Integrand size = 26, antiderivative size = 25 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\frac {\left (a+b x^3+c x^6\right )^{1+p}}{3 (1+p)} \]
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Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1482, 643} \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\frac {\left (a+b x^3+c x^6\right )^{p+1}}{3 (p+1)} \]
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Rule 643
Rule 1482
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int (b+2 c x) \left (a+b x+c x^2\right )^p \, dx,x,x^3\right ) \\ & = \frac {\left (a+b x^3+c x^6\right )^{1+p}}{3 (1+p)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\frac {\left (a+b x^3+c x^6\right )^{1+p}}{3 (1+p)} \]
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
gosper | \(\frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{1+p}}{3+3 p}\) | \(24\) |
risch | \(\frac {\left (c \,x^{6}+b \,x^{3}+a \right ) \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{3+3 p}\) | \(34\) |
parallelrisch | \(\frac {x^{6} \left (c \,x^{6}+b \,x^{3}+a \right )^{p} c^{2}+x^{3} \left (c \,x^{6}+b \,x^{3}+a \right )^{p} b c +\left (c \,x^{6}+b \,x^{3}+a \right )^{p} a c}{3 c \left (1+p \right )}\) | \(70\) |
norman | \(\frac {a \,{\mathrm e}^{p \ln \left (c \,x^{6}+b \,x^{3}+a \right )}}{3+3 p}+\frac {b \,x^{3} {\mathrm e}^{p \ln \left (c \,x^{6}+b \,x^{3}+a \right )}}{3+3 p}+\frac {c \,x^{6} {\mathrm e}^{p \ln \left (c \,x^{6}+b \,x^{3}+a \right )}}{3+3 p}\) | \(80\) |
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none
Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\frac {{\left (c x^{6} + b x^{3} + a\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{3 \, {\left (p + 1\right )}} \]
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Timed out. \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\text {Timed out} \]
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none
Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\frac {{\left (c x^{6} + b x^{3} + a\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p}}{3 \, {\left (p + 1\right )}} \]
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none
Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\frac {{\left (c x^{6} + b x^{3} + a\right )}^{p + 1}}{3 \, {\left (p + 1\right )}} \]
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Time = 8.74 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx={\left (c\,x^6+b\,x^3+a\right )}^p\,\left (\frac {a}{3\,p+3}+\frac {b\,x^3}{3\,p+3}+\frac {c\,x^6}{3\,p+3}\right ) \]
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