Optimal. Leaf size=61 \[ \frac {F^a (c+d x)^{m+1} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{\frac {m+1}{2}} \Gamma \left (\frac {1}{2} (-m-1),-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]
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Rubi [A] time = 0.05, 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} \[ \frac {F^a (c+d x)^{m+1} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{\frac {m+1}{2}} \text {Gamma}\left (\frac {1}{2} (-m-1),-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2218
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 = 0.04, size = 61, normalized size = 1.00 \[ \frac {F^a (c+d x)^{m+1} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{\frac {m+1}{2}} \Gamma \left (\frac {1}{2} (-m-1),-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x + c\right )}^{m} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}, x\right ) \]
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 \[ \int {\left (d x + c\right )}^{m} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, 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 \[ \int {\left (d x + c\right )}^{m} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}\,{d x} \]
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 \[ \frac {F^a\,{\mathrm {e}}^{\frac {b\,\ln \relax (F)}{2\,{\left (c+d\,x\right )}^2}}\,{\left (c+d\,x\right )}^{m+1}\,{\mathrm {M}}_{\frac {m}{4}+\frac {3}{4},-\frac {m}{4}-\frac {1}{4}}\left (\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )\,{\left (\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}^{\frac {m}{4}-\frac {1}{4}}}{d\,\left (m+1\right )} \]
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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