Integrand size = 25, antiderivative size = 281 \[ \int (e x)^m (A+B x) \sqrt {a+b x+c x^2} \, dx=\frac {A (e x)^{1+m} \sqrt {a+b x+c x^2} \operatorname {AppellF1}\left (1+m,-\frac {1}{2},-\frac {1}{2},2+m,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (1+m) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}}}+\frac {B (e x)^{2+m} \sqrt {a+b x+c x^2} \operatorname {AppellF1}\left (2+m,-\frac {1}{2},-\frac {1}{2},3+m,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (2+m) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}}} \]
[Out]
Time = 0.16 (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} \[ \int (e x)^m (A+B x) \sqrt {a+b x+c x^2} \, dx=\frac {A (e x)^{m+1} \sqrt {a+b x+c x^2} \operatorname {AppellF1}\left (m+1,-\frac {1}{2},-\frac {1}{2},m+2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (m+1) \sqrt {\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1}}+\frac {B (e x)^{m+2} \sqrt {a+b x+c x^2} \operatorname {AppellF1}\left (m+2,-\frac {1}{2},-\frac {1}{2},m+3,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (m+2) \sqrt {\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1}} \]
[In]
[Out]
Rule 138
Rule 773
Rule 857
Rubi steps \begin{align*} \text {integral}& = A \int (e x)^m \sqrt {a+b x+c x^2} \, dx+\frac {B \int (e x)^{1+m} \sqrt {a+b x+c x^2} \, dx}{e} \\ & = \frac {\left (B \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int x^{1+m} \sqrt {1+\frac {2 c x}{\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1+\frac {2 c x}{\left (b+\sqrt {b^2-4 a c}\right ) e}} \, dx,x,e x\right )}{e^2 \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}}}+\frac {\left (A \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int x^m \sqrt {1+\frac {2 c x}{\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1+\frac {2 c x}{\left (b+\sqrt {b^2-4 a c}\right ) e}} \, dx,x,e x\right )}{e \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}}} \\ & = \frac {A (e x)^{1+m} \sqrt {a+b x+c x^2} F_1\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 )}{e (1+m) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}}}+\frac {B (e x)^{2+m} \sqrt {a+b x+c x^2} F_1\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 )}{e^2 (2+m) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}}} \\ \end{align*}
Time = 1.34 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.83 \[ \int (e x)^m (A+B x) \sqrt {a+b x+c x^2} \, dx=\frac {x (e x)^m \sqrt {a+x (b+c x)} \left (A (2+m) \operatorname {AppellF1}\left (1+m,-\frac {1}{2},-\frac {1}{2},2+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+B (1+m) x \operatorname {AppellF1}\left (2+m,-\frac {1}{2},-\frac {1}{2},3+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )\right )}{(1+m) (2+m) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}}} \]
[In]
[Out]
\[\int \left (e x \right )^{m} \left (B x +A \right ) \sqrt {c \,x^{2}+b x +a}d x\]
[In]
[Out]
\[ \int (e x)^m (A+B x) \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (B x + A\right )} \left (e x\right )^{m} \,d x } \]
[In]
[Out]
\[ \int (e x)^m (A+B x) \sqrt {a+b x+c x^2} \, dx=\int \left (e x\right )^{m} \left (A + B x\right ) \sqrt {a + b x + c x^{2}}\, dx \]
[In]
[Out]
\[ \int (e x)^m (A+B x) \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (B x + A\right )} \left (e x\right )^{m} \,d x } \]
[In]
[Out]
\[ \int (e x)^m (A+B x) \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (B x + A\right )} \left (e x\right )^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int (e x)^m (A+B x) \sqrt {a+b x+c x^2} \, dx=\int {\left (e\,x\right )}^m\,\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a} \,d x \]
[In]
[Out]