Optimal. Leaf size=29 \[ \frac {x^{m+1}}{a (m+1) \sqrt {a+b x^{2 (m+1)}}} \]
[Out]
________________________________________________________________________________________
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 = {264} \[ \frac {x^{m+1}}{a (m+1) \sqrt {a+b x^{2 (m+1)}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \[ \frac {x^{m+1}}{a (m+1) \sqrt {a+b x^{2 m+2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 47, normalized size = 1.62 \[ \frac {\sqrt {b x^{2} x^{2 \, m} + a} x x^{m}}{{\left (a b m + a b\right )} x^{2} x^{2 \, m} + a^{2} m + a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{{\left (b x^{2 \, m + 2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (b \,x^{2 m +2}+a \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{{\left (b x^{2 \, m + 2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.22, size = 27, normalized size = 0.93 \[ \frac {x^{m+1}}{a\,\sqrt {a+b\,x^{2\,m+2}}\,\left (m+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 21.63, size = 121, normalized size = 4.17 \[ \frac {\sqrt {\pi } x x^{m} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {1}{2} \\ \frac {m}{2 m + 2} + 1 + \frac {1}{2 m + 2} \end {matrix}\middle | {\frac {b x^{2} x^{2 m} e^{i \pi }}{a}} \right )}}{2 a a^{\frac {m}{2 m + 2}} a^{\frac {1}{2 m + 2}} m \Gamma \left (\frac {m}{2 m + 2} + 1 + \frac {1}{2 m + 2}\right ) + 2 a a^{\frac {m}{2 m + 2}} a^{\frac {1}{2 m + 2}} \Gamma \left (\frac {m}{2 m + 2} + 1 + \frac {1}{2 m + 2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________