Optimal. Leaf size=248 \[ -\frac {2^p \left (a+b x+c x^2\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt {b^2-4 a c}}+\frac {e (p+2) (2 c d-b e) \left (a+b x+c x^2\right )^{p+1}}{2 c^2 (p+1) (2 p+3)}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{p+1}}{c (2 p+3)} \]
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Rubi [A] time = 0.21, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {742, 640, 624} \[ -\frac {2^p \left (a+b x+c x^2\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt {b^2-4 a c}}+\frac {e (p+2) (2 c d-b e) \left (a+b x+c x^2\right )^{p+1}}{2 c^2 (p+1) (2 p+3)}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{p+1}}{c (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 624
Rule 640
Rule 742
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^p \, dx &=\frac {e (d+e x) \left (a+b x+c x^2\right )^{1+p}}{c (3+2 p)}+\frac {\int \left (c d^2 (3+2 p)-e (a e+b d (1+p))+e (2 c d-b e) (2+p) x\right ) \left (a+b x+c x^2\right )^p \, dx}{c (3+2 p)}\\ &=\frac {e (2 c d-b e) (2+p) \left (a+b x+c x^2\right )^{1+p}}{2 c^2 (1+p) (3+2 p)}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{1+p}}{c (3+2 p)}+\frac {\left (b^2 e^2 (2+p)+2 c^2 d^2 (3+2 p)-2 c e (a e+b d (3+2 p))\right ) \int \left (a+b x+c x^2\right )^p \, dx}{2 c^2 (3+2 p)}\\ &=\frac {e (2 c d-b e) (2+p) \left (a+b x+c x^2\right )^{1+p}}{2 c^2 (1+p) (3+2 p)}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{1+p}}{c (3+2 p)}-\frac {2^p \left (b^2 e^2 (2+p)+2 c^2 d^2 (3+2 p)-2 c e (a e+b d (3+2 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^2 \sqrt {b^2-4 a c} (1+p) (3+2 p)}\\ \end {align*}
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Mathematica [C] time = 0.74, size = 414, normalized size = 1.67 \[ \frac {1}{6} (a+x (b+c x))^p \left (6 d e x^2 \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}+b}\right )^{-p} F_1\left (2;-p,-p;3;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )+2 e^2 x^3 \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}+b}\right )^{-p} F_1\left (3;-p,-p;4;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )+\frac {3 d^2 2^p \left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p} \, _2F_1\left (-p,p+1;p+2;\frac {-b-2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c (p+1)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} {\left (c x^{2} + b x + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{2} {\left (c x^{2} + b x + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.80, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{2} \left (c \,x^{2}+b x +a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{2} {\left (c x^{2} + b x + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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