Integrand size = 18, antiderivative size = 122 \[ \int \frac {x^2 (a+b x)^n}{(c+d x)^2} \, dx=\frac {(a+b x)^{1+n}}{b d^2 (1+n)}+\frac {c^2 (a+b x)^{1+n}}{d^2 (b c-a d) (c+d x)}+\frac {c (2 a d-b c (2+n)) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 (b c-a d)^2 (1+n)} \]
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Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {91, 81, 70} \[ \int \frac {x^2 (a+b x)^n}{(c+d x)^2} \, dx=\frac {c^2 (a+b x)^{n+1}}{d^2 (c+d x) (b c-a d)}+\frac {c (a+b x)^{n+1} (2 a d-b c (n+2)) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 (n+1) (b c-a d)^2}+\frac {(a+b x)^{n+1}}{b d^2 (n+1)} \]
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Rule 70
Rule 81
Rule 91
Rubi steps \begin{align*} \text {integral}& = \frac {c^2 (a+b x)^{1+n}}{d^2 (b c-a d) (c+d x)}-\frac {\int \frac {(a+b x)^n (-c (a d-b c (1+n))-d (b c-a d) x)}{c+d x} \, dx}{d^2 (b c-a d)} \\ & = \frac {(a+b x)^{1+n}}{b d^2 (1+n)}+\frac {c^2 (a+b x)^{1+n}}{d^2 (b c-a d) (c+d x)}+\frac {(c (2 a d-b c (2+n))) \int \frac {(a+b x)^n}{c+d x} \, dx}{d^2 (b c-a d)} \\ & = \frac {(a+b x)^{1+n}}{b d^2 (1+n)}+\frac {c^2 (a+b x)^{1+n}}{d^2 (b c-a d) (c+d x)}+\frac {c (2 a d-b c (2+n)) (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)^2 (1+n)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.94 \[ \int \frac {x^2 (a+b x)^n}{(c+d x)^2} \, dx=\frac {(a+b x)^{1+n} \left ((b c-a d) (-a d (c+d x)+b c (c (2+n)+d x))-b c (-2 a d+b c (2+n)) (c+d x) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )\right )}{b d^2 (b c-a d)^2 (1+n) (c+d x)} \]
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\[\int \frac {x^{2} \left (b x +a \right )^{n}}{\left (d x +c \right )^{2}}d x\]
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\[ \int \frac {x^2 (a+b x)^n}{(c+d x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {x^2 (a+b x)^n}{(c+d x)^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {x^2 (a+b x)^n}{(c+d x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2 (a+b x)^n}{(c+d x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b x)^n}{(c+d x)^2} \, dx=\int \frac {x^2\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^2} \,d x \]
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