Integrand size = 11, antiderivative size = 53 \[ \int \left (b x+d x^3\right )^n \, dx=\frac {x \left (b+d x^2\right ) \left (b x+d x^3\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {3 (1+n)}{2},\frac {3+n}{2},-\frac {d x^2}{b}\right )}{b (1+n)} \]
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Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2036, 372, 371} \[ \int \left (b x+d x^3\right )^n \, dx=\frac {x \left (\frac {d x^2}{b}+1\right )^{-n} \left (b x+d x^3\right )^n \operatorname {Hypergeometric2F1}\left (-n,\frac {n+1}{2},\frac {n+3}{2},-\frac {d x^2}{b}\right )}{n+1} \]
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Rule 371
Rule 372
Rule 2036
Rubi steps \begin{align*} \text {integral}& = \left (x^{-n} \left (b+d x^2\right )^{-n} \left (b x+d x^3\right )^n\right ) \int x^n \left (b+d x^2\right )^n \, dx \\ & = \left (x^{-n} \left (1+\frac {d x^2}{b}\right )^{-n} \left (b x+d x^3\right )^n\right ) \int x^n \left (1+\frac {d x^2}{b}\right )^n \, dx \\ & = \frac {x \left (1+\frac {d x^2}{b}\right )^{-n} \left (b x+d x^3\right )^n \, _2F_1\left (-n,\frac {1+n}{2};\frac {3+n}{2};-\frac {d x^2}{b}\right )}{1+n} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.15 \[ \int \left (b x+d x^3\right )^n \, dx=\frac {x \left (x \left (b+d x^2\right )\right )^n \left (1+\frac {d x^2}{b}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {1+n}{2},1+\frac {1+n}{2},-\frac {d x^2}{b}\right )}{1+n} \]
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\[\int \left (x^{3} d +b x \right )^{n}d x\]
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\[ \int \left (b x+d x^3\right )^n \, dx=\int { {\left (d x^{3} + b x\right )}^{n} \,d x } \]
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\[ \int \left (b x+d x^3\right )^n \, dx=\int \left (b x + d x^{3}\right )^{n}\, dx \]
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\[ \int \left (b x+d x^3\right )^n \, dx=\int { {\left (d x^{3} + b x\right )}^{n} \,d x } \]
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\[ \int \left (b x+d x^3\right )^n \, dx=\int { {\left (d x^{3} + b x\right )}^{n} \,d x } \]
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Time = 10.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06 \[ \int \left (b x+d x^3\right )^n \, dx=\frac {x\,{\left (d\,x^3+b\,x\right )}^n\,{{}}_2{\mathrm {F}}_1\left (\frac {n}{2}+\frac {1}{2},-n;\ \frac {n}{2}+\frac {3}{2};\ -\frac {d\,x^2}{b}\right )}{{\left (\frac {d\,x^2}{b}+1\right )}^n\,\left (n+1\right )} \]
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