3.97 \(\int \frac{1}{x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx\)

Optimal. Leaf size=122 \[ -\frac{a+b x^3}{3 a x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b \log (x) \left (a+b x^3\right )}{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

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

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Rubi [A]  time = 0.124639, antiderivative size = 122, 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 \[ -\frac{a+b x^3}{3 a x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b \log (x) \left (a+b x^3\right )}{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/((b*x**3+a)**2)**(1/2),x)

[Out]

Integral(1/(x**4*sqrt((a + b*x**3)**2)), x)

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Mathematica [A]  time = 0.0280344, size = 54, normalized size = 0.44 \[ -\frac{\left (a+b x^3\right ) \left (-b x^3 \log \left (a+b x^3\right )+a+3 b x^3 \log (x)\right )}{3 a^2 x^3 \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]

[Out]

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

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Maple [A]  time = 0.018, size = 52, normalized size = 0.4 \[{\frac{ \left ( b{x}^{3}+a \right ) \left ( b\ln \left ( b{x}^{3}+a \right ){x}^{3}-3\,b\ln \left ( x \right ){x}^{3}-a \right ) }{3\,{a}^{2}{x}^{3}}{\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/((b*x^3+a)^2)^(1/2),x)

[Out]

1/3*(b*x^3+a)*(b*ln(b*x^3+a)*x^3-3*b*ln(x)*x^3-a)/((b*x^3+a)^2)^(1/2)/a^2/x^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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260216, size = 45, normalized size = 0.37 \[ \frac{b x^{3} \log \left (b x^{3} + a\right ) - 3 \, b x^{3} \log \left (x\right ) - a}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x^3 + a)^2)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.94945, size = 31, normalized size = 0.25 \[ - \frac{1}{3 a x^{3}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/((b*x**3+a)**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.279935, size = 68, normalized size = 0.56 \[ \frac{1}{3} \,{\left (\frac{b{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{a^{2}} - \frac{3 \, b{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{b x^{3} - a}{a^{2} x^{3}}\right )}{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x^3 + a)^2)*x^4),x, algorithm="giac")

[Out]

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