Optimal. Leaf size=251 \[ \frac{b^5 x^{15} \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac{5 a b^4 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^4 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.171591, antiderivative size = 251, 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 \[ \frac{b^5 x^{15} \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 \left (a+b x^3\right )}+\frac{5 a b^4 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^4 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x,x]
[Out]
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Rubi in Sympy [A] time = 26.4812, size = 178, normalized size = 0.71 \[ \frac{a^{5} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (x \right )}}{a + b x^{3}} + \frac{a^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{3} + \frac{a^{3} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{6} + \frac{a^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{9} + \frac{a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{12} + \frac{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x,x)
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Mathematica [A] time = 0.0505826, size = 82, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (180 a^5 \log (x)+b x^3 \left (300 a^4+300 a^3 b x^3+200 a^2 b^2 x^6+75 a b^3 x^9+12 b^4 x^{12}\right )\right )}{180 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x,x]
[Out]
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Maple [A] time = 0.012, size = 79, normalized size = 0.3 \[{\frac{12\,{b}^{5}{x}^{15}+75\,a{b}^{4}{x}^{12}+200\,{a}^{2}{b}^{3}{x}^{9}+300\,{a}^{3}{b}^{2}{x}^{6}+300\,{a}^{4}b{x}^{3}+180\,{a}^{5}\ln \left ( x \right ) }{180\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x,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((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272122, size = 74, normalized size = 0.29 \[ \frac{1}{15} \, b^{5} x^{15} + \frac{5}{12} \, a b^{4} x^{12} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{5}{3} \, a^{3} b^{2} x^{6} + \frac{5}{3} \, a^{4} b x^{3} + a^{5} \log \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x,x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.296822, size = 140, normalized size = 0.56 \[ \frac{1}{15} \, b^{5} x^{15}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{12} \, a b^{4} x^{12}{\rm sign}\left (b x^{3} + a\right ) + \frac{10}{9} \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{3} \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{3} \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a^{5}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x^{3} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x,x, algorithm="giac")
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