Integrand size = 19, antiderivative size = 85 \[ \int F^{a+\frac {b}{c+d x}} (c+d x) \, dx=\frac {F^{a+\frac {b}{c+d x}} (c+d x)^2}{2 d}+\frac {b F^{a+\frac {b}{c+d x}} (c+d x) \log (F)}{2 d}-\frac {b^2 F^a \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{c+d x}\right ) \log ^2(F)}{2 d} \]
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Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2245, 2237, 2241} \[ \int F^{a+\frac {b}{c+d x}} (c+d x) \, dx=-\frac {b^2 F^a \log ^2(F) \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{c+d x}\right )}{2 d}+\frac {(c+d x)^2 F^{a+\frac {b}{c+d x}}}{2 d}+\frac {b \log (F) (c+d x) F^{a+\frac {b}{c+d x}}}{2 d} \]
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Rule 2237
Rule 2241
Rule 2245
Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+\frac {b}{c+d x}} (c+d x)^2}{2 d}+\frac {1}{2} (b \log (F)) \int F^{a+\frac {b}{c+d x}} \, dx \\ & = \frac {F^{a+\frac {b}{c+d x}} (c+d x)^2}{2 d}+\frac {b F^{a+\frac {b}{c+d x}} (c+d x) \log (F)}{2 d}+\frac {1}{2} \left (b^2 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx \\ & = \frac {F^{a+\frac {b}{c+d x}} (c+d x)^2}{2 d}+\frac {b F^{a+\frac {b}{c+d x}} (c+d x) \log (F)}{2 d}-\frac {b^2 F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right ) \log ^2(F)}{2 d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.68 \[ \int F^{a+\frac {b}{c+d x}} (c+d x) \, dx=\frac {F^a \left (-b^2 \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{c+d x}\right ) \log ^2(F)+F^{\frac {b}{c+d x}} (c+d x) (c+d x+b \log (F))\right )}{2 d} \]
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Time = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.56
method | result | size |
risch | \(\frac {d \,F^{a} F^{\frac {b}{d x +c}} x^{2}}{2}+F^{a} F^{\frac {b}{d x +c}} c x +\frac {F^{a} F^{\frac {b}{d x +c}} c^{2}}{2 d}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} x}{2}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c}{2 d}+\frac {b^{2} \ln \left (F \right )^{2} F^{a} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (F \right )}{d x +c}\right )}{2 d}\) | \(133\) |
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Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int F^{a+\frac {b}{c+d x}} (c+d x) \, dx=-\frac {F^{a} b^{2} {\rm Ei}\left (\frac {b \log \left (F\right )}{d x + c}\right ) \log \left (F\right )^{2} - {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + {\left (b d x + b c\right )} \log \left (F\right )\right )} F^{\frac {a d x + a c + b}{d x + c}}}{2 \, d} \]
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\[ \int F^{a+\frac {b}{c+d x}} (c+d x) \, dx=\int F^{a + \frac {b}{c + d x}} \left (c + d x\right )\, dx \]
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\[ \int F^{a+\frac {b}{c+d x}} (c+d x) \, dx=\int { {\left (d x + c\right )} F^{a + \frac {b}{d x + c}} \,d x } \]
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\[ \int F^{a+\frac {b}{c+d x}} (c+d x) \, dx=\int { {\left (d x + c\right )} F^{a + \frac {b}{d x + c}} \,d x } \]
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Time = 2.53 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96 \[ \int F^{a+\frac {b}{c+d x}} (c+d x) \, dx=\frac {F^a\,F^{\frac {b}{c+d\,x}}\,{\left (c+d\,x\right )}^2}{2\,d}+\frac {F^a\,b^2\,{\ln \left (F\right )}^2\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{c+d\,x}\right )}{2\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b\,\ln \left (F\right )\,\left (c+d\,x\right )}{2\,d} \]
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