Integrand size = 24, antiderivative size = 68 \[ \int \frac {(d+e x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {(d-e x) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,-5+m,-\frac {3}{2}+m,\frac {d+e x}{2 d}\right )}{d e (5-2 m) \left (d^2-e^2 x^2\right )^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {694, 692, 71} \[ \int \frac {(d+e x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2^{m-\frac {5}{2}} (d+e x)^m \left (\frac {e x}{d}+1\right )^{\frac {5}{2}-m} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {7}{2}-m,-\frac {3}{2},\frac {d-e x}{2 d}\right )}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rule 71
Rule 692
Rule 694
Rubi steps \begin{align*} \text {integral}& = \left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{-m}\right ) \int \frac {\left (1+\frac {e x}{d}\right )^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx \\ & = \frac {\left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{\frac {5}{2}-m} \left (d^2-d e x\right )^{5/2}\right ) \int \frac {\left (1+\frac {e x}{d}\right )^{-\frac {7}{2}+m}}{\left (d^2-d e x\right )^{7/2}} \, dx}{\left (d^2-e^2 x^2\right )^{5/2}} \\ & = \frac {2^{-\frac {5}{2}+m} (d+e x)^m \left (1+\frac {e x}{d}\right )^{\frac {5}{2}-m} \, _2F_1\left (-\frac {5}{2},\frac {7}{2}-m;-\frac {3}{2};\frac {d-e x}{2 d}\right )}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \\ \end{align*}
Time = 0.98 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.34 \[ \int \frac {(d+e x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2^{-\frac {5}{2}+m} (d+e x)^m \left (1+\frac {e x}{d}\right )^{\frac {1}{2}-m} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {7}{2}-m,-\frac {3}{2},\frac {d-e x}{2 d}\right )}{5 d^3 e (d-e x)^2 \sqrt {d^2-e^2 x^2}} \]
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\[\int \frac {\left (e x +d \right )^{m}}{\left (-x^{2} e^{2}+d^{2}\right )^{\frac {7}{2}}}d x\]
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\[ \int \frac {(d+e x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(d+e x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{m}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {(d+e x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(d+e x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
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