Integrand size = 22, antiderivative size = 68 \[ \int \frac {(d x)^m (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=-\frac {d^2 x (d x)^{-2+m} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-2+m,-n,-1+m,-\frac {b x}{a}\right )}{c (2-m) \sqrt {c x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 16, 68, 66} \[ \int \frac {(d x)^m (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=-\frac {d^2 x (d x)^{m-2} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (m-2,-n,m-1,-\frac {b x}{a}\right )}{c (2-m) \sqrt {c x^2}} \]
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Rule 15
Rule 16
Rule 66
Rule 68
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {(d x)^m (a+b x)^n}{x^3} \, dx}{c \sqrt {c x^2}} \\ & = \frac {\left (d^3 x\right ) \int (d x)^{-3+m} (a+b x)^n \, dx}{c \sqrt {c x^2}} \\ & = \frac {\left (d^3 x (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int (d x)^{-3+m} \left (1+\frac {b x}{a}\right )^n \, dx}{c \sqrt {c x^2}} \\ & = -\frac {d^2 x (d x)^{-2+m} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (-2+m,-n;-1+m;-\frac {b x}{a}\right )}{c (2-m) \sqrt {c x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.84 \[ \int \frac {(d x)^m (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {x (d x)^m (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-2+m,-n,-1+m,-\frac {b x}{a}\right )}{(-2+m) \left (c x^2\right )^{3/2}} \]
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\[\int \frac {\left (d x \right )^{m} \left (b x +a \right )^{n}}{\left (c \,x^{2}\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(d x)^m (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(d x)^m (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\int \frac {\left (d x\right )^{m} \left (a + b x\right )^{n}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(d x)^m (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(d x)^m (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d x)^m (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d\,x\right )}^m\,{\left (a+b\,x\right )}^n}{{\left (c\,x^2\right )}^{3/2}} \,d x \]
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