Integrand size = 22, antiderivative size = 32 \[ \int x^2 \left (c x^2\right )^p (a+b x)^{-4-2 p} \, dx=\frac {x^3 \left (c x^2\right )^p (a+b x)^{-3-2 p}}{a (3+2 p)} \]
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Time = 0.01 (sec) , antiderivative size = 32, 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 = {15, 37} \[ \int x^2 \left (c x^2\right )^p (a+b x)^{-4-2 p} \, dx=\frac {x^3 \left (c x^2\right )^p (a+b x)^{-2 p-3}}{a (2 p+3)} \]
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Rule 15
Rule 37
Rubi steps \begin{align*} \text {integral}& = \left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{2+2 p} (a+b x)^{-4-2 p} \, dx \\ & = \frac {x^3 \left (c x^2\right )^p (a+b x)^{-3-2 p}}{a (3+2 p)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int x^2 \left (c x^2\right )^p (a+b x)^{-4-2 p} \, dx=\frac {x^3 \left (c x^2\right )^p (a+b x)^{1-2 (2+p)}}{a (3+2 p)} \]
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Time = 0.39 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03
method | result | size |
gosper | \(\frac {x^{3} \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-3-2 p}}{a \left (3+2 p \right )}\) | \(33\) |
parallelrisch | \(\frac {x^{4} \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-4-2 p} b +x^{3} \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-4-2 p} a}{a \left (3+2 p \right )}\) | \(59\) |
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none
Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int x^2 \left (c x^2\right )^p (a+b x)^{-4-2 p} \, dx=\frac {{\left (b x^{4} + a x^{3}\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 4}}{2 \, a p + 3 \, a} \]
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\[ \int x^2 \left (c x^2\right )^p (a+b x)^{-4-2 p} \, dx=\int x^{2} \left (c x^{2}\right )^{p} \left (a + b x\right )^{- 2 p - 4}\, dx \]
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\[ \int x^2 \left (c x^2\right )^p (a+b x)^{-4-2 p} \, dx=\int { \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 4} x^{2} \,d x } \]
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\[ \int x^2 \left (c x^2\right )^p (a+b x)^{-4-2 p} \, dx=\int { \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 4} x^{2} \,d x } \]
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Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int x^2 \left (c x^2\right )^p (a+b x)^{-4-2 p} \, dx=\frac {x^3\,{\left (c\,x^2\right )}^p}{a\,\left (2\,p+3\right )\,{\left (a+b\,x\right )}^{2\,p+3}} \]
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