Integrand size = 27, antiderivative size = 388 \[ \int (d+e x)^m (f+g x) \sqrt {a+b x+c x^2} \, dx=\frac {(e f-d g) (d+e x)^{1+m} \sqrt {a+b x+c x^2} \operatorname {AppellF1}\left (1+m,-\frac {1}{2},-\frac {1}{2},2+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (1+m) \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}}+\frac {g (d+e x)^{2+m} \sqrt {a+b x+c x^2} \operatorname {AppellF1}\left (2+m,-\frac {1}{2},-\frac {1}{2},3+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (2+m) \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}} \]
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Time = 0.18 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {857, 773, 138} \[ \int (d+e x)^m (f+g x) \sqrt {a+b x+c x^2} \, dx=\frac {\sqrt {a+b x+c x^2} (e f-d g) (d+e x)^{m+1} \operatorname {AppellF1}\left (m+1,-\frac {1}{2},-\frac {1}{2},m+2,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (m+1) \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {g \sqrt {a+b x+c x^2} (d+e x)^{m+2} \operatorname {AppellF1}\left (m+2,-\frac {1}{2},-\frac {1}{2},m+3,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (m+2) \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}} \]
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Rule 138
Rule 773
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {g \int (d+e x)^{1+m} \sqrt {a+b x+c x^2} \, dx}{e}+\frac {(e f-d g) \int (d+e x)^m \sqrt {a+b x+c x^2} \, dx}{e} \\ & = \frac {\left (g \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int x^{1+m} \sqrt {1-\frac {2 c x}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c x}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \, dx,x,d+e x\right )}{e^2 \sqrt {1-\frac {d+e x}{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}}} \sqrt {1-\frac {d+e x}{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}}}}+\frac {\left ((e f-d g) \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int x^m \sqrt {1-\frac {2 c x}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c x}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \, dx,x,d+e x\right )}{e^2 \sqrt {1-\frac {d+e x}{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}}} \sqrt {1-\frac {d+e x}{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}}}} \\ & = \frac {(e f-d g) (d+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 (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (1+m) \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}}+\frac {g (d+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 (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (2+m) \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}} \\ \end{align*}
\[ \int (d+e x)^m (f+g x) \sqrt {a+b x+c x^2} \, dx=\int (d+e x)^m (f+g x) \sqrt {a+b x+c x^2} \, dx \]
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\[\int \left (e x +d \right )^{m} \left (g x +f \right ) \sqrt {c \,x^{2}+b x +a}d x\]
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\[ \int (d+e x)^m (f+g x) \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (g x + f\right )} {\left (e x + d\right )}^{m} \,d x } \]
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\[ \int (d+e x)^m (f+g x) \sqrt {a+b x+c x^2} \, dx=\int \left (d + e x\right )^{m} \left (f + g x\right ) \sqrt {a + b x + c x^{2}}\, dx \]
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\[ \int (d+e x)^m (f+g x) \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (g x + f\right )} {\left (e x + d\right )}^{m} \,d x } \]
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\[ \int (d+e x)^m (f+g x) \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (g x + f\right )} {\left (e x + d\right )}^{m} \,d x } \]
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Timed out. \[ \int (d+e x)^m (f+g x) \sqrt {a+b x+c x^2} \, dx=\int \left (f+g\,x\right )\,{\left (d+e\,x\right )}^m\,\sqrt {c\,x^2+b\,x+a} \,d x \]
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