Optimal. Leaf size=29 \[ \frac {x^{1+m}}{a (1+m) \sqrt {a+b x^{2 (1+m)}}} \]
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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {270}
\begin {gather*} \frac {x^{m+1}}{a (m+1) \sqrt {a+b x^{2 (m+1)}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {x^m}{\left (a+b x^{2+2 m}\right )^{3/2}} \, dx &=\frac {x^{1+m}}{a (1+m) \sqrt {a+b x^{2 (1+m)}}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 29, normalized size = 1.00 \begin {gather*} \frac {x^{1+m}}{a (1+m) \sqrt {a+b x^{2+2 m}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (a +b \,x^{2+2 m}\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 47, normalized size = 1.62 \begin {gather*} \frac {\sqrt {b x^{2} x^{2 \, m} + a} x x^{m}}{{\left (a b m + a b\right )} x^{2} x^{2 \, m} + a^{2} m + a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 13.50, size = 121, normalized size = 4.17 \begin {gather*} \frac {\sqrt {\pi } x x^{m} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {1}{2} \\ \frac {m}{2 m + 2} + 1 + \frac {1}{2 m + 2} \end {matrix}\middle | {\frac {b x^{2} x^{2 m} e^{i \pi }}{a}} \right )}}{2 a a^{\frac {m}{2 m + 2}} a^{\frac {1}{2 m + 2}} m \Gamma \left (\frac {m}{2 m + 2} + 1 + \frac {1}{2 m + 2}\right ) + 2 a a^{\frac {m}{2 m + 2}} a^{\frac {1}{2 m + 2}} \Gamma \left (\frac {m}{2 m + 2} + 1 + \frac {1}{2 m + 2}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 27, normalized size = 0.93 \begin {gather*} \frac {x^{m+1}}{a\,\sqrt {a+b\,x^{2\,m+2}}\,\left (m+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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