3.351 \(\int \frac{1}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=151 \[ -\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac{5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} \sqrt [3]{b}}+\frac{5 x}{18 a^2 \left (a+b x^3\right )}+\frac{x}{6 a \left (a+b x^3\right )^2} \]

[Out]

x/(6*a*(a + b*x^3)^2) + (5*x)/(18*a^2*(a + b*x^3)) - (5*ArcTan[(a^(1/3) - 2*b^(1
/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(1/3)) + (5*Log[a^(1/3) + b^(1/3
)*x])/(27*a^(8/3)*b^(1/3)) - (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/
(54*a^(8/3)*b^(1/3))

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Rubi [A]  time = 0.158257, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778 \[ -\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac{5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} \sqrt [3]{b}}+\frac{5 x}{18 a^2 \left (a+b x^3\right )}+\frac{x}{6 a \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(-3),x]

[Out]

x/(6*a*(a + b*x^3)^2) + (5*x)/(18*a^2*(a + b*x^3)) - (5*ArcTan[(a^(1/3) - 2*b^(1
/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(1/3)) + (5*Log[a^(1/3) + b^(1/3
)*x])/(27*a^(8/3)*b^(1/3)) - (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/
(54*a^(8/3)*b^(1/3))

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Rubi in Sympy [A]  time = 34.1366, size = 143, normalized size = 0.95 \[ \frac{x}{6 a \left (a + b x^{3}\right )^{2}} + \frac{5 x}{18 a^{2} \left (a + b x^{3}\right )} + \frac{5 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{8}{3}} \sqrt [3]{b}} - \frac{5 \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{8}{3}} \sqrt [3]{b}} - \frac{5 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**3+a)**3,x)

[Out]

x/(6*a*(a + b*x**3)**2) + 5*x/(18*a**2*(a + b*x**3)) + 5*log(a**(1/3) + b**(1/3)
*x)/(27*a**(8/3)*b**(1/3)) - 5*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**
2)/(54*a**(8/3)*b**(1/3)) - 5*sqrt(3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)
/a**(1/3))/(27*a**(8/3)*b**(1/3))

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Mathematica [A]  time = 0.121147, size = 135, normalized size = 0.89 \[ \frac{-\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac{9 a^{5/3} x}{\left (a+b x^3\right )^2}+\frac{15 a^{2/3} x}{a+b x^3}+\frac{10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{10 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}}{54 a^{8/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^(-3),x]

[Out]

((9*a^(5/3)*x)/(a + b*x^3)^2 + (15*a^(2/3)*x)/(a + b*x^3) - (10*Sqrt[3]*ArcTan[(
1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (10*Log[a^(1/3) + b^(1/3)*x])/b^(
1/3) - (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3))/(54*a^(8/3))

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Maple [A]  time = 0.006, size = 130, normalized size = 0.9 \[{\frac{x}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,x}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{5}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5}{54\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,\sqrt{3}}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^3+a)^3,x)

[Out]

1/6*x/a/(b*x^3+a)^2+5/18*x/a^2/(b*x^3+a)+5/27/a^2/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3)
)-5/54/a^2/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+5/27/a^2/b/(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((b*x^3 + a)^(-3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239543, size = 259, normalized size = 1.72 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 10 \, \sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 30 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (5 \, b x^{4} + 8 \, a x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(5*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) - 10*sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)*log((a^2*b)^(1/3
)*x + a) - 30*(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)*(5*b*x^4 + 8*a*x)*(a^2*b)^(1/3))/((a^2*b^2*x^6 + 2*a
^3*b*x^3 + a^4)*(a^2*b)^(1/3))

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Sympy [A]  time = 2.44892, size = 63, normalized size = 0.42 \[ \frac{8 a x + 5 b x^{4}}{18 a^{4} + 36 a^{3} b x^{3} + 18 a^{2} b^{2} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{8} b - 125, \left ( t \mapsto t \log{\left (\frac{27 t a^{3}}{5} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**3+a)**3,x)

[Out]

(8*a*x + 5*b*x**4)/(18*a**4 + 36*a**3*b*x**3 + 18*a**2*b**2*x**6) + RootSum(1968
3*_t**3*a**8*b - 125, Lambda(_t, _t*log(27*_t*a**3/5 + x)))

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GIAC/XCAS [A]  time = 0.249143, size = 185, normalized size = 1.23 \[ -\frac{5 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3}} + \frac{5 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b} + \frac{5 \, \left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b} + \frac{5 \, b x^{4} + 8 \, a x}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/27*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^3 + 5/27*sqrt(3)*(-a*b^2)^(1/3)*a
rctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b) + 5/54*(-a*b^2)^(1/
3)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b) + 1/18*(5*b*x^4 + 8*a*x)/((b*
x^3 + a)^2*a^2)