Optimal. Leaf size=160 \[ \frac {e \left (a+b x+c x^2\right )^{p+1}}{2 c (p+1)}-\frac {2^p (2 c d-b e) \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c (p+1) \sqrt {b^2-4 a c}} \]
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Rubi [A] time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, 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} \[ \frac {e \left (a+b x+c x^2\right )^{p+1}}{2 c (p+1)}-\frac {2^p (2 c d-b e) \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c (p+1) \sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 624
Rule 640
Rubi steps
\begin {align*} \int (d+e x) \left (a+b x+c x^2\right )^p \, dx &=\frac {e \left (a+b x+c x^2\right )^{1+p}}{2 c (1+p)}+\frac {(2 c d-b e) \int \left (a+b x+c x^2\right )^p \, dx}{2 c}\\ &=\frac {e \left (a+b x+c x^2\right )^{1+p}}{2 c (1+p)}-\frac {2^p (2 c d-b e) \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*}
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Mathematica [C] time = 0.35, size = 268, normalized size = 1.68 \[ \frac {1}{2} (a+x (b+c x))^p \left (e x^2 \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}+b}\right )^{-p} F_1\left (2;-p,-p;3;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )+\frac {d 2^p \left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p} \, _2F_1\left (-p,p+1;p+2;\frac {-b-2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c (p+1)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.17, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x + d\right )} {\left (c x^{2} + b x + a\right )}^{p}, x\right ) \]
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 \[ \int {\left (e x + d\right )} {\left (c x^{2} + b x + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.32, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right ) \left (c \,x^{2}+b x +a \right )^{p}\, dx \]
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 \[ \int {\left (e x + d\right )} {\left (c x^{2} + b x + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^p \,d x \]
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 \[ \int \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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