Integrand size = 25, antiderivative size = 281 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {A (e x)^{1+m} \left (a+b x+c x^2\right )^{5/2} \operatorname {AppellF1}\left (1+m,-\frac {5}{2},-\frac {5}{2},2+m,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (1+m) \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}}+\frac {B (e x)^{2+m} \left (a+b x+c x^2\right )^{5/2} \operatorname {AppellF1}\left (2+m,-\frac {5}{2},-\frac {5}{2},3+m,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (2+m) \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}} \]
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Time = 0.25 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {857, 773, 138} \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {A (e x)^{m+1} \left (a+b x+c x^2\right )^{5/2} \operatorname {AppellF1}\left (m+1,-\frac {5}{2},-\frac {5}{2},m+2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (m+1) \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{5/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{5/2}}+\frac {B (e x)^{m+2} \left (a+b x+c x^2\right )^{5/2} \operatorname {AppellF1}\left (m+2,-\frac {5}{2},-\frac {5}{2},m+3,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (m+2) \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{5/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{5/2}} \]
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Rule 138
Rule 773
Rule 857
Rubi steps \begin{align*} \text {integral}& = A \int (e x)^m \left (a+b x+c x^2\right )^{5/2} \, dx+\frac {B \int (e x)^{1+m} \left (a+b x+c x^2\right )^{5/2} \, dx}{e} \\ & = \frac {\left (B \left (a+b x+c x^2\right )^{5/2}\right ) \text {Subst}\left (\int x^{1+m} \left (1+\frac {2 c x}{\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \left (1+\frac {2 c x}{\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \, dx,x,e x\right )}{e^2 \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}}+\frac {\left (A \left (a+b x+c x^2\right )^{5/2}\right ) \text {Subst}\left (\int x^m \left (1+\frac {2 c x}{\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \left (1+\frac {2 c x}{\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \, dx,x,e x\right )}{e \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}} \\ & = \frac {A (e x)^{1+m} \left (a+b x+c x^2\right )^{5/2} F_1\left (1+m;-\frac {5}{2},-\frac {5}{2};2+m;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (1+m) \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}}+\frac {B (e x)^{2+m} \left (a+b x+c x^2\right )^{5/2} F_1\left (2+m;-\frac {5}{2},-\frac {5}{2};3+m;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (2+m) \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(618\) vs. \(2(281)=562\).
Time = 2.89 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.20 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {x (e x)^m \sqrt {a+x (b+c x)} \left (a^2 A \left (720+1044 m+580 m^2+155 m^3+20 m^4+m^5\right ) \operatorname {AppellF1}\left (1+m,-\frac {1}{2},-\frac {1}{2},2+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+(1+m) x \left (a (2 A b+a B) \left (360+342 m+119 m^2+18 m^3+m^4\right ) \operatorname {AppellF1}\left (2+m,-\frac {1}{2},-\frac {1}{2},3+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+(2+m) x \left (\left (2 a b B+A \left (b^2+2 a c\right )\right ) \left (120+74 m+15 m^2+m^3\right ) \operatorname {AppellF1}\left (3+m,-\frac {1}{2},-\frac {1}{2},4+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+(3+m) x \left (\left (b^2 B+2 A b c+2 a B c\right ) \left (30+11 m+m^2\right ) \operatorname {AppellF1}\left (4+m,-\frac {1}{2},-\frac {1}{2},5+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+c (4+m) x \left ((2 b B+A c) (6+m) \operatorname {AppellF1}\left (5+m,-\frac {1}{2},-\frac {1}{2},6+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+B c (5+m) x \operatorname {AppellF1}\left (6+m,-\frac {1}{2},-\frac {1}{2},7+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )\right )\right )\right )}{(1+m) (2+m) (3+m) (4+m) (5+m) (6+m) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}}} \]
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\[\int \left (e x \right )^{m} \left (B x +A \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}d x\]
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\[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (B x + A\right )} \left (e x\right )^{m} \,d x } \]
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\[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int \left (e x\right )^{m} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \]
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\[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (B x + A\right )} \left (e x\right )^{m} \,d x } \]
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\[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (B x + A\right )} \left (e x\right )^{m} \,d x } \]
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Timed out. \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int {\left (e\,x\right )}^m\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]
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