Optimal. Leaf size=120 \[ \frac{32 b^3 x^{-n/2} \sqrt{a+b x^n}}{35 a^4 n}-\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n} \]
[Out]
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Rubi [A] time = 0.126124, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{32 b^3 x^{-n/2} \sqrt{a+b x^n}}{35 a^4 n}-\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - (7*n)/2)/Sqrt[a + b*x^n],x]
[Out]
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Rubi in Sympy [A] time = 13.4536, size = 105, normalized size = 0.88 \[ - \frac{2 x^{- \frac{7 n}{2}} \sqrt{a + b x^{n}}}{7 a n} + \frac{12 b x^{- \frac{5 n}{2}} \sqrt{a + b x^{n}}}{35 a^{2} n} - \frac{16 b^{2} x^{- \frac{3 n}{2}} \sqrt{a + b x^{n}}}{35 a^{3} n} + \frac{32 b^{3} x^{- \frac{n}{2}} \sqrt{a + b x^{n}}}{35 a^{4} n} \]
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)
[Out]
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Mathematica [A] time = 0.0671014, size = 64, normalized size = 0.53 \[ \frac{2 x^{-7 n/2} \sqrt{a+b x^n} \left (-5 a^3+6 a^2 b x^n-8 a b^2 x^{2 n}+16 b^3 x^{3 n}\right )}{35 a^4 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - (7*n)/2)/Sqrt[a + b*x^n],x]
[Out]
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Maple [F] time = 0.085, size = 0, normalized size = 0. \[ \int{1{x}^{-1-{\frac{7\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-7/2*n)/(a+b*x^n)^(1/2),x)
[Out]
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Maxima [A] time = 1.5498, size = 130, normalized size = 1.08 \[ \frac{2 \, \sqrt{b x^{n} + a} b^{3} x^{-\frac{1}{2} \, n}}{a^{4} n} - \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}} b^{2} x^{-\frac{3}{2} \, n}}{a^{4} n} + \frac{6 \,{\left (b x^{n} + a\right )}^{\frac{5}{2}} b x^{-\frac{5}{2} \, n}}{5 \, a^{4} n} - \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{7}{2}} x^{-\frac{7}{2} \, n}}{7 \, a^{4} n} \]
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")
[Out]
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Fricas [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^(-7/2*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")
[Out]
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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)
[Out]
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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")
[Out]