3.1330 \(\int \frac{x^7}{\left (a+b x^6\right )^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.222056, 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 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{2/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{2/3} b^{4/3}}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{2/3} b^{4/3}}-\frac{x^2}{6 b \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^6)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 35.7098, size = 128, normalized size = 0.9 \[ - \frac{x^{2}}{6 b \left (a + b x^{6}\right )} + \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} x^{2} \right )}}{18 a^{\frac{2}{3}} b^{\frac{4}{3}}} - \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{2} + b^{\frac{2}{3}} x^{4} \right )}}{36 a^{\frac{2}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x^{2}}{3}\right )}{\sqrt [3]{a}} \right )}}{18 a^{\frac{2}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**6+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.383589, size = 197, normalized size = 1.39 \[ \frac{\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{2/3}}-\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{2/3}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{2/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{a^{2/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{a^{2/3}}-\frac{6 \sqrt [3]{b} x^2}{a+b x^6}}{36 b^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x^6)^2,x]

[Out]

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

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Maple [A]  time = 0.005, size = 114, normalized size = 0.8 \[ -{\frac{{x}^{2}}{6\,b \left ( b{x}^{6}+a \right ) }}+{\frac{1}{18\,{b}^{2}}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{36\,{b}^{2}}\ln \left ({x}^{4}-{x}^{2}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{18\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{x}^{2}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^6+a)^2,x)

[Out]

-1/6*x^2/b/(b*x^6+a)+1/18/b^2/(a/b)^(2/3)*ln(x^2+(a/b)^(1/3))-1/36/b^2/(a/b)^(2/
3)*ln(x^4-x^2*(a/b)^(1/3)+(a/b)^(2/3))+1/18/b^2/(a/b)^(2/3)*3^(1/2)*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^7/(b*x^6 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227628, size = 194, normalized size = 1.37 \[ -\frac{\sqrt{3}{\left (6 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x^{2} + \sqrt{3}{\left (b x^{6} + a\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{4} - \left (a^{2} b\right )^{\frac{1}{3}} a x^{2} + a^{2}\right ) - 2 \, \sqrt{3}{\left (b x^{6} + a\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x^{2} + a\right ) - 6 \,{\left (b x^{6} + a\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x^{2} - \sqrt{3} a}{3 \, a}\right )\right )}}{108 \,{\left (b^{2} x^{6} + a b\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^6 + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.36956, size = 42, normalized size = 0.3 \[ - \frac{x^{2}}{6 a b + 6 b^{2} x^{6}} + \operatorname{RootSum}{\left (5832 t^{3} a^{2} b^{4} - 1, \left ( t \mapsto t \log{\left (18 t a b + x^{2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**6+a)**2,x)

[Out]

-x**2/(6*a*b + 6*b**2*x**6) + RootSum(5832*_t**3*a**2*b**4 - 1, Lambda(_t, _t*lo
g(18*_t*a*b + x**2)))

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GIAC/XCAS [A]  time = 0.227327, size = 186, normalized size = 1.31 \[ -\frac{x^{2}}{6 \,{\left (b x^{6} + a\right )} b} - \frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{18 \, a b} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{18 \, a b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{36 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^6 + a)^2,x, algorithm="giac")

[Out]

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