Optimal. Leaf size=29 \[ -\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{c+d x}\right ) \log ^5(F)}{d} \]
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Rubi [A]
time = 0.03, 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 = {2250}
\begin {gather*} -\frac {b^5 F^a \log ^5(F) \text {Gamma}\left (-5,-\frac {b \log (F)}{c+d x}\right )}{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}} (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 \begin {gather*} -\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{c+d x}\right ) \log ^5(F)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(533\) vs.
\(2(28)=56\).
time = 0.12, size = 534, normalized size = 18.41
method | result | size |
risch | \(\frac {d^{4} F^{a} F^{\frac {b}{d x +c}} x^{5}}{5}+d^{3} F^{a} F^{\frac {b}{d x +c}} c \,x^{4}+2 d^{2} F^{a} F^{\frac {b}{d x +c}} c^{2} x^{3}+2 d \,F^{a} F^{\frac {b}{d x +c}} c^{3} x^{2}+F^{a} F^{\frac {b}{d x +c}} c^{4} x +\frac {F^{a} F^{\frac {b}{d x +c}} c^{5}}{5 d}+\frac {d^{3} b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} x^{4}}{20}+\frac {d^{2} b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c \,x^{3}}{5}+\frac {3 d b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{2} x^{2}}{10}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{3} x}{5}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{4}}{20 d}+\frac {d^{2} b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} x^{3}}{60}+\frac {d \,b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c \,x^{2}}{20}+\frac {b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c^{2} x}{20}+\frac {b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c^{3}}{60 d}+\frac {d \,b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} x^{2}}{120}+\frac {b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} c x}{60}+\frac {b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} c^{2}}{120 d}+\frac {b^{4} \ln \left (F \right )^{4} F^{a} F^{\frac {b}{d x +c}} x}{120}+\frac {b^{4} \ln \left (F \right )^{4} F^{a} F^{\frac {b}{d x +c}} c}{120 d}+\frac {b^{5} \ln \left (F \right )^{5} F^{a} \expIntegral \left (1, -\frac {b \ln \left (F \right )}{d x +c}\right )}{120 d}\) | \(534\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 244 vs.
\(2 (28) = 56\).
time = 0.08, size = 244, normalized size = 8.41 \begin {gather*} -\frac {F^{a} b^{5} {\rm Ei}\left (\frac {b \log \left (F\right )}{d x + c}\right ) \log \left (F\right )^{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 \left (F\right )^{4} + {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \left (F\right )^{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 \left (F\right )^{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 \left (F\right )\right )} F^{\frac {a d x + a c + b}{d x + c}}}{120 \, d} \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 F^{a + \frac {b}{c + d x}} \left (c + d x\right )^{4}\, 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 [B]
time = 3.67, size = 181, normalized size = 6.24 \begin {gather*} \frac {F^a\,F^{\frac {b}{c+d\,x}}\,{\left (c+d\,x\right )}^5}{5\,d}+\frac {F^a\,b^5\,{\ln \left (F\right )}^5\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{c+d\,x}\right )}{120\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^3}{60\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^3\,{\ln \left (F\right )}^3\,{\left (c+d\,x\right )}^2}{120\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^4}{20\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^4\,{\ln \left (F\right )}^4\,\left (c+d\,x\right )}{120\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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