Integrand size = 29, antiderivative size = 216 \[ \int \frac {(g x)^m (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 (g x)^{1+m} (d+e x)}{5 d g \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(3-2 m) (g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^4 g (1+m) \sqrt {d^2-e^2 x^2}}+\frac {2 e (3-m) (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^5 g^2 (2+m) \sqrt {d^2-e^2 x^2}} \]
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Time = 0.13 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1820, 822, 372, 371} \[ \int \frac {(g x)^m (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 (d+e x) (g x)^{m+1}}{5 d g \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 e (3-m) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^5 g^2 (m+2) \sqrt {d^2-e^2 x^2}}+\frac {(3-2 m) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^4 g (m+1) \sqrt {d^2-e^2 x^2}} \]
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Rule 371
Rule 372
Rule 822
Rule 1820
Rubi steps \begin{align*} \text {integral}& = \frac {2 (g x)^{1+m} (d+e x)}{5 d g \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(g x)^m \left (-d^2 (3-2 m)-2 d e (3-m) x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2} \\ & = \frac {2 (g x)^{1+m} (d+e x)}{5 d g \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e (3-m)) \int \frac {(g x)^{1+m}}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d g}-\frac {1}{5} (-3+2 m) \int \frac {(g x)^m}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx \\ & = \frac {2 (g x)^{1+m} (d+e x)}{5 d g \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\left (2 e (3-m) \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {(g x)^{1+m}}{\left (1-\frac {e^2 x^2}{d^2}\right )^{5/2}} \, dx}{5 d^5 g \sqrt {d^2-e^2 x^2}}-\frac {\left ((-3+2 m) \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {(g x)^m}{\left (1-\frac {e^2 x^2}{d^2}\right )^{5/2}} \, dx}{5 d^4 \sqrt {d^2-e^2 x^2}} \\ & = \frac {2 (g x)^{1+m} (d+e x)}{5 d g \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(3-2 m) (g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {5}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^4 g (1+m) \sqrt {d^2-e^2 x^2}}+\frac {2 e (3-m) (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {5}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^5 g^2 (2+m) \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.88 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.81 \[ \int \frac {(g x)^m (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x (g x)^m \sqrt {1-\frac {e^2 x^2}{d^2}} \left (d^2 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )+e (1+m) x \left (2 d (3+m) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )+e (2+m) x \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {3+m}{2},\frac {5+m}{2},\frac {e^2 x^2}{d^2}\right )\right )\right )}{d^6 (1+m) (2+m) (3+m) \sqrt {d^2-e^2 x^2}} \]
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\[\int \frac {\left (g x \right )^{m} \left (e x +d \right )^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}d x\]
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\[ \int \frac {(g x)^m (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(g x)^m (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (g x\right )^{m} \left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {(g x)^m (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(g x)^m (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(g x)^m (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^2}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
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