Optimal. Leaf size=62 \[ -\frac{(a+b x)^{n+1} (a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)}-\frac{c (a+b x)^{n+1}}{a x} \]
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Rubi [A] time = 0.0183483, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 65} \[ -\frac{(a+b x)^{n+1} (a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)}-\frac{c (a+b x)^{n+1}}{a x} \]
Antiderivative was successfully verified.
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Rule 78
Rule 65
Rubi steps
\begin{align*} \int \frac{(a+b x)^n (c+d x)}{x^2} \, dx &=-\frac{c (a+b x)^{1+n}}{a x}+\frac{(a d+b c n) \int \frac{(a+b x)^n}{x} \, dx}{a}\\ &=-\frac{c (a+b x)^{1+n}}{a x}-\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 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0180129, size = 55, normalized size = 0.89 \[ -\frac{(a+b x)^{n+1} \left (x (a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )+a c (n+1)\right )}{a^2 (n+1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) }{{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 (d x + c\right )}{\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 (d x + c\right )}{\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 = 8.73961, size = 493, normalized size = 7.95 \begin{align*} \text{result too large to display} \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 (d x + c\right )}{\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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