Optimal. Leaf size=287 \[ -\frac{2^{p-1} \left (a+b x+c x^2\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (-4 a A c^2+6 a b B c+2 A b^2 c (p+2)+b^3 (-B) (p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^3 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{\left (a+b x+c x^2\right )^{p+1} (2 a B c (2 p+3)-2 c (p+1) x (2 A c (p+2)-b B (p+3))+b (p+2) (2 A c (p+2)-b B (p+3)))}{4 c^3 (p+1) (p+2) (2 p+3)}+\frac{B x^2 \left (a+b x+c x^2\right )^{p+1}}{2 c (p+2)} \]
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Rubi [A] time = 0.286622, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {832, 779, 624} \[ -\frac{2^{p-1} \left (a+b x+c x^2\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (-4 a A c^2+6 a b B c+2 A b^2 c (p+2)+b^3 (-B) (p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^3 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{\left (a+b x+c x^2\right )^{p+1} (2 a B c (2 p+3)-2 c (p+1) x (2 A c (p+2)-b B (p+3))+b (p+2) (2 A c (p+2)-b B (p+3)))}{4 c^3 (p+1) (p+2) (2 p+3)}+\frac{B x^2 \left (a+b x+c x^2\right )^{p+1}}{2 c (p+2)} \]
Antiderivative was successfully verified.
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Rule 832
Rule 779
Rule 624
Rubi steps
\begin{align*} \int x^2 (A+B x) \left (a+b x+c x^2\right )^p \, dx &=\frac{B x^2 \left (a+b x+c x^2\right )^{1+p}}{2 c (2+p)}+\frac{\int x (-2 a B+(2 A c (2+p)-b B (3+p)) x) \left (a+b x+c x^2\right )^p \, dx}{2 c (2+p)}\\ &=\frac{B x^2 \left (a+b x+c x^2\right )^{1+p}}{2 c (2+p)}-\frac{(2 a B c (3+2 p)+b (2+p) (2 A c (2+p)-b B (3+p))-2 c (1+p) (2 A c (2+p)-b B (3+p)) x) \left (a+b x+c x^2\right )^{1+p}}{4 c^3 (1+p) (2+p) (3+2 p)}+\frac{\left (6 a b B c-4 a A c^2+2 A b^2 c (2+p)-b^3 B (3+p)\right ) \int \left (a+b x+c x^2\right )^p \, dx}{4 c^3 (3+2 p)}\\ &=\frac{B x^2 \left (a+b x+c x^2\right )^{1+p}}{2 c (2+p)}-\frac{(2 a B c (3+2 p)+b (2+p) (2 A c (2+p)-b B (3+p))-2 c (1+p) (2 A c (2+p)-b B (3+p)) x) \left (a+b x+c x^2\right )^{1+p}}{4 c^3 (1+p) (2+p) (3+2 p)}-\frac{2^{-1+p} \left (6 a b B c-4 a A c^2+2 A b^2 c (2+p)-b^3 B (3+p)\right ) \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^3 \sqrt{b^2-4 a c} (1+p) (3+2 p)}\\ \end{align*}
Mathematica [C] time = 0.326921, size = 210, normalized size = 0.73 \[ \frac{1}{12} x^3 \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )^{-p} (a+x (b+c x))^p \left (4 A F_1\left (3;-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+3 B x F_1\left (4;-p,-p;5;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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} 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 ({\left (B x^{3} + A x^{2}\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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