Integrand size = 22, antiderivative size = 61 \[ \int \frac {(b x)^{5/2} (c+d x)^n}{(e+f x)^2} \, dx=\frac {2 (b x)^{7/2} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (\frac {7}{2},-n,2,\frac {9}{2},-\frac {d x}{c},-\frac {f x}{e}\right )}{7 b e^2} \]
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Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {129, 525, 524} \[ \int \frac {(b x)^{5/2} (c+d x)^n}{(e+f x)^2} \, dx=\frac {2 (b x)^{7/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {7}{2},-n,2,\frac {9}{2},-\frac {d x}{c},-\frac {f x}{e}\right )}{7 b e^2} \]
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Rule 129
Rule 524
Rule 525
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^6 \left (c+\frac {d x^2}{b}\right )^n}{\left (e+\frac {f x^2}{b}\right )^2} \, dx,x,\sqrt {b x}\right )}{b} \\ & = \frac {\left (2 (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \text {Subst}\left (\int \frac {x^6 \left (1+\frac {d x^2}{b c}\right )^n}{\left (e+\frac {f x^2}{b}\right )^2} \, dx,x,\sqrt {b x}\right )}{b} \\ & = \frac {2 (b x)^{7/2} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} F_1\left (\frac {7}{2};-n,2;\frac {9}{2};-\frac {d x}{c},-\frac {f x}{e}\right )}{7 b e^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(133\) vs. \(2(61)=122\).
Time = 0.72 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.18 \[ \int \frac {(b x)^{5/2} (c+d x)^n}{(e+f x)^2} \, dx=\frac {2 b^2 \sqrt {b x} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \left (9 e \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},-\frac {d x}{c},-\frac {f x}{e}\right )-3 e \operatorname {AppellF1}\left (\frac {1}{2},-n,2,\frac {3}{2},-\frac {d x}{c},-\frac {f x}{e}\right )-6 e \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},-\frac {d x}{c}\right )+f x \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},-\frac {d x}{c}\right )\right )}{3 f^3} \]
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\[\int \frac {\left (b x \right )^{\frac {5}{2}} \left (d x +c \right )^{n}}{\left (f x +e \right )^{2}}d x\]
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\[ \int \frac {(b x)^{5/2} (c+d x)^n}{(e+f x)^2} \, dx=\int { \frac {\left (b x\right )^{\frac {5}{2}} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(b x)^{5/2} (c+d x)^n}{(e+f x)^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(b x)^{5/2} (c+d x)^n}{(e+f x)^2} \, dx=\int { \frac {\left (b x\right )^{\frac {5}{2}} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{2}} \,d x } \]
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\[ \int \frac {(b x)^{5/2} (c+d x)^n}{(e+f x)^2} \, dx=\int { \frac {\left (b x\right )^{\frac {5}{2}} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(b x)^{5/2} (c+d x)^n}{(e+f x)^2} \, dx=\int \frac {{\left (b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^n}{{\left (e+f\,x\right )}^2} \,d x \]
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