Optimal. Leaf size=129 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} n}+\frac{5 a^2 x^{n/2} \sqrt{a+b x^n}}{8 b^3 n}-\frac{5 a x^{3 n/2} \sqrt{a+b x^n}}{12 b^2 n}+\frac{x^{5 n/2} \sqrt{a+b x^n}}{3 b n} \]
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Rubi [A] time = 0.160444, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} n}+\frac{5 a^2 x^{n/2} \sqrt{a+b x^n}}{8 b^3 n}-\frac{5 a x^{3 n/2} \sqrt{a+b x^n}}{12 b^2 n}+\frac{x^{5 n/2} \sqrt{a+b x^n}}{3 b n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + (7*n)/2)/Sqrt[a + b*x^n],x]
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Rubi in Sympy [A] time = 20.6, size = 160, normalized size = 1.24 \[ \frac{a^{3} x^{\frac{5 n}{2}}}{3 b n \left (a + b x^{n}\right )^{\frac{5}{2}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )^{3}} - \frac{5 a^{3} x^{\frac{3 n}{2}}}{12 b^{2} n \left (a + b x^{n}\right )^{\frac{3}{2}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )^{2}} + \frac{5 a^{3} x^{\frac{n}{2}}}{8 b^{3} n \sqrt{a + b x^{n}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )} - \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a + b x^{n}}} \right )}}{8 b^{\frac{7}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+7/2*n)/(a+b*x**n)**(1/2),x)
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Mathematica [A] time = 0.121634, size = 93, normalized size = 0.72 \[ \frac{\sqrt{b} x^{n/2} \sqrt{a+b x^n} \left (15 a^2-10 a b x^n+8 b^2 x^{2 n}\right )-15 a^3 \log \left (\sqrt{b} \sqrt{a+b x^n}+b x^{n/2}\right )}{24 b^{7/2} n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + (7*n)/2)/Sqrt[a + b*x^n],x]
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Maple [A] time = 0.088, size = 98, normalized size = 0.8 \[{\frac{1}{24\,{b}^{3}n}{{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \left ( 8\, \left ({{\rm e}^{1/2\,n\ln \left ( x \right ) }} \right ) ^{4}{b}^{2}-10\,a \left ({{\rm e}^{1/2\,n\ln \left ( x \right ) }} \right ) ^{2}b+15\,{a}^{2} \right ) \sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}}}-{\frac{5\,{a}^{3}}{8\,n}\ln \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}}\sqrt{b}+\sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}} \right ){b}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+7/2*n)/(a+b*x^n)^(1/2),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(x^(7/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")
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Fricas [A] time = 0.265074, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \log \left (2 \, \sqrt{b x^{n} + a} b x^{\frac{1}{2} \, n} - 2 \, b^{\frac{3}{2}} x^{n} - a \sqrt{b}\right ) + 2 \,{\left (8 \, b^{\frac{5}{2}} x^{\frac{5}{2} \, n} - 10 \, a b^{\frac{3}{2}} x^{\frac{3}{2} \, n} + 15 \, a^{2} \sqrt{b} x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{48 \, b^{\frac{7}{2}} n}, -\frac{15 \, a^{3} \arctan \left (\frac{\sqrt{-b} x^{\frac{1}{2} \, n}}{\sqrt{b x^{n} + a}}\right ) -{\left (8 \, \sqrt{-b} b^{2} x^{\frac{5}{2} \, n} - 10 \, a \sqrt{-b} b x^{\frac{3}{2} \, n} + 15 \, a^{2} \sqrt{-b} x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{24 \, \sqrt{-b} b^{3} n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(7/2*n - 1)/sqrt(b*x^n + a),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(x**(-1+7/2*n)/(a+b*x**n)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{7}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(7/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")
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