Integrand size = 21, antiderivative size = 22 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x} \, dx=-\frac {F^a \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]
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Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2241} \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x} \, dx=-\frac {F^a \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]
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Rule 2241
Rubi steps \begin{align*} \text {integral}& = -\frac {F^a \text {Ei}\left (\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x} \, dx=-\frac {F^a \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]
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\[\int \frac {F^{a +\frac {b}{\left (d x +c \right )^{3}}}}{d x +c}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x} \, dx=-\frac {F^{a} {\rm Ei}\left (\frac {b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{3 \, d} \]
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\[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x} \, dx=\int \frac {F^{a + \frac {b}{\left (c + d x\right )^{3}}}}{c + d x}\, dx \]
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\[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{d x + c} \,d x } \]
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\[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{d x + c} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x} \, dx=-\frac {F^a\,\mathrm {ei}\left (\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )}{3\,d} \]
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