Optimal. Leaf size=161 \[ -\frac {F^{c (a+b x)}}{4 e (d+e x)^4}-\frac {b c F^{c (a+b x)} \log (F)}{12 e^2 (d+e x)^3}-\frac {b^2 c^2 F^{c (a+b x)} \log ^2(F)}{24 e^3 (d+e x)^2}-\frac {b^3 c^3 F^{c (a+b x)} \log ^3(F)}{24 e^4 (d+e x)}+\frac {b^4 c^4 F^{c \left (a-\frac {b d}{e}\right )} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right ) \log ^4(F)}{24 e^5} \]
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Rubi [A]
time = 0.09, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2208, 2209}
\begin {gather*} \frac {b^4 c^4 \log ^4(F) F^{c \left (a-\frac {b d}{e}\right )} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right )}{24 e^5}-\frac {b^3 c^3 \log ^3(F) F^{c (a+b x)}}{24 e^4 (d+e x)}-\frac {b^2 c^2 \log ^2(F) F^{c (a+b x)}}{24 e^3 (d+e x)^2}-\frac {b c \log (F) F^{c (a+b x)}}{12 e^2 (d+e x)^3}-\frac {F^{c (a+b x)}}{4 e (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2208
Rule 2209
Rubi steps
\begin {align*} \int \frac {F^{c (a+b x)}}{(d+e x)^5} \, dx &=-\frac {F^{c (a+b x)}}{4 e (d+e x)^4}+\frac {(b c \log (F)) \int \frac {F^{c (a+b x)}}{(d+e x)^4} \, dx}{4 e}\\ &=-\frac {F^{c (a+b x)}}{4 e (d+e x)^4}-\frac {b c F^{c (a+b x)} \log (F)}{12 e^2 (d+e x)^3}+\frac {\left (b^2 c^2 \log ^2(F)\right ) \int \frac {F^{c (a+b x)}}{(d+e x)^3} \, dx}{12 e^2}\\ &=-\frac {F^{c (a+b x)}}{4 e (d+e x)^4}-\frac {b c F^{c (a+b x)} \log (F)}{12 e^2 (d+e x)^3}-\frac {b^2 c^2 F^{c (a+b x)} \log ^2(F)}{24 e^3 (d+e x)^2}+\frac {\left (b^3 c^3 \log ^3(F)\right ) \int \frac {F^{c (a+b x)}}{(d+e x)^2} \, dx}{24 e^3}\\ &=-\frac {F^{c (a+b x)}}{4 e (d+e x)^4}-\frac {b c F^{c (a+b x)} \log (F)}{12 e^2 (d+e x)^3}-\frac {b^2 c^2 F^{c (a+b x)} \log ^2(F)}{24 e^3 (d+e x)^2}-\frac {b^3 c^3 F^{c (a+b x)} \log ^3(F)}{24 e^4 (d+e x)}+\frac {\left (b^4 c^4 \log ^4(F)\right ) \int \frac {F^{c (a+b x)}}{d+e x} \, dx}{24 e^4}\\ &=-\frac {F^{c (a+b x)}}{4 e (d+e x)^4}-\frac {b c F^{c (a+b x)} \log (F)}{12 e^2 (d+e x)^3}-\frac {b^2 c^2 F^{c (a+b x)} \log ^2(F)}{24 e^3 (d+e x)^2}-\frac {b^3 c^3 F^{c (a+b x)} \log ^3(F)}{24 e^4 (d+e x)}+\frac {b^4 c^4 F^{c \left (a-\frac {b d}{e}\right )} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right ) \log ^4(F)}{24 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 121, normalized size = 0.75 \begin {gather*} \frac {F^{a c} \left (b^4 c^4 F^{-\frac {b c d}{e}} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right ) \log ^4(F)-\frac {e F^{b c x} \left (6 e^3+2 b c e^2 (d+e x) \log (F)+b^2 c^2 e (d+e x)^2 \log ^2(F)+b^3 c^3 (d+e x)^3 \log ^3(F)\right )}{(d+e x)^4}\right )}{24 e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 243, normalized size = 1.51
method | result | size |
risch | \(-\frac {c^{4} b^{4} \ln \left (F \right )^{4} F^{b c x} F^{c a}}{4 e^{5} \left (b c x \ln \left (F \right )+\frac {\ln \left (F \right ) b c d}{e}\right )^{4}}-\frac {c^{4} b^{4} \ln \left (F \right )^{4} F^{b c x} F^{c a}}{12 e^{5} \left (b c x \ln \left (F \right )+\frac {\ln \left (F \right ) b c d}{e}\right )^{3}}-\frac {c^{4} b^{4} \ln \left (F \right )^{4} F^{b c x} F^{c a}}{24 e^{5} \left (b c x \ln \left (F \right )+\frac {\ln \left (F \right ) b c d}{e}\right )^{2}}-\frac {c^{4} b^{4} \ln \left (F \right )^{4} F^{b c x} F^{c a}}{24 e^{5} \left (b c x \ln \left (F \right )+\frac {\ln \left (F \right ) b c d}{e}\right )}-\frac {c^{4} b^{4} \ln \left (F \right )^{4} F^{\frac {c \left (a e -b d \right )}{e}} \expIntegral \left (1, -b c x \ln \left (F \right )-c a \ln \left (F \right )-\frac {-\ln \left (F \right ) a c e +\ln \left (F \right ) b c d}{e}\right )}{24 e^{5}}\) | \(243\) |
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 [A]
time = 0.40, size = 285, normalized size = 1.77 \begin {gather*} \frac {\frac {{\left (b^{4} c^{4} x^{4} e^{4} + 4 \, b^{4} c^{4} d x^{3} e^{3} + 6 \, b^{4} c^{4} d^{2} x^{2} e^{2} + 4 \, b^{4} c^{4} d^{3} x e + b^{4} c^{4} d^{4}\right )} {\rm Ei}\left ({\left (b c x e + b c d\right )} e^{\left (-1\right )} \log \left (F\right )\right ) \log \left (F\right )^{4}}{F^{{\left (b c d - a c e\right )} e^{\left (-1\right )}}} - {\left ({\left (b^{3} c^{3} x^{3} e^{4} + 3 \, b^{3} c^{3} d x^{2} e^{3} + 3 \, b^{3} c^{3} d^{2} x e^{2} + b^{3} c^{3} d^{3} e\right )} \log \left (F\right )^{3} + {\left (b^{2} c^{2} x^{2} e^{4} + 2 \, b^{2} c^{2} d x e^{3} + b^{2} c^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} + 2 \, {\left (b c x e^{4} + b c d e^{3}\right )} \log \left (F\right ) + 6 \, e^{4}\right )} F^{b c x + a c}}{24 \, {\left (x^{4} e^{9} + 4 \, d x^{3} e^{8} + 6 \, d^{2} x^{2} e^{7} + 4 \, d^{3} x e^{6} + d^{4} e^{5}\right )}} \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 \frac {F^{c \left (a + b x\right )}}{\left (d + e x\right )^{5}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (d+e\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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