Integrand size = 30, antiderivative size = 115 \[ \int (d+e x)^{-3-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {b (a+b x) (d+e x)^{-1-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p}{2 (b d-a e)^2 (1+p) (1+2 p)}+\frac {(a+b x) (d+e x)^{-2 (1+p)} \left (a^2+2 a b x+b^2 x^2\right )^p}{2 (b d-a e) (1+p)} \]
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Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {660, 47, 37} \[ \int (d+e x)^{-3-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 p-1}}{2 (p+1) (2 p+1) (b d-a e)^2}+\frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)} \]
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Rule 37
Rule 47
Rule 660
Rubi steps \begin{align*} \text {integral}& = \left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (a b+b^2 x\right )^{2 p} (d+e x)^{-3-2 p} \, dx \\ & = \frac {(a+b x) (d+e x)^{-2 (1+p)} \left (a^2+2 a b x+b^2 x^2\right )^p}{2 (b d-a e) (1+p)}+\frac {\left (b \left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (a b+b^2 x\right )^{2 p} (d+e x)^{-2 (1+p)} \, dx}{2 (b d-a e) (1+p)} \\ & = \frac {b (a+b x) (d+e x)^{-1-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p}{2 (b d-a e)^2 (1+p) (1+2 p)}+\frac {(a+b x) (d+e x)^{-2 (1+p)} \left (a^2+2 a b x+b^2 x^2\right )^p}{2 (b d-a e) (1+p)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.63 \[ \int (d+e x)^{-3-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {(a+b x) \left ((a+b x)^2\right )^p (d+e x)^{-2 (1+p)} (2 b d (1+p)-a e (1+2 p)+b e x)}{2 (b d-a e)^2 (1+p) (1+2 p)} \]
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Time = 2.47 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.21
method | result | size |
gosper | \(-\frac {\left (b x +a \right ) \left (e x +d \right )^{-2-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} \left (2 a e p -2 b d p -b e x +a e -2 b d \right )}{2 \left (2 a^{2} e^{2} p^{2}-4 a b d e \,p^{2}+2 b^{2} d^{2} p^{2}+3 a^{2} e^{2} p -6 a b d e p +3 b^{2} d^{2} p +a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}\) | \(139\) |
parallelrisch | \(\frac {x^{3} \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} b^{3} e^{3}-2 x^{2} \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{2} e^{3} p +2 x^{2} \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} b^{3} d \,e^{2} p +3 x^{2} \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} b^{3} d \,e^{2}-2 x \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b \,e^{3} p +2 x \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} b^{3} d^{2} e p -x \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b \,e^{3}+2 x \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{2} d \,e^{2}+2 x \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} b^{3} d^{2} e -2 \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b d \,e^{2} p +2 \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{2} d^{2} e p -\left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b d \,e^{2}+2 \left (e x +d \right )^{-3-2 p} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{2} d^{2} e}{2 b e \left (2 a^{2} e^{2} p^{2}-4 a b d e \,p^{2}+2 b^{2} d^{2} p^{2}+3 a^{2} e^{2} p -6 a b d e p +3 b^{2} d^{2} p +a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}\) | \(609\) |
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Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.90 \[ \int (d+e x)^{-3-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {{\left (b^{2} e^{2} x^{3} + 2 \, a b d^{2} - a^{2} d e + {\left (3 \, b^{2} d e + 2 \, {\left (b^{2} d e - a b e^{2}\right )} p\right )} x^{2} + 2 \, {\left (a b d^{2} - a^{2} d e\right )} p + {\left (2 \, b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} + 2 \, {\left (b^{2} d^{2} - a^{2} e^{2}\right )} p\right )} x\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}}{2 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2} + 2 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} p^{2} + 3 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} p\right )}} \]
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\[ \int (d+e x)^{-3-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int \left (d + e x\right )^{- 2 p - 3} \left (\left (a + b x\right )^{2}\right )^{p}\, dx \]
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\[ \int (d+e x)^{-3-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int { {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3} \,d x } \]
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\[ \int (d+e x)^{-3-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int { {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3} \,d x } \]
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Time = 9.76 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.25 \[ \int (d+e x)^{-3-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx={\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {x\,\left (2\,b^2\,d^2-a^2\,e^2-2\,a^2\,e^2\,p+2\,b^2\,d^2\,p+2\,a\,b\,d\,e\right )}{2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (2\,p^2+3\,p+1\right )}+\frac {b^2\,e^2\,x^3}{2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (2\,p^2+3\,p+1\right )}-\frac {a\,d\,\left (a\,e-2\,b\,d+2\,a\,e\,p-2\,b\,d\,p\right )}{2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (2\,p^2+3\,p+1\right )}+\frac {b\,e\,x^2\,\left (3\,b\,d-2\,a\,e\,p+2\,b\,d\,p\right )}{2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (2\,p^2+3\,p+1\right )}\right ) \]
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