Optimal. Leaf size=44 \[ \frac {(c x)^{1+m} \, _2F_1\left (-\frac {3}{2},\frac {1}{2} (-1-m);\frac {1-m}{2};-\frac {b}{x^2}\right )}{c (1+m)} \]
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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 = {346, 371}
\begin {gather*} \frac {(c x)^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};-\frac {b}{x^2}\right )}{c (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 346
Rule 371
Rubi steps
\begin {align*} \int \left (1+\frac {b}{x^2}\right )^{3/2} (c x)^m \, dx &=-\frac {\left (\left (\frac {1}{x}\right )^{1+m} (c x)^{1+m}\right ) \text {Subst}\left (\int x^{-2-m} \left (1+b x^2\right )^{3/2} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {(c x)^{1+m} \, _2F_1\left (-\frac {3}{2},\frac {1}{2} (-1-m);\frac {1-m}{2};-\frac {b}{x^2}\right )}{c (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 63, normalized size = 1.43 \begin {gather*} \frac {b \sqrt {1+\frac {b}{x^2}} (c x)^m \, _2F_1\left (-\frac {3}{2},-1+\frac {m}{2};\frac {m}{2};-\frac {x^2}{b}\right )}{(-2+m) x \sqrt {\frac {b+x^2}{b}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (1+\frac {b}{x^{2}}\right )^{\frac {3}{2}} \left (c x \right )^{m}\, dx\]
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 [F]
time = 0.39, size = 27, normalized size = 0.61 \begin {gather*} {\rm integral}\left (\frac {{\left (x^{2} + b\right )} \left (c x\right )^{m} \sqrt {\frac {x^{2} + b}{x^{2}}}}{x^{2}}, x\right ) \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 = 2.49, size = 56, normalized size = 1.27 \begin {gather*} - \frac {c^{m} x x^{m} \Gamma \left (- \frac {m}{2} - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{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 )} \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (c\,x\right )}^m\,{\left (\frac {b}{x^2}+1\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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