Optimal. Leaf size=35 \[ \frac {b (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {b x}{a}+1\right )}{a^2 (n+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {65} \[ \frac {b (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {b x}{a}+1\right )}{a^2 (n+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rubi steps
\begin {align*} \int \frac {(a+b x)^n}{x^2} \, dx &=\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*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.00 \[ \frac {b (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {b x}{a}+1\right )}{a^2 (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{n}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a+b\,x\right )}^n}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.09, size = 354, normalized size = 10.11 \[ \frac {a b^{2} b^{n} n^{2} \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} + \frac {a b^{2} b^{n} n \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {a b^{2} b^{n} n \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {a b^{2} b^{n} \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {b^{3} b^{n} n^{2} \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {b^{3} b^{n} n \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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