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 = 0.10, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 65
Rule 266
Rule 694
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*}
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Mathematica [A] time = 0.03, 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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fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x + a\right )}^{p}}{2 \, c d x + b d}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{p}}{2 \, c d x + b d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \,x^{2}+b x +a \right )^{p}}{2 c d x +b d}\, dx \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{p}}{2 \, c d x + b d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^p}{b\,d+2\,c\,d\,x} \,d x \]
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 \[ \frac {\int \frac {\left (a + b x + c x^{2}\right )^{p}}{b + 2 c x}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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