Integrand size = 25, antiderivative size = 179 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\frac {d^6 \left (d^2-e^2 x^2\right )^{-1+p}}{e^6 (1-p)}+\frac {5 d^4 \left (d^2-e^2 x^2\right )^p}{2 e^6 p}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{1+p}}{e^6 (1+p)}+\frac {\left (d^2-e^2 x^2\right )^{2+p}}{2 e^6 (2+p)}-\frac {2 e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},2-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )}{7 d^3} \]
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Time = 0.12 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {866, 1666, 457, 78, 12, 372, 371} \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{2 e^6 (p+2)}+\frac {d^6 \left (d^2-e^2 x^2\right )^{p-1}}{e^6 (1-p)}+\frac {5 d^4 \left (d^2-e^2 x^2\right )^p}{2 e^6 p}-\frac {2 e x^7 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {7}{2},2-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )}{7 d^3} \]
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Rule 12
Rule 78
Rule 371
Rule 372
Rule 457
Rule 866
Rule 1666
Rubi steps \begin{align*} \text {integral}& = \int x^5 (d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p} \, dx \\ & = \int -2 d e x^6 \left (d^2-e^2 x^2\right )^{-2+p} \, dx+\int x^5 \left (d^2-e^2 x^2\right )^{-2+p} \left (d^2+e^2 x^2\right ) \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int x^2 \left (d^2-e^2 x\right )^{-2+p} \left (d^2+e^2 x\right ) \, dx,x,x^2\right )-(2 d e) \int x^6 \left (d^2-e^2 x^2\right )^{-2+p} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {2 d^6 \left (d^2-e^2 x\right )^{-2+p}}{e^4}-\frac {5 d^4 \left (d^2-e^2 x\right )^{-1+p}}{e^4}+\frac {4 d^2 \left (d^2-e^2 x\right )^p}{e^4}-\frac {\left (d^2-e^2 x\right )^{1+p}}{e^4}\right ) \, dx,x,x^2\right )-\frac {\left (2 e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int x^6 \left (1-\frac {e^2 x^2}{d^2}\right )^{-2+p} \, dx}{d^3} \\ & = \frac {d^6 \left (d^2-e^2 x^2\right )^{-1+p}}{e^6 (1-p)}+\frac {5 d^4 \left (d^2-e^2 x^2\right )^p}{2 e^6 p}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{1+p}}{e^6 (1+p)}+\frac {\left (d^2-e^2 x^2\right )^{2+p}}{2 e^6 (2+p)}-\frac {2 e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {7}{2},2-p;\frac {9}{2};\frac {e^2 x^2}{d^2}\right )}{7 d^3} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.34 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.37 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\frac {x^6 (d-e x)^p (d+e x)^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {AppellF1}\left (6,-p,2-p,7,\frac {e x}{d},-\frac {e x}{d}\right )}{6 d^2} \]
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\[\int \frac {x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{2}}d x\]
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\[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {x^{5} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{2}}\, dx \]
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\[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {x^5\,{\left (d^2-e^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \]
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