Integrand size = 18, antiderivative size = 108 \[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=-\frac {(b c+a d) (a+b x)^{1+n}}{b^2 d^2 (1+n)}+\frac {(a+b x)^{2+n}}{b^2 d (2+n)}+\frac {c^2 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 (b c-a d) (1+n)} \]
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Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {90, 70} \[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=-\frac {(a d+b c) (a+b x)^{n+1}}{b^2 d^2 (n+1)}+\frac {(a+b x)^{n+2}}{b^2 d (n+2)}+\frac {c^2 (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 (n+1) (b c-a d)} \]
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Rule 70
Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c-a d) (a+b x)^n}{b d^2}+\frac {(a+b x)^{1+n}}{b d}+\frac {c^2 (a+b x)^n}{d^2 (c+d x)}\right ) \, dx \\ & = -\frac {(b c+a d) (a+b x)^{1+n}}{b^2 d^2 (1+n)}+\frac {(a+b x)^{2+n}}{b^2 d (2+n)}+\frac {c^2 \int \frac {(a+b x)^n}{c+d x} \, dx}{d^2} \\ & = -\frac {(b c+a d) (a+b x)^{1+n}}{b^2 d^2 (1+n)}+\frac {(a+b x)^{2+n}}{b^2 d (2+n)}+\frac {c^2 (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{d^2 (b c-a d) (1+n)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\frac {(a+b x)^{1+n} \left (-((b c-a d) (a d+b c (2+n)-b d (1+n) x))+b^2 c^2 (2+n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )\right )}{b^2 d^2 (b c-a d) (1+n) (2+n)} \]
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\[\int \frac {x^{2} \left (b x +a \right )^{n}}{d x +c}d x\]
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\[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{d x + c} \,d x } \]
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\[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\int \frac {x^{2} \left (a + b x\right )^{n}}{c + d x}\, dx \]
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\[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{d x + c} \,d x } \]
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\[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{d x + c} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\int \frac {x^2\,{\left (a+b\,x\right )}^n}{c+d\,x} \,d x \]
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