3.1331 \(\int \frac{x^6}{\left (a+b x^6\right )^2} \, dx\)

Optimal. Leaf size=232 \[ -\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}-\frac{x}{6 b \left (a+b x^6\right )} \]

[Out]

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

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Rubi [A]  time = 0.776916, antiderivative size = 232, 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 \[ -\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}-\frac{x}{6 b \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x^6)^2,x]

[Out]

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

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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**6/(b*x**6+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.290074, size = 191, normalized size = 0.82 \[ \frac{-\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{5/6}}+\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{5/6}}+\frac{4 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{a^{5/6}}-\frac{2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{a^{5/6}}+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{a^{5/6}}-\frac{12 \sqrt [6]{b} x}{a+b x^6}}{72 b^{7/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x^6)^2,x]

[Out]

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

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Maple [A]  time = 0.053, size = 192, normalized size = 0.8 \[ -{\frac{x}{6\,b \left ( b{x}^{6}+a \right ) }}+{\frac{1}{18\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{\sqrt{3}}{72\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ( \sqrt{3}\sqrt [6]{{\frac{a}{b}}}x-{x}^{2}-\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{36\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{\sqrt{3}}{72\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{36\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^6+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236712, size = 567, normalized size = 2.44 \[ -\frac{4 \, \sqrt{3}{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \arctan \left (\frac{\sqrt{3} a b \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}}}{a b \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + 2 \, x + 2 \, \sqrt{a^{2} b^{2} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{3}} + a b x \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x^{2}}}\right ) + 4 \, \sqrt{3}{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \arctan \left (-\frac{\sqrt{3} a b \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}}}{a b \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} - 2 \, x - 2 \, \sqrt{a^{2} b^{2} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{3}} - a b x \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x^{2}}}\right ) -{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \log \left (a^{2} b^{2} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{3}} + a b x \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) +{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \log \left (a^{2} b^{2} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{3}} - a b x \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) - 2 \,{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \log \left (a b \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x\right ) + 2 \,{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \log \left (-a b \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x\right ) + 12 \, x}{72 \,{\left (b^{2} x^{6} + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.16358, size = 39, normalized size = 0.17 \[ - \frac{x}{6 a b + 6 b^{2} x^{6}} + \operatorname{RootSum}{\left (2176782336 t^{6} a^{5} b^{7} + 1, \left ( t \mapsto t \log{\left (36 t a b + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**6+a)**2,x)

[Out]

-x/(6*a*b + 6*b**2*x**6) + RootSum(2176782336*_t**6*a**5*b**7 + 1, Lambda(_t, _t
*log(36*_t*a*b + x)))

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GIAC/XCAS [A]  time = 0.224353, size = 277, normalized size = 1.19 \[ -\frac{x}{6 \,{\left (b x^{6} + a\right )} b} + \frac{\sqrt{3} \left (a b^{5}\right )^{\frac{1}{6}}{\rm ln}\left (x^{2} + \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{6}} + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{72 \, a b^{2}} - \frac{\sqrt{3} \left (a b^{5}\right )^{\frac{1}{6}}{\rm ln}\left (x^{2} - \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{6}} + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{72 \, a b^{2}} + \frac{\left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x + \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{36 \, a b^{2}} + \frac{\left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{36 \, a b^{2}} + \frac{\left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{x}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*x/((b*x^6 + a)*b) + 1/72*sqrt(3)*(a*b^5)^(1/6)*ln(x^2 + sqrt(3)*x*(a/b)^(1/
6) + (a/b)^(1/3))/(a*b^2) - 1/72*sqrt(3)*(a*b^5)^(1/6)*ln(x^2 - sqrt(3)*x*(a/b)^
(1/6) + (a/b)^(1/3))/(a*b^2) + 1/36*(a*b^5)^(1/6)*arctan((2*x + sqrt(3)*(a/b)^(1
/6))/(a/b)^(1/6))/(a*b^2) + 1/36*(a*b^5)^(1/6)*arctan((2*x - sqrt(3)*(a/b)^(1/6)
)/(a/b)^(1/6))/(a*b^2) + 1/18*(a*b^5)^(1/6)*arctan(x/(a/b)^(1/6))/(a*b^2)