Integrand size = 21, antiderivative size = 105 \[ \int \frac {(d+e x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {\left (-\frac {e x}{d}\right )^{3/2} (d+e x)^{1+m} \left (1-\frac {c (d+e x)}{c d-b e}\right )^{3/2} \operatorname {AppellF1}\left (1+m,\frac {3}{2},\frac {3}{2},2+m,\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e (1+m) \left (b x+c x^2\right )^{3/2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {773, 138} \[ \int \frac {(d+e x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {\left (-\frac {e x}{d}\right )^{3/2} (d+e x)^{m+1} \left (1-\frac {c (d+e x)}{c d-b e}\right )^{3/2} \operatorname {AppellF1}\left (m+1,\frac {3}{2},\frac {3}{2},m+2,\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e (m+1) \left (b x+c x^2\right )^{3/2}} \]
[In]
[Out]
Rule 138
Rule 773
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1-\frac {d+e x}{d}\right )^{3/2} \left (1-\frac {d+e x}{d-\frac {b e}{c}}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {x^m}{\left (1-\frac {x}{d}\right )^{3/2} \left (1-\frac {c x}{c d-b e}\right )^{3/2}} \, dx,x,d+e x\right )}{e \left (b x+c x^2\right )^{3/2}} \\ & = \frac {\left (-\frac {e x}{d}\right )^{3/2} (d+e x)^{1+m} \left (1-\frac {c (d+e x)}{c d-b e}\right )^{3/2} F_1\left (1+m;\frac {3}{2},\frac {3}{2};2+m;\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e (1+m) \left (b x+c x^2\right )^{3/2}} \\ \end{align*}
Time = 1.75 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {x (b+c x)} (d+e x)^m \left (1+\frac {e x}{d}\right )^{-m} \left (b \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-m,\frac {1}{2},-\frac {c x}{b},-\frac {e x}{d}\right )+c x \left (\operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m,\frac {3}{2},-\frac {c x}{b},-\frac {e x}{d}\right )+\operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{2},-m,\frac {3}{2},-\frac {c x}{b},-\frac {e x}{d}\right )\right )\right )}{b^3 x \sqrt {1+\frac {c x}{b}}} \]
[In]
[Out]
\[\int \frac {\left (e x +d \right )^{m}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}d x\]
[In]
[Out]
\[ \int \frac {(d+e x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(d+e x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{m}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {(d+e x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(d+e x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(d+e x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]
[In]
[Out]