Optimal. Leaf size=44 \[ \frac {(c x)^{m+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};-\frac {b}{x^2}\right )}{c (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {339, 364} \[ \frac {(c x)^{m+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};-\frac {b}{x^2}\right )}{c (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 339
Rule 364
Rubi steps
\begin {align*} \int \frac {(c x)^m}{\sqrt {1+\frac {b}{x^2}}} \, dx &=-\frac {\left (\left (\frac {1}{x}\right )^{1+m} (c x)^{1+m}\right ) \operatorname {Subst}\left (\int \frac {x^{-2-m}}{\sqrt {1+b x^2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {(c x)^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1-m);\frac {1-m}{2};-\frac {b}{x^2}\right )}{c (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 64, normalized size = 1.45 \[ \frac {x \sqrt {\frac {x^2}{b}+1} (c x)^m \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+2}{2}+1;-\frac {x^2}{b}\right )}{(m+2) \sqrt {\frac {b}{x^2}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c x\right )^{m} x^{2} \sqrt {\frac {x^{2} + b}{x^{2}}}}{x^{2} + b}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{m}}{\sqrt {\frac {b}{x^{2}} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{m}}{\sqrt {\frac {b}{x^{2}}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{m}}{\sqrt {\frac {b}{x^{2}} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,x\right )}^m}{\sqrt {\frac {b}{x^2}+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.41, size = 54, normalized size = 1.23 \[ - \frac {c^{m} x x^{m} \Gamma \left (- \frac {m}{2} - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - \frac {m}{2} - \frac {1}{2} \\ \frac {1}{2} - \frac {m}{2} \end {matrix}\middle | {\frac {b e^{i \pi }}{x^{2}}} \right )}}{2 \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________