Optimal. Leaf size=66 \[ \frac{2 a^2 \sqrt{a+b x^n}}{b^3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 b^3 n}-\frac{4 a \left (a+b x^n\right )^{3/2}}{3 b^3 n} \]
[Out]
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Rubi [A] time = 0.0895722, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 a^2 \sqrt{a+b x^n}}{b^3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 b^3 n}-\frac{4 a \left (a+b x^n\right )^{3/2}}{3 b^3 n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 3*n)/Sqrt[a + b*x^n],x]
[Out]
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Rubi in Sympy [A] time = 12.3223, size = 58, normalized size = 0.88 \[ \frac{2 a^{2} \sqrt{a + b x^{n}}}{b^{3} n} - \frac{4 a \left (a + b x^{n}\right )^{\frac{3}{2}}}{3 b^{3} n} + \frac{2 \left (a + b x^{n}\right )^{\frac{5}{2}}}{5 b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+3*n)/(a+b*x**n)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0423846, size = 44, normalized size = 0.67 \[ \frac{2 \sqrt{a+b x^n} \left (8 a^2-4 a b x^n+3 b^2 x^{2 n}\right )}{15 b^3 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 3*n)/Sqrt[a + b*x^n],x]
[Out]
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Maple [A] time = 0.032, size = 41, normalized size = 0.6 \[{\frac{6\,{b}^{2} \left ({x}^{n} \right ) ^{2}-8\,a{x}^{n}b+16\,{a}^{2}}{15\,{b}^{3}n}\sqrt{a+b{x}^{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+3*n)/(a+b*x^n)^(1/2),x)
[Out]
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Maxima [A] time = 1.46928, size = 72, normalized size = 1.09 \[ \frac{2 \,{\left (3 \, b^{3} x^{3 \, n} - a b^{2} x^{2 \, n} + 4 \, a^{2} b x^{n} + 8 \, a^{3}\right )}}{15 \, \sqrt{b x^{n} + a} b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231316, size = 54, normalized size = 0.82 \[ \frac{2 \,{\left (3 \, b^{2} x^{2 \, n} - 4 \, a b x^{n} + 8 \, a^{2}\right )} \sqrt{b x^{n} + a}}{15 \, b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1+3*n)/(a+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")
[Out]