Optimal. Leaf size=77 \[ -\frac{a d (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d (a+b x)^{n+2}}{b^2 (n+2)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]
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Rubi [A] time = 0.0508392, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {952, 80, 65} \[ -\frac{a d (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d (a+b x)^{n+2}}{b^2 (n+2)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]
Antiderivative was successfully verified.
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Rule 952
Rule 80
Rule 65
Rubi steps
\begin{align*} \int \frac{(a+b x)^n \left (c+d x^2\right )}{x} \, dx &=\frac{d (a+b x)^{2+n}}{b^2 (2+n)}+\frac{\int \frac{(a+b x)^n \left (b^2 c (2+n)-a b d (2+n) x\right )}{x} \, dx}{b^2 (2+n)}\\ &=-\frac{a d (a+b x)^{1+n}}{b^2 (1+n)}+\frac{d (a+b x)^{2+n}}{b^2 (2+n)}+c \int \frac{(a+b x)^n}{x} \, dx\\ &=-\frac{a d (a+b x)^{1+n}}{b^2 (1+n)}+\frac{d (a+b x)^{2+n}}{b^2 (2+n)}-\frac{c (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{b x}{a}\right )}{a (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0449481, size = 64, normalized size = 0.83 \[ -\frac{(a+b x)^{n+1} \left (b^2 c (n+2) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )+a d (a-b (n+1) x)\right )}{a b^2 (n+1) (n+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.365, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n} \left ( d{x}^{2}+c \right ) }{x}}\, 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^{2} + c\right )}{\left (b x + a\right )}^{n}}{x}\,{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^{2} + c\right )}{\left (b x + a\right )}^{n}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.48357, size = 347, normalized size = 4.51 \begin{align*} - \frac{b^{n} c n \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} - \frac{b^{n} c \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} + d \left (\begin{cases} \frac{a^{n} x^{2}}{2} & \text{for}\: b = 0 \\\frac{a \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac{b x \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} - \frac{b x}{a b^{2} + b^{3} x} & \text{for}\: n = -2 \\- \frac{a \log{\left (\frac{a}{b} + x \right )}}{b^{2}} + \frac{x}{b} & \text{for}\: n = -1 \\- \frac{a^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{a b n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text{otherwise} \end{cases}\right ) - \frac{b b^{n} c n x \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac{b b^{n} c x \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \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 (d x^{2} + c\right )}{\left (b x + a\right )}^{n}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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