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

Optimal. Leaf size=238 \[ -\frac{a+b x^3}{a x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

-((a + b*x^3)/(a*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])) + (b^(1/3)*(a + b*x^3)*ArcT
an[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(4/3)*Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6]) + (b^(1/3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(4/3)*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6]) - (b^(1/3)*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2])/(6*a^(4/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

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Rubi [A]  time = 0.246815, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{a+b x^3}{a x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

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

[Out]

-((a + b*x^3)/(a*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])) + (b^(1/3)*(a + b*x^3)*ArcT
an[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(4/3)*Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6]) + (b^(1/3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(4/3)*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6]) - (b^(1/3)*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2])/(6*a^(4/3)*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. \[ \int \frac{1}{x^{2} \sqrt{\left (a + b x^{3}\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*sqrt((a + b*x**3)**2)), x)

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Mathematica [A]  time = 0.0651402, size = 133, normalized size = 0.56 \[ -\frac{\left (a+b x^3\right ) \left (\sqrt [3]{b} x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt [3]{b} x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} \sqrt [3]{b} x \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+6 \sqrt [3]{a}\right )}{6 a^{4/3} x \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 111, normalized size = 0.5 \[ -{\frac{b{x}^{3}+a}{6\,ax} \left ( -2\,\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) \sqrt{3}x-2\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) x+\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) x+6\,\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271098, size = 171, normalized size = 0.72 \[ -\frac{\sqrt{3}{\left (\sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 6 \, x \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 6 \, \sqrt{3}\right )}}{18 \, a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.45133, size = 29, normalized size = 0.12 \[ \operatorname{RootSum}{\left (27 t^{3} a^{4} - b, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{3}}{b} + x \right )} \right )\right )} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*a**4 - b, Lambda(_t, _t*log(9*_t**2*a**3/b + x))) - 1/(a*x)

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GIAC/XCAS [A]  time = 0.279593, size = 177, normalized size = 0.74 \[ \frac{1}{6} \,{\left (\frac{2 \, b \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a^{2}} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a^{2} b} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{a^{2} b} - \frac{6}{a x}\right )}{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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