Integrand size = 18, antiderivative size = 124 \[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=-\frac {(a+b x)^{1+n}}{a c x}+\frac {d^2 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{c^2 (b c-a d) (1+n)}+\frac {(a d-b c n) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a^2 c^2 (1+n)} \]
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Time = 0.05 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {105, 162, 67, 70} \[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\frac {(a+b x)^{n+1} (a d-b c n) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a^2 c^2 (n+1)}+\frac {d^2 (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{c^2 (n+1) (b c-a d)}-\frac {(a+b x)^{n+1}}{a c x} \]
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Rule 67
Rule 70
Rule 105
Rule 162
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{1+n}}{a c x}-\frac {\int \frac {(a+b x)^n (a d-b c n-b d n x)}{x (c+d x)} \, dx}{a c} \\ & = -\frac {(a+b x)^{1+n}}{a c x}+\frac {d^2 \int \frac {(a+b x)^n}{c+d x} \, dx}{c^2}-\frac {(a d-b c n) \int \frac {(a+b x)^n}{x} \, dx}{a c^2} \\ & = -\frac {(a+b x)^{1+n}}{a c x}+\frac {d^2 (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{c^2 (b c-a d) (1+n)}+\frac {(a d-b c n) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 c^2 (1+n)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=-\frac {(a+b x)^{1+n} \left (a^2 d^2 x \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )+(-b c+a d) \left (a c (1+n)+(-a d x+b c n x) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )\right )}{a^2 c^2 (-b c+a d) (1+n) x} \]
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\[\int \frac {\left (b x +a \right )^{n}}{x^{2} \left (d x +c \right )}d x\]
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\[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x^{2}} \,d x } \]
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\[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx=\int \frac {{\left (a+b\,x\right )}^n}{x^2\,\left (c+d\,x\right )} \,d x \]
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