Integrand size = 29, antiderivative size = 163 \[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^7 g (1+m) \sqrt {d^2-e^2 x^2}}-\frac {e (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^8 g^2 (2+m) \sqrt {d^2-e^2 x^2}} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 163, 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 = {906, 83, 127, 372, 371} \[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{d^7 g (m+1) \sqrt {d^2-e^2 x^2}}-\frac {e \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{d^8 g^2 (m+2) \sqrt {d^2-e^2 x^2}} \]
[In]
[Out]
Rule 83
Rule 127
Rule 371
Rule 372
Rule 906
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {d-e x} \sqrt {d+e x}\right ) \int \frac {(g x)^m}{(d-e x)^{7/2} (d+e x)^{9/2}} \, dx}{\sqrt {d^2-e^2 x^2}} \\ & = \frac {\left (d \sqrt {d-e x} \sqrt {d+e x}\right ) \int \frac {(g x)^m}{(d-e x)^{9/2} (d+e x)^{9/2}} \, dx}{\sqrt {d^2-e^2 x^2}}-\frac {\left (e \sqrt {d-e x} \sqrt {d+e x}\right ) \int \frac {(g x)^{1+m}}{(d-e x)^{9/2} (d+e x)^{9/2}} \, dx}{g \sqrt {d^2-e^2 x^2}} \\ & = d \int \frac {(g x)^m}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx-\frac {e \int \frac {(g x)^{1+m}}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx}{g} \\ & = \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {(g x)^m}{\left (1-\frac {e^2 x^2}{d^2}\right )^{9/2}} \, dx}{d^7 \sqrt {d^2-e^2 x^2}}-\frac {\left (e \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 )^{9/2}} \, dx}{d^8 g \sqrt {d^2-e^2 x^2}} \\ & = \frac {(g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {9}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^7 g (1+m) \sqrt {d^2-e^2 x^2}}-\frac {e (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {9}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^8 g^2 (2+m) \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 1.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.75 \[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x (g x)^m \sqrt {1-\frac {e^2 x^2}{d^2}} \left (-e (1+m) x \operatorname {Hypergeometric2F1}\left (\frac {9}{2},1+\frac {m}{2},2+\frac {m}{2},\frac {e^2 x^2}{d^2}\right )+d (2+m) \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )\right )}{d^8 (1+m) (2+m) \sqrt {d^2-e^2 x^2}} \]
[In]
[Out]
\[\int \frac {\left (g x \right )^{m}}{\left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}d x\]
[In]
[Out]
\[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (g x\right )^{m}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )}\, dx \]
[In]
[Out]
\[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (g\,x\right )}^m}{{\left (d^2-e^2\,x^2\right )}^{7/2}\,\left (d+e\,x\right )} \,d x \]
[In]
[Out]