Integrand size = 19, antiderivative size = 154 \[ \int \frac {(d+e x)^m}{\sqrt {a+c x^2}} \, dx=\frac {(d+e x)^{1+m} \sqrt {1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}} \sqrt {1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}} \operatorname {AppellF1}\left (1+m,\frac {1}{2},\frac {1}{2},2+m,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (1+m) \sqrt {a+c x^2}} \]
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Time = 0.06 (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} \[ \int \frac {(d+e x)^m}{\sqrt {a+c x^2}} \, dx=\frac {(d+e x)^{m+1} \sqrt {1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}} \sqrt {1-\frac {d+e x}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}} \operatorname {AppellF1}\left (m+1,\frac {1}{2},\frac {1}{2},m+2,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (m+1) \sqrt {a+c x^2}} \]
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Rule 138
Rule 774
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}} \sqrt {1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}}\right ) \text {Subst}\left (\int \frac {x^m}{\sqrt {1-\frac {x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}} \sqrt {1-\frac {x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}}} \, dx,x,d+e x\right )}{e \sqrt {a+c x^2}} \\ & = \frac {(d+e x)^{1+m} \sqrt {1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}} \sqrt {1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}} F_1\left (1+m;\frac {1}{2},\frac {1}{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) \sqrt {a+c x^2}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^m}{\sqrt {a+c x^2}} \, dx=\frac {\sqrt {\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{d+\sqrt {-\frac {a}{c}} e}} \sqrt {\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{-d+\sqrt {-\frac {a}{c}} e}} (d+e x)^{1+m} \operatorname {AppellF1}\left (1+m,\frac {1}{2},\frac {1}{2},2+m,\frac {d+e x}{d-\sqrt {-\frac {a}{c}} e},\frac {d+e x}{d+\sqrt {-\frac {a}{c}} e}\right )}{e (1+m) \sqrt {a+c x^2}} \]
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\[\int \frac {\left (e x +d \right )^{m}}{\sqrt {c \,x^{2}+a}}d x\]
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\[ \int \frac {(d+e x)^m}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {c x^{2} + a}} \,d x } \]
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\[ \int \frac {(d+e x)^m}{\sqrt {a+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{m}}{\sqrt {a + c x^{2}}}\, dx \]
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\[ \int \frac {(d+e x)^m}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {c x^{2} + a}} \,d x } \]
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\[ \int \frac {(d+e x)^m}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {c x^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m}{\sqrt {a+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{\sqrt {c\,x^2+a}} \,d x \]
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