Integrand size = 19, antiderivative size = 154 \[ \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx=\frac {(d+e x)^{1+m} \left (a+c x^2\right )^{3/2} \operatorname {AppellF1}\left (1+m,-\frac {3}{2},-\frac {3}{2},2+m,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (1+m) \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2}} \]
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Time = 0.11 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {774, 138} \[ \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx=\frac {\left (a+c x^2\right )^{3/2} (d+e x)^{m+1} \operatorname {AppellF1}\left (m+1,-\frac {3}{2},-\frac {3}{2},m+2,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (m+1) \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}\right )^{3/2}} \]
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Rule 138
Rule 774
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+c x^2\right )^{3/2} \text {Subst}\left (\int x^m \left (1-\frac {x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \, dx,x,d+e x\right )}{e \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2}} \\ & = \frac {(d+e x)^{1+m} \left (a+c x^2\right )^{3/2} F_1\left (1+m;-\frac {3}{2},-\frac {3}{2};2+m;\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (1+m) \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2}} \\ \end{align*}
\[ \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx=\int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx \]
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\[\int \left (e x +d \right )^{m} \left (c \,x^{2}+a \right )^{\frac {3}{2}}d x\]
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\[ \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{m} \,d x } \]
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\[ \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx=\int \left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{m}\, dx \]
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\[ \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{m} \,d x } \]
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\[ \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{m} \,d x } \]
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Timed out. \[ \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^m \,d x \]
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