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

Optimal. Leaf size=153 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{7/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{7/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{2/3} b^{7/3}}-\frac{2 x}{9 b^2 \left (a+b x^3\right )}-\frac{x^4}{6 b \left (a+b x^3\right )^2} \]

[Out]

-x^4/(6*b*(a + b*x^3)^2) - (2*x)/(9*b^2*(a + b*x^3)) - (2*ArcTan[(a^(1/3) - 2*b^
(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(2/3)*b^(7/3)) + (2*Log[a^(1/3) + b^(1
/3)*x])/(27*a^(2/3)*b^(7/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2
7*a^(2/3)*b^(7/3))

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Rubi [A]  time = 0.181624, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{7/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{7/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{2/3} b^{7/3}}-\frac{2 x}{9 b^2 \left (a+b x^3\right )}-\frac{x^4}{6 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x^3)^3,x]

[Out]

-x^4/(6*b*(a + b*x^3)^2) - (2*x)/(9*b^2*(a + b*x^3)) - (2*ArcTan[(a^(1/3) - 2*b^
(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(2/3)*b^(7/3)) + (2*Log[a^(1/3) + b^(1
/3)*x])/(27*a^(2/3)*b^(7/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2
7*a^(2/3)*b^(7/3))

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Rubi in Sympy [A]  time = 38.213, size = 143, normalized size = 0.93 \[ - \frac{x^{4}}{6 b \left (a + b x^{3}\right )^{2}} - \frac{2 x}{9 b^{2} \left (a + b x^{3}\right )} + \frac{2 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{27 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{2}{3}} b^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**4/(6*b*(a + b*x**3)**2) - 2*x/(9*b**2*(a + b*x**3)) + 2*log(a**(1/3) + b**(1
/3)*x)/(27*a**(2/3)*b**(7/3)) - log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x*
*2)/(27*a**(2/3)*b**(7/3)) - 2*sqrt(3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3
)/a**(1/3))/(27*a**(2/3)*b**(7/3))

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Mathematica [A]  time = 0.137863, size = 136, normalized size = 0.89 \[ \frac{-\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{21 \sqrt [3]{b} x}{a+b x^3}+\frac{9 a \sqrt [3]{b} x}{\left (a+b x^3\right )^2}}{54 b^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x^3)^3,x]

[Out]

((9*a*b^(1/3)*x)/(a + b*x^3)^2 - (21*b^(1/3)*x)/(a + b*x^3) - (4*Sqrt[3]*ArcTan[
(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (4*Log[a^(1/3) + b^(1/3)*x])/a^(
2/3) - (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(54*b^(7/3))

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Maple [A]  time = 0.013, size = 117, normalized size = 0.8 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ( -{\frac{7\,{x}^{4}}{18\,b}}-{\frac{2\,ax}{9\,{b}^{2}}} \right ) }+{\frac{2}{27\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{27\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}}{27\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^3+a)^3,x)

[Out]

(-7/18/b*x^4-2/9*a*x/b^2)/(b*x^3+a)^2+2/27/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/2
7/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/27/b^3/(a/b)^(2/3)*3^(1/2)
*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-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^6/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245386, size = 261, normalized size = 1.71 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 4 \, \sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 12 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (7 \, b x^{4} + 4 \, a x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{162 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(2*sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)*log((a^2*b)^(2/3)*x^2 - (a
^2*b)^(1/3)*a*x + a^2) - 4*sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)*log((a^2*b)^(1/3)
*x + a) - 12*(b^2*x^6 + 2*a*b*x^3 + a^2)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x -
sqrt(3)*a)/a) + 3*sqrt(3)*(7*b*x^4 + 4*a*x)*(a^2*b)^(1/3))/((b^4*x^6 + 2*a*b^3*
x^3 + a^2*b^2)*(a^2*b)^(1/3))

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Sympy [A]  time = 2.52239, size = 66, normalized size = 0.43 \[ - \frac{4 a x + 7 b x^{4}}{18 a^{2} b^{2} + 36 a b^{3} x^{3} + 18 b^{4} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{2} b^{7} - 8, \left ( t \mapsto t \log{\left (\frac{27 t a b^{2}}{2} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**3+a)**3,x)

[Out]

-(4*a*x + 7*b*x**4)/(18*a**2*b**2 + 36*a*b**3*x**3 + 18*b**4*x**6) + RootSum(196
83*_t**3*a**2*b**7 - 8, Lambda(_t, _t*log(27*_t*a*b**2/2 + x)))

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GIAC/XCAS [A]  time = 0.225175, size = 189, normalized size = 1.24 \[ -\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{2}} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a b^{3}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{27 \, a b^{3}} - \frac{7 \, b x^{4} + 4 \, a x}{18 \,{\left (b x^{3} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/27*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^2) + 2/27*sqrt(3)*(-a*b^2)^(1/
3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^3) + 1/27*(-a*b^2)
^(1/3)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^3) - 1/18*(7*b*x^4 + 4*a*x)/
((b*x^3 + a)^2*b^2)