Integrand size = 29, antiderivative size = 250 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=-\frac {3 d (g x)^{1+m} \sqrt {d^2-e^2 x^2}}{g (2+m)}+\frac {e (g x)^{2+m} \sqrt {d^2-e^2 x^2}}{g^2 (3+m)}+\frac {d^3 (5+4 m) (g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{g (1+m) (2+m) \sqrt {d^2-e^2 x^2}}-\frac {d^2 e (11+4 m) (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) (3+m) \sqrt {d^2-e^2 x^2}} \]
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Time = 0.25 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {866, 1823, 822, 372, 371} \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=-\frac {d^2 e (4 m+11) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+3) \sqrt {d^2-e^2 x^2}}+\frac {e \sqrt {d^2-e^2 x^2} (g x)^{m+2}}{g^2 (m+3)}-\frac {3 d \sqrt {d^2-e^2 x^2} (g x)^{m+1}}{g (m+2)}+\frac {d^3 (4 m+5) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{g (m+1) (m+2) \sqrt {d^2-e^2 x^2}} \]
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Rule 371
Rule 372
Rule 822
Rule 866
Rule 1823
Rubi steps \begin{align*} \text {integral}& = \int \frac {(g x)^m (d-e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {e (g x)^{2+m} \sqrt {d^2-e^2 x^2}}{g^2 (3+m)}-\frac {\int \frac {(g x)^m \left (-d^3 e^2 (3+m)+d^2 e^3 (11+4 m) x-3 d e^4 (3+m) x^2\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{e^2 (3+m)} \\ & = -\frac {3 d (g x)^{1+m} \sqrt {d^2-e^2 x^2}}{g (2+m)}+\frac {e (g x)^{2+m} \sqrt {d^2-e^2 x^2}}{g^2 (3+m)}+\frac {\int \frac {(g x)^m \left (d^3 e^4 (3+m) (5+4 m)-d^2 e^5 (2+m) (11+4 m) x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{e^4 (2+m) (3+m)} \\ & = -\frac {3 d (g x)^{1+m} \sqrt {d^2-e^2 x^2}}{g (2+m)}+\frac {e (g x)^{2+m} \sqrt {d^2-e^2 x^2}}{g^2 (3+m)}+\frac {\left (d^3 (5+4 m)\right ) \int \frac {(g x)^m}{\sqrt {d^2-e^2 x^2}} \, dx}{2+m}-\frac {\left (d^2 e (11+4 m)\right ) \int \frac {(g x)^{1+m}}{\sqrt {d^2-e^2 x^2}} \, dx}{g (3+m)} \\ & = -\frac {3 d (g x)^{1+m} \sqrt {d^2-e^2 x^2}}{g (2+m)}+\frac {e (g x)^{2+m} \sqrt {d^2-e^2 x^2}}{g^2 (3+m)}+\frac {\left (d^3 (5+4 m) \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {(g x)^m}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{(2+m) \sqrt {d^2-e^2 x^2}}-\frac {\left (d^2 e (11+4 m) \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {(g x)^{1+m}}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{g (3+m) \sqrt {d^2-e^2 x^2}} \\ & = -\frac {3 d (g x)^{1+m} \sqrt {d^2-e^2 x^2}}{g (2+m)}+\frac {e (g x)^{2+m} \sqrt {d^2-e^2 x^2}}{g^2 (3+m)}+\frac {d^3 (5+4 m) (g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) (2+m) \sqrt {d^2-e^2 x^2}}-\frac {d^2 e (11+4 m) (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) (3+m) \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.98 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {x (g x)^m \sqrt {d^2-e^2 x^2} \sqrt {1-\frac {e^2 x^2}{d^2}} \left (d^3 \left (24+26 m+9 m^2+m^3\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )-e (1+m) x \left (3 d^2 \left (12+7 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )+e (2+m) x \left (-3 d (4+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},\frac {e^2 x^2}{d^2}\right )+e (3+m) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},\frac {e^2 x^2}{d^2}\right )\right )\right )\right )}{(1+m) (2+m) (3+m) (4+m) (d-e x) (d+e x)} \]
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\[\int \frac {\left (g x \right )^{m} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{\left (e x +d \right )^{3}}d x\]
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\[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Exception generated. \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} \left (g x\right )^{m}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (g\,x\right )}^m}{{\left (d+e\,x\right )}^3} \,d x \]
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