Optimal. Leaf size=188 \[ -\frac{b}{6 a^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{3 b \log (x) \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{3 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.214629, antiderivative size = 188, 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 \[ -\frac{b}{6 a^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{3 b \log (x) \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{3 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 35.7636, size = 180, normalized size = 0.96 \[ \frac{2 a + 2 b x^{3}}{12 a x^{3} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}} + \frac{1}{2 a^{2} x^{3} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}} - \frac{b \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (x^{3} \right )}}{a^{4} \left (a + b x^{3}\right )} + \frac{b \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (a + b x^{3} \right )}}{a^{4} \left (a + b x^{3}\right )} - \frac{\sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)
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Mathematica [A] time = 0.0613676, size = 97, normalized size = 0.52 \[ \frac{-a \left (2 a^2+9 a b x^3+6 b^2 x^6\right )-18 b x^3 \log (x) \left (a+b x^3\right )^2+6 b x^3 \left (a+b x^3\right )^2 \log \left (a+b x^3\right )}{6 a^4 x^3 \left (a+b x^3\right ) \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]
[Out]
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Maple [A] time = 0.023, size = 133, normalized size = 0.7 \[{\frac{ \left ( 6\,\ln \left ( b{x}^{3}+a \right ){x}^{9}{b}^{3}-18\,{b}^{3}\ln \left ( x \right ){x}^{9}+12\,\ln \left ( b{x}^{3}+a \right ){x}^{6}a{b}^{2}-36\,a{b}^{2}\ln \left ( x \right ){x}^{6}-6\,a{x}^{6}{b}^{2}+6\,\ln \left ( b{x}^{3}+a \right ){x}^{3}{a}^{2}b-18\,{a}^{2}b\ln \left ( x \right ){x}^{3}-9\,{x}^{3}{a}^{2}b-2\,{a}^{3} \right ) \left ( b{x}^{3}+a \right ) }{6\,{x}^{3}{a}^{4}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)
[Out]
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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^4),x, algorithm="maxima")
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Fricas [A] time = 0.274706, size = 161, normalized size = 0.86 \[ -\frac{6 \, a b^{2} x^{6} + 9 \, a^{2} b x^{3} + 2 \, a^{3} - 6 \,{\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \left (b x^{3} + a\right ) + 18 \,{\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}} \]
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^4),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)
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GIAC/XCAS [A] time = 0.69738, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
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^4),x, algorithm="giac")
[Out]