Integrand size = 19, antiderivative size = 66 \[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 x \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} (d x)^m (b+c x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{2}-m,\frac {1}{2},1+\frac {c x}{b}\right )}{b \left (b x+c x^2\right )^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {688, 69, 67} \[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 x (b+c x) (d x)^m \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{2}-m,\frac {1}{2},\frac {c x}{b}+1\right )}{b \left (b x+c x^2\right )^{3/2}} \]
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Rule 67
Rule 69
Rule 688
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{\frac {3}{2}-m} (d x)^m (b+c x)^{3/2}\right ) \int \frac {x^{-\frac {3}{2}+m}}{(b+c x)^{3/2}} \, dx}{\left (b x+c x^2\right )^{3/2}} \\ & = -\frac {\left (c x \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} (d x)^m (b+c x)^{3/2}\right ) \int \frac {\left (-\frac {c x}{b}\right )^{-\frac {3}{2}+m}}{(b+c x)^{3/2}} \, dx}{b \left (b x+c x^2\right )^{3/2}} \\ & = \frac {2 x \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} (d x)^m (b+c x) \, _2F_1\left (-\frac {1}{2},\frac {3}{2}-m;\frac {1}{2};1+\frac {c x}{b}\right )}{b \left (b x+c x^2\right )^{3/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} (d x)^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{2}-m,\frac {1}{2},1+\frac {c x}{b}\right )}{b \sqrt {x (b+c x)}} \]
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\[\int \frac {\left (d x \right )^{m}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d x\right )^{m}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]
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