Optimal. Leaf size=61 \[ -\frac {F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^2\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},-b (c+d x)^2 \log (F)\right )}{2 d} \]
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Rubi [A] time = 0.07, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac {F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^2\right )^{\frac {1}{2} (-m-1)} \text {Gamma}\left (\frac {m+1}{2},-b \log (F) (c+d x)^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^2} (c+d x)^m \, dx &=-\frac {F^a (c+d x)^{1+m} \Gamma \left (\frac {1+m}{2},-b (c+d x)^2 \log (F)\right ) \left (-b (c+d x)^2 \log (F)\right )^{\frac {1}{2} (-1-m)}}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 61, normalized size = 1.00 \[ -\frac {F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^2\right )^{\frac {1}{2} (-m-1)} \Gamma \left (\frac {m+1}{2},-b (c+d x)^2 \log (F)\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 59, normalized size = 0.97 \[ \frac {e^{\left (-\frac {1}{2} \, {\left (m - 1\right )} \log \left (-b \log \relax (F)\right ) + a \log \relax (F)\right )} \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F)\right )}{2 \, b d \log \relax (F)} \]
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 \[ \int {\left (d x + c\right )}^{m} F^{{\left (d x + c\right )}^{2} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int F^{a +\left (d x +c \right )^{2} b} \left (d x +c \right )^{m}\, dx \]
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 \[ \int {\left (d x + c\right )}^{m} F^{{\left (d x + c\right )}^{2} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.81, size = 75, normalized size = 1.23 \[ \frac {F^a\,{\mathrm {e}}^{\frac {b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2}{2}}\,{\left (c+d\,x\right )}^{m+1}\,{\mathrm {M}}_{\frac {1}{4}-\frac {m}{4},\frac {m}{4}+\frac {1}{4}}\left (b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2\right )}{d\,\left (m+1\right )\,{\left (b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2\right )}^{\frac {m}{4}+\frac {3}{4}}} \]
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 \[ \int F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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