Integrand size = 16, antiderivative size = 132 \[ \int \left (b x+c x^2+d x^3\right )^n \, dx=\frac {x \left (1+\frac {2 d x}{c-\sqrt {c^2-4 b d}}\right )^{-n} \left (1+\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )^{-n} \left (b x+c x^2+d x^3\right )^n \operatorname {AppellF1}\left (1+n,-n,-n,2+n,-\frac {2 d x}{c-\sqrt {c^2-4 b d}},-\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )}{1+n} \]
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Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1922, 773, 138} \[ \int \left (b x+c x^2+d x^3\right )^n \, dx=\frac {x \left (\frac {2 d x}{c-\sqrt {c^2-4 b d}}+1\right )^{-n} \left (\frac {2 d x}{\sqrt {c^2-4 b d}+c}+1\right )^{-n} \left (b x+c x^2+d x^3\right )^n \operatorname {AppellF1}\left (n+1,-n,-n,n+2,-\frac {2 d x}{c-\sqrt {c^2-4 b d}},-\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )}{n+1} \]
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Rule 138
Rule 773
Rule 1922
Rubi steps \begin{align*} \text {integral}& = \left (x^{-n} \left (b+c x+d x^2\right )^{-n} \left (b x+c x^2+d x^3\right )^n\right ) \int x^n \left (b+c x+d x^2\right )^n \, dx \\ & = \left (x^{-n} \left (1+\frac {2 d x}{c-\sqrt {c^2-4 b d}}\right )^{-n} \left (1+\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )^{-n} \left (b x+c x^2+d x^3\right )^n\right ) \text {Subst}\left (\int x^n \left (1+\frac {2 d x}{c-\sqrt {c^2-4 b d}}\right )^n \left (1+\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )^n \, dx,x,x\right ) \\ & = \frac {x \left (1+\frac {2 d x}{c-\sqrt {c^2-4 b d}}\right )^{-n} \left (1+\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )^{-n} \left (b x+c x^2+d x^3\right )^n F_1\left (1+n;-n,-n;2+n;-\frac {2 d x}{c-\sqrt {c^2-4 b d}},-\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )}{1+n} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int \left (b x+c x^2+d x^3\right )^n \, dx=\frac {x \left (1+\frac {2 d x}{c-\sqrt {c^2-4 b d}}\right )^{-n} \left (1+\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )^{-n} (x (b+x (c+d x)))^n \operatorname {AppellF1}\left (1+n,-n,-n,2+n,-\frac {2 d x}{c+\sqrt {c^2-4 b d}},\frac {2 d x}{-c+\sqrt {c^2-4 b d}}\right )}{1+n} \]
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\[\int \left (x^{3} d +c \,x^{2}+b x \right )^{n}d x\]
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\[ \int \left (b x+c x^2+d x^3\right )^n \, dx=\int { {\left (d x^{3} + c x^{2} + b x\right )}^{n} \,d x } \]
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\[ \int \left (b x+c x^2+d x^3\right )^n \, dx=\int \left (b x + c x^{2} + d x^{3}\right )^{n}\, dx \]
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\[ \int \left (b x+c x^2+d x^3\right )^n \, dx=\int { {\left (d x^{3} + c x^{2} + b x\right )}^{n} \,d x } \]
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\[ \int \left (b x+c x^2+d x^3\right )^n \, dx=\int { {\left (d x^{3} + c x^{2} + b x\right )}^{n} \,d x } \]
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Timed out. \[ \int \left (b x+c x^2+d x^3\right )^n \, dx=\int {\left (d\,x^3+c\,x^2+b\,x\right )}^n \,d x \]
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