Optimal. Leaf size=63 \[ \frac{\left (a+b x+c x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (p+1) \left (b^2-4 a c\right )} \]
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Rubi [A] time = 0.102314, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {694, 266, 65} \[ \frac{\left (a+b x+c x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (p+1) \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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Rule 694
Rule 266
Rule 65
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^p}{b d+2 c d x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}\right )^p}{x} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-\frac{b^2}{4 c}+\frac{x}{4 c d^2}\right )^p}{x} \, dx,x,(b d+2 c d x)^2\right )}{4 c d}\\ &=\frac{(a+x (b+c x))^{1+p} \, _2F_1\left (1,1+p;2+p;1-\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) d (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0269272, size = 64, normalized size = 1.02 \[ \frac{(a+x (b+c x))^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right )}{d (p+1) \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.157, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{p}}{2\,cdx+bd}}\, 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 (c x^{2} + b x + a\right )}^{p}}{2 \, c d x + b d}\,{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 (c x^{2} + b x + a\right )}^{p}}{2 \, c d x + b d}, 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*} \frac{\int \frac{\left (a + b x + c x^{2}\right )^{p}}{b + 2 c x}\, dx}{d} \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 (c x^{2} + b x + a\right )}^{p}}{2 \, c d x + b d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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