Integrand size = 21, antiderivative size = 49 \[ \int F^{a+b (c+d x)^3} (c+d x)^3 \, dx=-\frac {F^a (c+d x)^4 \Gamma \left (\frac {4}{3},-b (c+d x)^3 \log (F)\right )}{3 d \left (-b (c+d x)^3 \log (F)\right )^{4/3}} \]
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Time = 0.04 (sec) , antiderivative size = 49, 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} \[ \int F^{a+b (c+d x)^3} (c+d x)^3 \, dx=-\frac {F^a (c+d x)^4 \Gamma \left (\frac {4}{3},-b (c+d x)^3 \log (F)\right )}{3 d \left (-b \log (F) (c+d x)^3\right )^{4/3}} \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = -\frac {F^a (c+d x)^4 \Gamma \left (\frac {4}{3},-b (c+d x)^3 \log (F)\right )}{3 d \left (-b (c+d x)^3 \log (F)\right )^{4/3}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int F^{a+b (c+d x)^3} (c+d x)^3 \, dx=-\frac {F^a (c+d x)^4 \Gamma \left (\frac {4}{3},-b (c+d x)^3 \log (F)\right )}{3 d \left (-b (c+d x)^3 \log (F)\right )^{4/3}} \]
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\[\int F^{a +b \left (d x +c \right )^{3}} \left (d x +c \right )^{3}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (43) = 86\).
Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.41 \[ \int F^{a+b (c+d x)^3} (c+d x)^3 \, dx=\frac {3 \, {\left (b d^{3} x + b c d^{2}\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a} \log \left (F\right ) - \left (-b d^{3} \log \left (F\right )\right )^{\frac {2}{3}} F^{a} \Gamma \left (\frac {1}{3}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right )}{9 \, b^{2} d^{3} \log \left (F\right )^{2}} \]
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\[ \int F^{a+b (c+d x)^3} (c+d x)^3 \, dx=\int F^{a + b \left (c + d x\right )^{3}} \left (c + d x\right )^{3}\, dx \]
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\[ \int F^{a+b (c+d x)^3} (c+d x)^3 \, dx=\int { {\left (d x + c\right )}^{3} F^{{\left (d x + c\right )}^{3} b + a} \,d x } \]
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\[ \int F^{a+b (c+d x)^3} (c+d x)^3 \, dx=\int { {\left (d x + c\right )}^{3} F^{{\left (d x + c\right )}^{3} b + a} \,d x } \]
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Time = 0.57 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.29 \[ \int F^{a+b (c+d x)^3} (c+d x)^3 \, dx=\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^3}\,\left (c+d\,x\right )}{3\,b\,d\,\ln \left (F\right )}-\frac {F^a\,\Gamma \left (\frac {1}{3},-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )\,{\left (c+d\,x\right )}^4}{9\,d\,{\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}^{4/3}}+\frac {2\,\pi \,\sqrt {3}\,F^a\,{\left (c+d\,x\right )}^4}{27\,d\,\Gamma \left (\frac {2}{3}\right )\,{\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}^{4/3}} \]
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