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 = 0.0068788, antiderivative size = 35, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0057215, size = 35, normalized size = 1. \[ \frac{b (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.90122, size = 354, normalized size = 10.11 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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