6 + a)*log((-a*b^2)^(2/3
)*x^2 + a*b) - 6*(b*x^6 + a)*arctan(1/3*(2*sqrt(3)*(-a*b^2)^(2/3)*x^2 - sqrt(3)*
a*b)/(a*b)))/((a*b*x^6 + a^2)*(-a*b^2)^(1/3))

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Sympy [A]  time = 4.14417, size = 46, normalized size = 0.32 \[ \frac{x^{4}}{6 a^{2} + 6 a b x^{6}} + \operatorname{RootSum}{\left (5832 t^{3} a^{4} b^{2} + 1, \left ( t \mapsto t \log{\left (324 t^{2} a^{3} b + x^{2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**6+a)**2,x)

[Out]

x**4/(6*a**2 + 6*a*b*x**6) + RootSum(5832*_t**3*a**4*b**2 + 1, Lambda(_t, _t*log
(324*_t**2*a**3*b + x**2)))

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GIAC/XCAS [A]  time = 0.225969, size = 182, normalized size = 1.28 \[ \frac{x^{4}}{6 \,{\left (b x^{6} + a\right )} a} - \frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{18 \, a^{2}} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{18 \, a^{2} b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{36 \, a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^6 + a)^2,x, algorithm="giac")

[Out]

1/6*x^4/((b*x^6 + a)*a) - 1/18*(-a/b)^(2/3)*ln(abs(x^2 - (-a/b)^(1/3)))/a^2 - 1/
18*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x^2 + (-a/b)^(1/3))/(-a/b)^(1/3)
)/(a^2*b^2) + 1/36*(-a*b^2)^(2/3)*ln(x^4 + x^2*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2
*b^2)