Integrand size = 21, antiderivative size = 20 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^n \, dx=\frac {\left (a+b x+d x^3\right )^{1+n}}{1+n} \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1602} \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^n \, dx=\frac {\left (a+b x+d x^3\right )^{n+1}}{n+1} \]
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Rule 1602
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+b x+d x^3\right )^{1+n}}{1+n} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^n \, dx=\frac {\left (a+b x+d x^3\right )^{1+n}}{1+n} \]
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Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05
method | result | size |
gosper | \(\frac {\left (x^{3} d +b x +a \right )^{1+n}}{1+n}\) | \(21\) |
derivativedivides | \(\frac {\left (x^{3} d +b x +a \right )^{1+n}}{1+n}\) | \(21\) |
default | \(\frac {\left (x^{3} d +b x +a \right )^{1+n}}{1+n}\) | \(21\) |
risch | \(\frac {\left (x^{3} d +b x +a \right ) \left (x^{3} d +b x +a \right )^{n}}{1+n}\) | \(29\) |
parallelrisch | \(\frac {x^{3} \left (x^{3} d +b x +a \right )^{n} d^{2}+x \left (x^{3} d +b x +a \right )^{n} b d +\left (x^{3} d +b x +a \right )^{n} a d}{d \left (1+n \right )}\) | \(61\) |
norman | \(\frac {a \,{\mathrm e}^{n \ln \left (x^{3} d +b x +a \right )}}{1+n}+\frac {b x \,{\mathrm e}^{n \ln \left (x^{3} d +b x +a \right )}}{1+n}+\frac {d \,x^{3} {\mathrm e}^{n \ln \left (x^{3} d +b x +a \right )}}{1+n}\) | \(69\) |
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none
Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^n \, dx=\frac {{\left (d x^{3} + b x + a\right )} {\left (d x^{3} + b x + a\right )}^{n}}{n + 1} \]
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Timed out. \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^n \, dx=\text {Timed out} \]
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none
Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^n \, dx=\frac {{\left (d x^{3} + b x + a\right )}^{n + 1}}{n + 1} \]
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none
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^n \, dx=\frac {{\left (d x^{3} + b x + a\right )}^{n + 1}}{n + 1} \]
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Time = 9.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^n \, dx=\left (\frac {a}{n+1}+\frac {b\,x}{n+1}+\frac {d\,x^3}{n+1}\right )\,{\left (d\,x^3+b\,x+a\right )}^n \]
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