Integrand size = 22, antiderivative size = 189 \[ \int \frac {(d+e x)^m}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {(d+e x)^{1+m} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \operatorname {AppellF1}\left (1+m,\frac {5}{2},\frac {5}{2},2+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e (1+m) \left (a+b x+c x^2\right )^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {773, 138} \[ \int \frac {(d+e x)^m}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {(d+e x)^{m+1} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{5/2} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{5/2} \operatorname {AppellF1}\left (m+1,\frac {5}{2},\frac {5}{2},m+2,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e (m+1) \left (a+b x+c x^2\right )^{5/2}} \]
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Rule 138
Rule 773
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1-\frac {d+e x}{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}}\right )^{5/2} \left (1-\frac {d+e x}{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {x^m}{\left (1-\frac {2 c x}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \left (1-\frac {2 c x}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{5/2}} \, dx,x,d+e x\right )}{e \left (a+b x+c x^2\right )^{5/2}} \\ & = \frac {(d+e x)^{1+m} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} F_1\left (1+m;\frac {5}{2},\frac {5}{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 (1+m) \left (a+b x+c x^2\right )^{5/2}} \\ \end{align*}
Time = 2.51 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^m}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {e^3 \sqrt {\frac {e \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} (d+e x)^{1+m} \operatorname {AppellF1}\left (1+m,\frac {5}{2},\frac {5}{2},2+m,\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )}{\left (c d^2+e (-b d+a e)\right )^2 (1+m) \sqrt {a+x (b+c x)}} \]
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\[\int \frac {\left (e x +d \right )^{m}}{\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(d+e x)^m}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(d+e x)^m}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{m}}{\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {(d+e x)^m}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(d+e x)^m}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
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