Integrand size = 23, antiderivative size = 86 \[ \int (d x)^m \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\frac {(d x)^{1+m} \sqrt {a+b \left (c x^2\right )^{3/2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{3},\frac {4+m}{3},-\frac {b \left (c x^2\right )^{3/2}}{a}\right )}{d (1+m) \sqrt {1+\frac {b \left (c x^2\right )^{3/2}}{a}}} \]
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Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {375, 372, 371} \[ \int (d x)^m \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\frac {(d x)^{m+1} \sqrt {a+b \left (c x^2\right )^{3/2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {m+1}{3},\frac {m+4}{3},-\frac {b \left (c x^2\right )^{3/2}}{a}\right )}{d (m+1) \sqrt {\frac {b \left (c x^2\right )^{3/2}}{a}+1}} \]
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Rule 371
Rule 372
Rule 375
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \text {Subst}\left (\int x^m \sqrt {a+b x^3} \, dx,x,\sqrt {c x^2}\right )}{d} \\ & = \frac {\left ((d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)} \sqrt {a+b \left (c x^2\right )^{3/2}}\right ) \text {Subst}\left (\int x^m \sqrt {1+\frac {b x^3}{a}} \, dx,x,\sqrt {c x^2}\right )}{d \sqrt {1+\frac {b \left (c x^2\right )^{3/2}}{a}}} \\ & = \frac {(d x)^{1+m} \sqrt {a+b \left (c x^2\right )^{3/2}} \, _2F_1\left (-\frac {1}{2},\frac {1+m}{3};\frac {4+m}{3};-\frac {b \left (c x^2\right )^{3/2}}{a}\right )}{d (1+m) \sqrt {1+\frac {b \left (c x^2\right )^{3/2}}{a}}} \\ \end{align*}
\[ \int (d x)^m \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\int (d x)^m \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx \]
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\[\int \left (d x \right )^{m} \sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}d x\]
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\[ \int (d x)^m \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} \left (d x\right )^{m} \,d x } \]
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\[ \int (d x)^m \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\int \left (d x\right )^{m} \sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int (d x)^m \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} \left (d x\right )^{m} \,d x } \]
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\[ \int (d x)^m \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} \left (d x\right )^{m} \,d x } \]
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Timed out. \[ \int (d x)^m \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}} \,d x \]
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