Optimal. Leaf size=61 \[ \frac {F^a (c+d x)^{1+m} \Gamma \left (\frac {1}{2} (-1-m),-\frac {b \log (F)}{(c+d x)^2}\right ) \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{\frac {1+m}{2}}}{2 d} \]
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Rubi [A]
time = 0.03, 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 = {2250}
\begin {gather*} \frac {F^a (c+d x)^{m+1} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{\frac {m+1}{2}} \text {Gamma}\left (\frac {1}{2} (-m-1),-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^m \, dx &=\frac {F^a (c+d x)^{1+m} \Gamma \left (\frac {1}{2} (-1-m),-\frac {b \log (F)}{(c+d x)^2}\right ) \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{\frac {1+m}{2}}}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 61, normalized size = 1.00 \begin {gather*} \frac {F^a (c+d x)^{1+m} \Gamma \left (\frac {1}{2} (-1-m),-\frac {b \log (F)}{(c+d x)^2}\right ) \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{\frac {1+m}{2}}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int F^{a +\frac {b}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{m}\, dx\]
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 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [B]
time = 3.76, size = 73, normalized size = 1.20 \begin {gather*} \frac {F^a\,{\mathrm {e}}^{\frac {b\,\ln \left (F\right )}{2\,{\left (c+d\,x\right )}^2}}\,{\left (c+d\,x\right )}^{m+1}\,{\mathrm {M}}_{\frac {m}{4}+\frac {3}{4},-\frac {m}{4}-\frac {1}{4}}\left (\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^2}\right )\,{\left (\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^2}\right )}^{\frac {m}{4}-\frac {1}{4}}}{d\,\left (m+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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