Optimal. Leaf size=29 \[ -\frac {b^5 F^a \log ^5(F) \Gamma \left (-5,-\frac {b \log (F)}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 29, 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} \[ -\frac {b^5 F^a \log ^5(F) \text {Gamma}\left (-5,-\frac {b \log (F)}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int F^{a+\frac {b}{c+d x}} (c+d x)^4 \, dx &=-\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{c+d x}\right ) \log ^5(F)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \[ -\frac {b^5 F^a \log ^5(F) \Gamma \left (-5,-\frac {b \log (F)}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 244, normalized size = 8.41 \[ -\frac {F^{a} b^{5} {\rm Ei}\left (\frac {b \log \relax (F)}{d x + c}\right ) \log \relax (F)^{5} - {\left (24 \, d^{5} x^{5} + 120 \, c d^{4} x^{4} + 240 \, c^{2} d^{3} x^{3} + 240 \, c^{3} d^{2} x^{2} + 120 \, c^{4} d x + 24 \, c^{5} + {\left (b^{4} d x + b^{4} c\right )} \log \relax (F)^{4} + {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \relax (F)^{3} + 2 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \relax (F)^{2} + 6 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \relax (F)\right )} F^{\frac {a d x + a c + b}{d x + c}}}{120 \, d} \]
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 \[ \int {\left (d x + c\right )}^{4} F^{a + \frac {b}{d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 534, normalized size = 18.41 \[ \frac {b^{5} F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{d x +c}\right ) \ln \relax (F )^{5}}{120 d}+\frac {b^{4} x \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{4}}{120}+\frac {b^{3} d \,x^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{3}}{120}+\frac {b^{2} d^{2} x^{3} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{60}+\frac {b \,d^{3} x^{4} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{20}+\frac {d^{4} x^{5} F^{a} F^{\frac {b}{d x +c}}}{5}+\frac {b^{4} c \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{4}}{120 d}+\frac {b^{3} c x \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{3}}{60}+\frac {b^{2} c d \,x^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{20}+\frac {b c \,d^{2} x^{3} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{5}+c \,d^{3} x^{4} F^{a} F^{\frac {b}{d x +c}}+\frac {b^{3} c^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{3}}{120 d}+\frac {b^{2} c^{2} x \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{20}+\frac {3 b \,c^{2} d \,x^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{10}+2 c^{2} d^{2} x^{3} F^{a} F^{\frac {b}{d x +c}}+\frac {b^{2} c^{3} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{60 d}+\frac {b \,c^{3} x \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{5}+2 c^{3} d \,x^{2} F^{a} F^{\frac {b}{d x +c}}+\frac {b \,c^{4} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{20 d}+c^{4} x \,F^{a} F^{\frac {b}{d x +c}}+\frac {c^{5} F^{a} F^{\frac {b}{d x +c}}}{5 d} \]
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 \[ \frac {1}{120} \, {\left (24 \, F^{a} d^{4} x^{5} + 6 \, {\left (F^{a} b d^{3} \log \relax (F) + 20 \, F^{a} c d^{3}\right )} x^{4} + 2 \, {\left (F^{a} b^{2} d^{2} \log \relax (F)^{2} + 12 \, F^{a} b c d^{2} \log \relax (F) + 120 \, F^{a} c^{2} d^{2}\right )} x^{3} + {\left (F^{a} b^{3} d \log \relax (F)^{3} + 6 \, F^{a} b^{2} c d \log \relax (F)^{2} + 36 \, F^{a} b c^{2} d \log \relax (F) + 240 \, F^{a} c^{3} d\right )} x^{2} + {\left (F^{a} b^{4} \log \relax (F)^{4} + 2 \, F^{a} b^{3} c \log \relax (F)^{3} + 6 \, F^{a} b^{2} c^{2} \log \relax (F)^{2} + 24 \, F^{a} b c^{3} \log \relax (F) + 120 \, F^{a} c^{4}\right )} x\right )} F^{\frac {b}{d x + c}} + \int \frac {{\left (F^{a} b^{5} d x \log \relax (F)^{5} - F^{a} b^{4} c^{2} \log \relax (F)^{4} - 2 \, F^{a} b^{3} c^{3} \log \relax (F)^{3} - 6 \, F^{a} b^{2} c^{4} \log \relax (F)^{2} - 24 \, F^{a} b c^{5} \log \relax (F)\right )} F^{\frac {b}{d x + c}}}{120 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.67, size = 181, normalized size = 6.24 \[ \frac {F^a\,F^{\frac {b}{c+d\,x}}\,{\left (c+d\,x\right )}^5}{5\,d}+\frac {F^a\,b^5\,{\ln \relax (F)}^5\,\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{c+d\,x}\right )}{120\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^2\,{\ln \relax (F)}^2\,{\left (c+d\,x\right )}^3}{60\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^3\,{\ln \relax (F)}^3\,{\left (c+d\,x\right )}^2}{120\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b\,\ln \relax (F)\,{\left (c+d\,x\right )}^4}{20\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^4\,{\ln \relax (F)}^4\,\left (c+d\,x\right )}{120\,d} \]
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 \[ \int F^{a + \frac {b}{c + d x}} \left (c + d x\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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