Optimal. Leaf size=85 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n}+\frac{2 a^2 \sqrt{a+b x^n}}{n}+\frac{2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 n} \]
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Rubi [A] time = 0.115577, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n}+\frac{2 a^2 \sqrt{a+b x^n}}{n}+\frac{2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 n} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(5/2)/x,x]
[Out]
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Rubi in Sympy [A] time = 12.8996, size = 73, normalized size = 0.86 \[ - \frac{2 a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{n} + \frac{2 a^{2} \sqrt{a + b x^{n}}}{n} + \frac{2 a \left (a + b x^{n}\right )^{\frac{3}{2}}}{3 n} + \frac{2 \left (a + b x^{n}\right )^{\frac{5}{2}}}{5 n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**(5/2)/x,x)
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Mathematica [A] time = 0.0926866, size = 69, normalized size = 0.81 \[ \frac{2 \sqrt{a+b x^n} \left (23 a^2+11 a b x^n+3 b^2 x^{2 n}\right )-30 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{15 n} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(5/2)/x,x]
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Maple [A] time = 0.006, size = 62, normalized size = 0.7 \[{\frac{1}{n} \left ({\frac{2}{5} \left ( a+b{x}^{n} \right ) ^{{\frac{5}{2}}}}+{\frac{2\,a}{3} \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}}+2\,\sqrt{a+b{x}^{n}}{a}^{2}-2\,{a}^{5/2}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^(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*x^n + a)^(5/2)/x,x, algorithm="maxima")
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Fricas [A] time = 0.230742, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{\frac{5}{2}} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \,{\left (3 \, b^{2} x^{2 \, n} + 11 \, a b x^{n} + 23 \, a^{2}\right )} \sqrt{b x^{n} + a}}{15 \, n}, -\frac{2 \,{\left (15 \, \sqrt{-a} a^{2} \arctan \left (\frac{\sqrt{b x^{n} + a}}{\sqrt{-a}}\right ) -{\left (3 \, b^{2} x^{2 \, n} + 11 \, a b x^{n} + 23 \, a^{2}\right )} \sqrt{b x^{n} + a}\right )}}{15 \, n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(5/2)/x,x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**n)**(5/2)/x,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(5/2)/x,x, algorithm="giac")
[Out]