Integrand size = 28, antiderivative size = 79 \[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {e^2 (a+b x) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3 (1+m) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 70} \[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {e^2 (a+b x) (d+e x)^{m+1} \operatorname {Hypergeometric2F1}\left (3,m+1,m+2,\frac {b (d+e x)}{b d-a e}\right )}{(m+1) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]
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Rule 70
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^m}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {e^2 (a+b x) (d+e x)^{1+m} \, _2F_1\left (3,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3 (1+m) \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e^2 (a+b x)^3 (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,-\frac {b (d+e x)}{-b d+a e}\right )}{(-b d+a e)^3 (1+m) \left ((a+b x)^2\right )^{3/2}} \]
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\[\int \frac {\left (e x +d \right )^{m}}{\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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