Integrand size = 26, antiderivative size = 96 \[ \int \frac {(b d+2 c d x)^m}{\sqrt {a+b x+c x^2}} \, dx=-\frac {(b d+2 c d x)^{1+m} \sqrt {4 a-\frac {b^2}{c}+\frac {(b+2 c x)^2}{c}} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {3+m}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) d (1+m)} \]
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Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {708, 372, 371} \[ \int \frac {(b d+2 c d x)^m}{\sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {1-\frac {(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d (m+1) \sqrt {a+b x+c x^2}} \]
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Rule 371
Rule 372
Rule 708
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^m}{\sqrt {a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}}} \, dx,x,b d+2 c d x\right )}{2 c d} \\ & = \frac {\sqrt {4+\frac {(b d+2 c d x)^2}{\left (a-\frac {b^2}{4 c}\right ) c d^2}} \text {Subst}\left (\int \frac {x^m}{\sqrt {1+\frac {x^2}{4 \left (a-\frac {b^2}{4 c}\right ) c d^2}}} \, dx,x,b d+2 c d x\right )}{4 c d \sqrt {a+b x+c x^2}} \\ & = \frac {(d (b+2 c x))^{1+m} \sqrt {1-\frac {(b+2 c x)^2}{b^2-4 a c}} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d (1+m) \sqrt {a+b x+c x^2}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.06 \[ \int \frac {(b d+2 c d x)^m}{\sqrt {a+b x+c x^2}} \, dx=\frac {(b+2 c x) (d (b+2 c x))^m \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c (1+m) \sqrt {a+x (b+c x)}} \]
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\[\int \frac {\left (2 c d x +b d \right )^{m}}{\sqrt {c \,x^{2}+b x +a}}d x\]
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\[ \int \frac {(b d+2 c d x)^m}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{m}}{\sqrt {c x^{2} + b x + a}} \,d x } \]
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\[ \int \frac {(b d+2 c d x)^m}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (d \left (b + 2 c x\right )\right )^{m}}{\sqrt {a + b x + c x^{2}}}\, dx \]
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\[ \int \frac {(b d+2 c d x)^m}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{m}}{\sqrt {c x^{2} + b x + a}} \,d x } \]
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\[ \int \frac {(b d+2 c d x)^m}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{m}}{\sqrt {c x^{2} + b x + a}} \,d x } \]
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Timed out. \[ \int \frac {(b d+2 c d x)^m}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^m}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
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