Optimal. Leaf size=273 \[ \frac{A 2^{2 p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{c x}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{c x}\right )^{-p} \left (a+b x+c x^2\right )^p F_1\left (-2 p;-p,-p;1-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x},-\frac{b+\sqrt{b^2-4 a c}}{2 c x}\right )}{p}-\frac{B 2^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{(p+1) \sqrt{b^2-4 a c}} \]
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Rubi [A] time = 0.148102, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {843, 624, 758, 133} \[ \frac{A 2^{2 p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{c x}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{c x}\right )^{-p} \left (a+b x+c x^2\right )^p F_1\left (-2 p;-p,-p;1-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x},-\frac{b+\sqrt{b^2-4 a c}}{2 c x}\right )}{p}-\frac{B 2^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{(p+1) \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 843
Rule 624
Rule 758
Rule 133
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )^p}{x} \, dx &=A \int \frac{\left (a+b x+c x^2\right )^p}{x} \, dx+B \int \left (a+b x+c x^2\right )^p \, dx\\ &=-\frac{2^{1+p} B \left (-\frac{b-\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}\right )^{-1-p} \left (a+b x+c x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac{b+\sqrt{b^2-4 a c}+2 c x}{2 \sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} (1+p)}-\left (2^{2 p} A \left (\frac{1}{x}\right )^{2 p} \left (\frac{b-\sqrt{b^2-4 a c}+2 c x}{c x}\right )^{-p} \left (\frac{b+\sqrt{b^2-4 a c}+2 c x}{c x}\right )^{-p} \left (a+b x+c x^2\right )^p\right ) \operatorname{Subst}\left (\int x^{1-2 (1+p)} \left (1+\frac{\left (b-\sqrt{b^2-4 a c}\right ) x}{2 c}\right )^p \left (1+\frac{\left (b+\sqrt{b^2-4 a c}\right ) x}{2 c}\right )^p \, dx,x,\frac{1}{x}\right )\\ &=\frac{2^{-1+2 p} A \left (\frac{b-\sqrt{b^2-4 a c}+2 c x}{c x}\right )^{-p} \left (\frac{b+\sqrt{b^2-4 a c}+2 c x}{c x}\right )^{-p} \left (a+b x+c x^2\right )^p F_1\left (-2 p;-p,-p;1-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x},-\frac{b+\sqrt{b^2-4 a c}}{2 c x}\right )}{p}-\frac{2^{1+p} B \left (-\frac{b-\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}\right )^{-1-p} \left (a+b x+c x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac{b+\sqrt{b^2-4 a c}+2 c x}{2 \sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} (1+p)}\\ \end{align*}
Mathematica [A] time = 0.411322, size = 263, normalized size = 0.96 \[ \frac{1}{2} (a+x (b+c x))^p \left (\frac{A 4^p \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{c x}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{c x}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x},\frac{\sqrt{b^2-4 a c}-b}{2 c x}\right )}{p}+\frac{B 2^p \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p} \, _2F_1\left (-p,p+1;p+2;\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c (p+1)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( Bx+A \right ) \left ( c{x}^{2}+bx+a \right ) ^{p}}{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 (B x + A\right )}{\left (c x^{2} + b x + a\right )}^{p}}{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 (B x + A\right )}{\left (c x^{2} + b x + a\right )}^{p}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{p}}{x}\, dx \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 )}{\left (c x^{2} + b x + a\right )}^{p}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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