Optimal. Leaf size=68 \[ \frac {8 c (d (b+2 c x))^{1+m} \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^2 d (1+m)} \]
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Rubi [A]
time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {708, 371}
\begin {gather*} \frac {8 c (d (b+2 c x))^{m+1} \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 708
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^m}{\left (a+b x+c x^2\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^m}{\left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )^2} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=\frac {8 c (d (b+2 c x))^{1+m} \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^2 d (1+m)}\\ \end {align*}
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Mathematica [F]
time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b d+2 c d x)^m}{\left (a+b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \frac {\left (2 c d x +b d \right )^{m}}{\left (c \,x^{2}+b x +a \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (d \left (b + 2 c x\right )\right )^{m}}{\left (a + b x + c x^{2}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (b\,d+2\,c\,d\,x\right )}^m}{{\left (c\,x^2+b\,x+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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