Optimal. Leaf size=90 \[ \frac {2 \left (\frac {1}{4} \left (4 a-\frac {b^2}{c}\right )+\frac {(b+2 c x)^2}{4 c}\right )^{p+1} \, _2F_1\left (1,p-\frac {1}{2};-\frac {1}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]
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Rubi [A] time = 0.07, antiderivative size = 85, normalized size of antiderivative = 0.94, 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, 365, 364} \[ -\frac {\left (a+b x+c x^2\right )^p \left (1-\frac {(b+2 c x)^2}{b^2-4 a c}\right )^{-p} \, _2F_1\left (-\frac {3}{2},-p;-\frac {1}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{6 c d^4 (b+2 c x)^3} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 694
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )^p}{x^4} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=\frac {\left (2^{-1+2 p} \left (a+b x+c x^2\right )^p \left (4+\frac {(b d+2 c d x)^2}{\left (a-\frac {b^2}{4 c}\right ) c d^2}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x^2}{4 \left (a-\frac {b^2}{4 c}\right ) c d^2}\right )^p}{x^4} \, dx,x,b d+2 c d x\right )}{c d}\\ &=-\frac {2^{-1+2 p} \left (a+b x+c x^2\right )^p \left (4-\frac {4 (b+2 c x)^2}{b^2-4 a c}\right )^{-p} \, _2F_1\left (-\frac {3}{2},-p;-\frac {1}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 c d^4 (b+2 c x)^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 92, normalized size = 1.02 \[ -\frac {2^{-2 p-1} (a+x (b+c x))^p \left (\frac {c (a+x (b+c x))}{4 a c-b^2}\right )^{-p} \, _2F_1\left (-\frac {3}{2},-p;-\frac {1}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 c d^4 (b+2 c x)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x + a\right )}^{p}}{16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{p}}{{\left (2 \, c d x + b d\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \,x^{2}+b x +a \right )^{p}}{\left (2 c d x +b d \right )^{4}}\, dx \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{p}}{{\left (2 \, c d x + b d\right )}^{4}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^p}{{\left (b\,d+2\,c\,d\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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