3.337 \(\int \frac{1}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=134 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x}{3 a \left (a+b x^3\right )} \]

[Out]

x/(3*a*(a + b*x^3)) - (2*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*S
qrt[3]*a^(5/3)*b^(1/3)) + (2*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(1/3)) - Log
[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(9*a^(5/3)*b^(1/3))

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Rubi [A]  time = 0.132016, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x}{3 a \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(-2),x]

[Out]

x/(3*a*(a + b*x^3)) - (2*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*S
qrt[3]*a^(5/3)*b^(1/3)) + (2*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(1/3)) - Log
[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(9*a^(5/3)*b^(1/3))

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Rubi in Sympy [A]  time = 29.9418, size = 124, normalized size = 0.93 \[ \frac{x}{3 a \left (a + b x^{3}\right )} + \frac{2 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**3+a)**2,x)

[Out]

x/(3*a*(a + b*x**3)) + 2*log(a**(1/3) + b**(1/3)*x)/(9*a**(5/3)*b**(1/3)) - log(
a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(9*a**(5/3)*b**(1/3)) - 2*sqrt(3
)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(5/3)*b**(1/3))

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Mathematica [A]  time = 0.128712, size = 118, normalized size = 0.88 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac{3 a^{2/3} x}{a+b x^3}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}}{9 a^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^(-2),x]

[Out]

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

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Maple [A]  time = 0.007, size = 115, normalized size = 0.9 \[{\frac{x}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{2}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{9\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^3+a)^2,x)

[Out]

1/3*x/a/(b*x^3+a)+2/9/a/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/9/a/b/(a/b)^(2/3)*ln(x
^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/9/a/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)^(-2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238362, size = 182, normalized size = 1.36 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (b x^{3} + a\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left (b x^{3} + a\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 6 \,{\left (b x^{3} + a\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x\right )}}{27 \,{\left (a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(-2),x, algorithm="fricas")

[Out]

-1/27*sqrt(3)*(sqrt(3)*(b*x^3 + a)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x + a
^2) - 2*sqrt(3)*(b*x^3 + a)*log((a^2*b)^(1/3)*x + a) - 6*(b*x^3 + a)*arctan(1/3*
(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) - 3*sqrt(3)*(a^2*b)^(1/3)*x)/((a*b*x^
3 + a^2)*(a^2*b)^(1/3))

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Sympy [A]  time = 1.62998, size = 39, normalized size = 0.29 \[ \frac{x}{3 a^{2} + 3 a b x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b - 8, \left ( t \mapsto t \log{\left (\frac{9 t a^{2}}{2} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**3+a)**2,x)

[Out]

x/(3*a**2 + 3*a*b*x**3) + RootSum(729*_t**3*a**5*b - 8, Lambda(_t, _t*log(9*_t*a
**2/2 + x)))

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GIAC/XCAS [A]  time = 0.217102, size = 171, normalized size = 1.28 \[ -\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2}} + \frac{x}{3 \,{\left (b x^{3} + a\right )} a} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(-2),x, algorithm="giac")

[Out]

-2/9*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^2 + 1/3*x/((b*x^3 + a)*a) + 2/9*sq
rt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*
b) + 1/9*(-a*b^2)^(1/3)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b)