3.93 \(\int \frac{1}{\sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx\)

Optimal. Leaf size=202 \[ \frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

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

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Rubi [A]  time = 0.254339, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (a + b x^{3}\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt((a + b*x**3)**2), x)

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Mathematica [A]  time = 0.039963, size = 109, normalized size = 0.54 \[ -\frac{\left (a+b x^3\right ) \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{6 a^{2/3} \sqrt [3]{b} \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

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

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Maple [A]  time = 0.006, size = 97, normalized size = 0.5 \[{\frac{b{x}^{3}+a}{6\,b} \left ( -2\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) +2\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) -\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \right ){\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((b*x^3+a)^2)^(1/2),x)

[Out]

1/6*(b*x^3+a)*(-2*3^(1/2)*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))+2*l
n(x+(a/b)^(1/3))-ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3)))/((b*x^3+a)^2)^(1/2)/b/(a/b)^
(2/3)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259872, size = 120, normalized size = 0.59 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 6 \, \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right )\right )}}{18 \, \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((b*x^3 + a)^2),x, algorithm="fricas")

[Out]

-1/18*sqrt(3)*(sqrt(3)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x + a^2) - 2*sqrt
(3)*log((a^2*b)^(1/3)*x + a) - 6*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)
*a)/a))/(a^2*b)^(1/3)

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Sympy [A]  time = 0.467147, size = 20, normalized size = 0.1 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log{\left (3 t a + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*a**2*b - 1, Lambda(_t, _t*log(3*_t*a + x)))

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GIAC/XCAS [A]  time = 0.293867, size = 165, normalized size = 0.82 \[ -\frac{1}{6} \,{\left (\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a} - \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a b} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{a b}\right )}{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((b*x^3 + a)^2),x, algorithm="giac")

[Out]

-1/6*(2*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a - 2*sqrt(3)*(-a*b^2)^(1/3)*arct
an(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b) - (-a*b^2)^(1/3)*ln(x^2
+ x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b))*sign(b*x^3 + a)