Integrand size = 21, antiderivative size = 28 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{c+d x}\right ) \log ^4(F)}{d} \]
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Time = 0.03 (sec) , antiderivative size = 28, 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 = {2250} \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\frac {b^4 F^a \log ^4(F) \Gamma \left (-4,-\frac {b \log (F)}{c+d x}\right )}{d} \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = \frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{c+d x}\right ) \log ^4(F)}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{c+d x}\right ) \log ^4(F)}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(367\) vs. \(2(28)=56\).
Time = 0.46 (sec) , antiderivative size = 368, normalized size of antiderivative = 13.14
method | result | size |
risch | \(\frac {d^{3} F^{a} F^{\frac {b}{d x +c}} x^{4}}{4}+d^{2} F^{a} F^{\frac {b}{d x +c}} c \,x^{3}+\frac {3 d \,F^{a} F^{\frac {b}{d x +c}} c^{2} x^{2}}{2}+F^{a} F^{\frac {b}{d x +c}} c^{3} x +\frac {F^{a} F^{\frac {b}{d x +c}} c^{4}}{4 d}+\frac {d^{2} b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} x^{3}}{12}+\frac {d b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c \,x^{2}}{4}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{2} x}{4}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{3}}{12 d}+\frac {d \,b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} x^{2}}{24}+\frac {b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c x}{12}+\frac {b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c^{2}}{24 d}+\frac {b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} x}{24}+\frac {b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} c}{24 d}+\frac {b^{4} \ln \left (F \right )^{4} F^{a} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (F \right )}{d x +c}\right )}{24 d}\) | \(368\) |
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Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 175, normalized size of antiderivative = 6.25 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=-\frac {F^{a} b^{4} {\rm Ei}\left (\frac {b \log \left (F\right )}{d x + c}\right ) \log \left (F\right )^{4} - {\left (6 \, d^{4} x^{4} + 24 \, c d^{3} x^{3} + 36 \, c^{2} d^{2} x^{2} + 24 \, c^{3} d x + 6 \, c^{4} + {\left (b^{3} d x + b^{3} c\right )} \log \left (F\right )^{3} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (F\right )^{2} + 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right )} F^{\frac {a d x + a c + b}{d x + c}}}{24 \, d} \]
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\[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\int F^{a + \frac {b}{c + d x}} \left (c + d x\right )^{3}\, dx \]
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\[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\int { {\left (d x + c\right )}^{3} F^{a + \frac {b}{d x + c}} \,d x } \]
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\[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\int { {\left (d x + c\right )}^{3} F^{a + \frac {b}{d x + c}} \,d x } \]
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Time = 0.37 (sec) , antiderivative size = 148, normalized size of antiderivative = 5.29 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\frac {F^a\,F^{\frac {b}{c+d\,x}}\,{\left (c+d\,x\right )}^4}{4\,d}+\frac {F^a\,b^4\,{\ln \left (F\right )}^4\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{c+d\,x}\right )}{24\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^2}{24\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}{12\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^3\,{\ln \left (F\right )}^3\,\left (c+d\,x\right )}{24\,d} \]
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