Integrand size = 11, antiderivative size = 35 \[ \int \frac {(a+b x)^n}{x^2} \, dx=\frac {b (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {b x}{a}\right )}{a^2 (1+n)} \]
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Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {67} \[ \int \frac {(a+b x)^n}{x^2} \, dx=\frac {b (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {b x}{a}+1\right )}{a^2 (n+1)} \]
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Rule 67
Rubi steps \begin{align*} \text {integral}& = \frac {b (a+b x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 (1+n)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^n}{x^2} \, dx=\frac {b (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {b x}{a}\right )}{a^2 (1+n)} \]
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\[\int \frac {\left (b x +a \right )^{n}}{x^{2}}d x\]
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\[ \int \frac {(a+b x)^n}{x^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (27) = 54\).
Time = 2.01 (sec) , antiderivative size = 333, normalized size of antiderivative = 9.51 \[ \int \frac {(a+b x)^n}{x^2} \, dx=\frac {a b^{n + 2} n^{2} \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} + \frac {a b^{n + 2} n \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {a b^{n + 2} n \left (\frac {a}{b} + x\right )^{n + 1} \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {a b^{n + 2} \left (\frac {a}{b} + x\right )^{n + 1} \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {b b^{n + 2} n^{2} \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {b b^{n + 2} n \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} \]
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\[ \int \frac {(a+b x)^n}{x^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
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\[ \int \frac {(a+b x)^n}{x^2} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^n}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^n}{x^2} \,d x \]
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