3.1318 \(\int \frac{x^6}{a+b x^6} \, dx\)

Optimal. Leaf size=220 \[ \frac{\sqrt [6]{a} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}+\frac{x}{b} \]

[Out]

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

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Rubi [A]  time = 0.795999, antiderivative size = 220, 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 \[ \frac{\sqrt [6]{a} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}+\frac{x}{b} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x^6),x]

[Out]

x/b - (a^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)])/(3*b^(7/6)) + (a^(1/6)*ArcTan[(Sqrt[
3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)])/(6*b^(7/6)) - (a^(1/6)*ArcTan[(Sqrt[3]*a^(1/
6) + 2*b^(1/6)*x)/a^(1/6)])/(6*b^(7/6)) + (a^(1/6)*Log[a^(1/3) - Sqrt[3]*a^(1/6)
*b^(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*b^(7/6)) - (a^(1/6)*Log[a^(1/3) + Sqrt[3]*
a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*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),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0672719, size = 182, normalized size = 0.83 \[ \frac{\sqrt{3} \sqrt [6]{a} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt{3} \sqrt [6]{a} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-4 \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt [6]{a} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt [6]{a} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )+12 \sqrt [6]{b} x}{12 b^{7/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x^6),x]

[Out]

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

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Maple [A]  time = 0.09, size = 167, normalized size = 0.8 \[{\frac{x}{b}}-{\frac{1}{3\,b}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{\sqrt{3}}{12\,b}\sqrt [6]{{\frac{a}{b}}}\ln \left ( \sqrt{3}\sqrt [6]{{\frac{a}{b}}}x-{x}^{2}-\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{1}{6\,b}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{\sqrt{3}}{12\,b}\sqrt [6]{{\frac{a}{b}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{1}{6\,b}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^6+a),x)

[Out]

x/b-1/3/b*(a/b)^(1/6)*arctan(x/(a/b)^(1/6))+1/12/b*3^(1/2)*(a/b)^(1/6)*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/b*3^(1/2)*(a/b)^(1/6)*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^6/(b*x^6 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2362, size = 389, normalized size = 1.77 \[ \frac{4 \, \sqrt{3} b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \arctan \left (\frac{\sqrt{3} b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}}}{b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + 2 \, x + 2 \, \sqrt{b^{2} \left (-\frac{a}{b^{7}}\right )^{\frac{1}{3}} + b x \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x^{2}}}\right ) + 4 \, \sqrt{3} b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \arctan \left (-\frac{\sqrt{3} b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}}}{b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} - 2 \, x - 2 \, \sqrt{b^{2} \left (-\frac{a}{b^{7}}\right )^{\frac{1}{3}} - b x \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x^{2}}}\right ) - b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \log \left (b^{2} \left (-\frac{a}{b^{7}}\right )^{\frac{1}{3}} + b x \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) + b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \log \left (b^{2} \left (-\frac{a}{b^{7}}\right )^{\frac{1}{3}} - b x \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) - 2 \, b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \log \left (b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x\right ) + 2 \, b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \log \left (-b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x\right ) + 12 \, x}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.38468, size = 22, normalized size = 0.1 \[ \operatorname{RootSum}{\left (46656 t^{6} b^{7} + a, \left ( t \mapsto t \log{\left (- 6 t b + x \right )} \right )\right )} + \frac{x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**6+a),x)

[Out]

RootSum(46656*_t**6*b**7 + a, Lambda(_t, _t*log(-6*_t*b + x))) + x/b

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GIAC/XCAS [A]  time = 0.229521, size = 243, normalized size = 1.1 \[ \frac{x}{b} - \frac{\sqrt{3} \left (a b^{5}\right )^{\frac{1}{6}}{\rm ln}\left (x^{2} + \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{6}} + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{12 \, b^{2}} + \frac{\sqrt{3} \left (a b^{5}\right )^{\frac{1}{6}}{\rm ln}\left (x^{2} - \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{6}} + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{12 \, b^{2}} - \frac{\left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x + \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{6 \, b^{2}} - \frac{\left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{6 \, b^{2}} - \frac{\left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{x}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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