Optimal. Leaf size=69 \[ -\frac {f^{a+b x+c x^2}}{4 c (b+2 c x)^2}+\frac {f^{a-\frac {b^2}{4 c}} \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right ) \log (f)}{16 c^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2271, 2270}
\begin {gather*} \frac {\log (f) f^{a-\frac {b^2}{4 c}} \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right )}{16 c^2}-\frac {f^{a+b x+c x^2}}{4 c (b+2 c x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2270
Rule 2271
Rubi steps
\begin {align*} \int \frac {f^{a+b x+c x^2}}{(b+2 c x)^3} \, dx &=-\frac {f^{a+b x+c x^2}}{4 c (b+2 c x)^2}+\frac {\log (f) \int \frac {f^{a+b x+c x^2}}{b+2 c x} \, dx}{4 c}\\ &=-\frac {f^{a+b x+c x^2}}{4 c (b+2 c x)^2}+\frac {f^{a-\frac {b^2}{4 c}} \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right ) \log (f)}{16 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 79, normalized size = 1.14 \begin {gather*} \frac {f^{a-\frac {b^2}{4 c}} \left (-4 c f^{\frac {(b+2 c x)^2}{4 c}}+(b+2 c x)^2 \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right ) \log (f)\right )}{16 c^2 (b+2 c x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 88, normalized size = 1.28
method | result | size |
risch | \(-\frac {f^{\frac {\left (2 c x +b \right )^{2}}{4 c}} f^{\frac {4 c a -b^{2}}{4 c}}}{4 c \left (2 c x +b \right )^{2}}-\frac {\ln \left (f \right ) f^{\frac {4 c a -b^{2}}{4 c}} \expIntegral \left (1, -\frac {\left (2 c x +b \right )^{2} \ln \left (f \right )}{4 c}\right )}{16 c^{2}}\) | \(88\) |
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 [A]
time = 0.40, size = 106, normalized size = 1.54 \begin {gather*} -\frac {4 \, c f^{c x^{2} + b x + a} - \frac {{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} {\rm Ei}\left (\frac {{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (f\right )}{4 \, c}\right ) \log \left (f\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{16 \, {\left (4 \, c^{4} x^{2} + 4 \, b c^{3} x + b^{2} c^{2}\right )}} \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 {f^{a + b x + c x^{2}}}{\left (b + 2 c x\right )^{3}}\, 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 {f^{c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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