Optimal. Leaf size=15 \[ \frac {x^m}{\sqrt {a+b x^n}} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {12, 449} \[ \frac {x^m}{\sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 449
Rubi steps
\begin {align*} \int \frac {x^{-1+m} \left (2 a m+b (2 m-n) x^n\right )}{2 \left (a+b x^n\right )^{3/2}} \, dx &=\frac {1}{2} \int \frac {x^{-1+m} \left (2 a m+b (2 m-n) x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx\\ &=\frac {x^m}{\sqrt {a+b x^n}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 111, normalized size = 7.40 \[ \frac {x^m \sqrt {\frac {b x^n}{a}+1} \left (b (2 m-n) x^n \, _2F_1\left (\frac {3}{2},\frac {m+n}{n};\frac {m}{n}+2;-\frac {b x^n}{a}\right )+2 a (m+n) \, _2F_1\left (\frac {3}{2},\frac {m}{n};\frac {m+n}{n};-\frac {b x^n}{a}\right )\right )}{2 a (m+n) \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 16, normalized size = 1.07 \[ \frac {x x^{m - 1}}{\sqrt {b x^{n} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b {\left (2 \, m - n\right )} x^{n} + 2 \, a m\right )} x^{m - 1}}{2 \, {\left (b x^{n} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (2 a m +\left (2 m -n \right ) b \,x^{n}\right ) x^{m -1}}{2 \left (b \,x^{n}+a \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 13, normalized size = 0.87 \[ \frac {x^{m}}{\sqrt {b x^{n} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.68, size = 13, normalized size = 0.87 \[ \frac {x^m}{\sqrt {a+b\,x^n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 98.05, size = 100, normalized size = 6.67 \[ \frac {m x^{m} \Gamma \left (\frac {m}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {m}{n} \\ \frac {m}{n} + 1 \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (\frac {m}{n} + 1\right )} + \frac {b x^{m} x^{n} \left (2 m - n\right ) \Gamma \left (\frac {m}{n} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {m}{n} + 1 \\ \frac {m}{n} + 2 \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} n \Gamma \left (\frac {m}{n} + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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