Optimal. Leaf size=158 \[ \frac{2^p (b B-2 A c) \left (a+b x+c x^2\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c (p+1) \sqrt{b^2-4 a c}}+\frac{B \left (a+b x+c x^2\right )^{p+1}}{2 c (p+1)} \]
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Rubi [A] time = 0.0492388, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {640, 624} \[ \frac{2^p (b B-2 A c) \left (a+b x+c x^2\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c (p+1) \sqrt{b^2-4 a c}}+\frac{B \left (a+b x+c x^2\right )^{p+1}}{2 c (p+1)} \]
Antiderivative was successfully verified.
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Rule 640
Rule 624
Rubi steps
\begin{align*} \int (A+B x) \left (a+b x+c x^2\right )^p \, dx &=\frac{B \left (a+b x+c x^2\right )^{1+p}}{2 c (1+p)}+\frac{(-b B+2 A c) \int \left (a+b x+c x^2\right )^p \, dx}{2 c}\\ &=\frac{B \left (a+b x+c x^2\right )^{1+p}}{2 c (1+p)}+\frac{2^p (b B-2 A c) \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 )}{c \sqrt{b^2-4 a c} (1+p)}\\ \end{align*}
Mathematica [C] time = 0.426119, size = 268, normalized size = 1.7 \[ \frac{1}{2} (a+x (b+c x))^p \left (\frac{A 2^p \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p} \, _2F_1\left (-p,p+1;p+2;\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c (p+1)}+B x^2 \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )^{-p} F_1\left (2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int \left ( Bx+A \right ) \left ( c{x}^{2}+bx+a \right ) ^{p}\, 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{\left (B x + A\right )}{\left (c x^{2} + b x + a\right )}^{p}\,{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 ({\left (B x + A\right )}{\left (c x^{2} + b x + a\right )}^{p}, 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 \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{p}\, 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{\left (B x + A\right )}{\left (c x^{2} + b x + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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