3.103 \(\int \frac{1}{x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=147 \[ \frac{1}{6 a \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{3 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\log (x) \left (a+b x^3\right )}{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

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

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Rubi [A]  time = 0.183795, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{1}{6 a \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{3 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\log (x) \left (a+b x^3\right )}{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.1173, size = 144, normalized size = 0.98 \[ \frac{2 a + 2 b x^{3}}{12 a \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}} + \frac{1}{3 a^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}} + \frac{\sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (x^{3} \right )}}{3 a^{3} \left (a + b x^{3}\right )} - \frac{\sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (a + b x^{3} \right )}}{3 a^{3} \left (a + b x^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a + 2*b*x**3)/(12*a*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)) + 1/(3*a**2*sqrt(
a**2 + 2*a*b*x**3 + b**2*x**6)) + sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)*log(x**3)/
(3*a**3*(a + b*x**3)) - sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)*log(a + b*x**3)/(3*a
**3*(a + b*x**3))

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Mathematica [A]  time = 0.0508981, size = 74, normalized size = 0.5 \[ \frac{a \left (3 a+2 b x^3\right )+6 \log (x) \left (a+b x^3\right )^2-2 \left (a+b x^3\right )^2 \log \left (a+b x^3\right )}{6 a^3 \left (a+b x^3\right ) \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 107, normalized size = 0.7 \[ -{\frac{ \left ( 2\,\ln \left ( b{x}^{3}+a \right ){x}^{6}{b}^{2}-6\,\ln \left ( x \right ){x}^{6}{b}^{2}+4\,\ln \left ( b{x}^{3}+a \right ){x}^{3}ab-12\,\ln \left ( x \right ){x}^{3}ab-2\,ab{x}^{3}+2\,\ln \left ( b{x}^{3}+a \right ){a}^{2}-6\,{a}^{2}\ln \left ( x \right ) -3\,{a}^{2} \right ) \left ( b{x}^{3}+a \right ) }{6\,{a}^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(2*ln(b*x^3+a)*x^6*b^2-6*ln(x)*x^6*b^2+4*ln(b*x^3+a)*x^3*a*b-12*ln(x)*x^3*a
*b-2*a*b*x^3+2*ln(b*x^3+a)*a^2-6*a^2*ln(x)-3*a^2)*(b*x^3+a)/a^3/((b*x^3+a)^2)^(3
/2)

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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/((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264618, size = 122, normalized size = 0.83 \[ \frac{2 \, a b x^{3} + 3 \, a^{2} - 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (x\right )}{6 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{3} + a^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*x),x, algorithm="fricas")

[Out]

1/6*(2*a*b*x^3 + 3*a^2 - 2*(b^2*x^6 + 2*a*b*x^3 + a^2)*log(b*x^3 + a) + 6*(b^2*x
^6 + 2*a*b*x^3 + a^2)*log(x))/(a^3*b^2*x^6 + 2*a^4*b*x^3 + a^5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.721415, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*x),x, algorithm="giac")

[Out]

sage0*x