Integrand size = 25, antiderivative size = 158 \[ \int \frac {x^2 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\frac {(e+f x)^{1+n}}{b d f (1+n)}-\frac {a^2 (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )}{b (b c-a d) (b e-a f) (1+n)}+\frac {c^2 (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )}{d (b c-a d) (d e-c f) (1+n)} \]
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Time = 0.08 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {186, 70} \[ \int \frac {x^2 (e+f x)^n}{(a+b x) (c+d x)} \, dx=-\frac {a^2 (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b (e+f x)}{b e-a f}\right )}{b (n+1) (b c-a d) (b e-a f)}+\frac {c^2 (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d (e+f x)}{d e-c f}\right )}{d (n+1) (b c-a d) (d e-c f)}+\frac {(e+f x)^{n+1}}{b d f (n+1)} \]
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Rule 70
Rule 186
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(e+f x)^n}{b d}+\frac {a^2 (e+f x)^n}{b (b c-a d) (a+b x)}+\frac {c^2 (e+f x)^n}{d (-b c+a d) (c+d x)}\right ) \, dx \\ & = \frac {(e+f x)^{1+n}}{b d f (1+n)}+\frac {a^2 \int \frac {(e+f x)^n}{a+b x} \, dx}{b (b c-a d)}-\frac {c^2 \int \frac {(e+f x)^n}{c+d x} \, dx}{d (b c-a d)} \\ & = \frac {(e+f x)^{1+n}}{b d f (1+n)}-\frac {a^2 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{b (b c-a d) (b e-a f) (1+n)}+\frac {c^2 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{d (b c-a d) (d e-c f) (1+n)} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.97 \[ \int \frac {x^2 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\frac {(e+f x)^{1+n} \left (a^2 d f (-d e+c f) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )+(b e-a f) \left (-((b c-a d) (-d e+c f))+b c^2 f \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )\right )\right )}{b d (b c-a d) f (b e-a f) (d e-c f) (1+n)} \]
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\[\int \frac {x^{2} \left (f x +e \right )^{n}}{\left (b x +a \right ) \left (d x +c \right )}d x\]
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\[ \int \frac {x^2 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{2}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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\[ \int \frac {x^2 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int \frac {x^{2} \left (e + f x\right )^{n}}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]
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\[ \int \frac {x^2 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{2}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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\[ \int \frac {x^2 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{2}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^2 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int \frac {x^2\,{\left (e+f\,x\right )}^n}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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