Integrand size = 21, antiderivative size = 87 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^5 \, dx=\frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^6}{6 d}+\frac {b F^{a+\frac {b}{(c+d x)^3}} (c+d x)^3 \log (F)}{6 d}-\frac {b^2 F^a \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^3}\right ) \log ^2(F)}{6 d} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2245, 2241} \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^5 \, dx=-\frac {b^2 F^a \log ^2(F) \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^3}\right )}{6 d}+\frac {(c+d x)^6 F^{a+\frac {b}{(c+d x)^3}}}{6 d}+\frac {b \log (F) (c+d x)^3 F^{a+\frac {b}{(c+d x)^3}}}{6 d} \]
[In]
[Out]
Rule 2241
Rule 2245
Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^6}{6 d}+\frac {1}{2} (b \log (F)) \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^2 \, dx \\ & = \frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^6}{6 d}+\frac {b F^{a+\frac {b}{(c+d x)^3}} (c+d x)^3 \log (F)}{6 d}+\frac {1}{2} \left (b^2 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x} \, dx \\ & = \frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^6}{6 d}+\frac {b F^{a+\frac {b}{(c+d x)^3}} (c+d x)^3 \log (F)}{6 d}-\frac {b^2 F^a \text {Ei}\left (\frac {b \log (F)}{(c+d x)^3}\right ) \log ^2(F)}{6 d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.82 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^5 \, dx=\frac {F^a \left (F^{\frac {b}{(c+d x)^3}} (c+d x)^6+b \log (F) \left (F^{\frac {b}{(c+d x)^3}} (c+d x)^3-b \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^3}\right ) \log (F)\right )\right )}{6 d} \]
[In]
[Out]
\[\int F^{a +\frac {b}{\left (d x +c \right )^{3}}} \left (d x +c \right )^{5}d x\]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (81) = 162\).
Time = 0.28 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.45 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^5 \, dx=-\frac {F^{a} b^{2} {\rm Ei}\left (\frac {b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \left (F\right )^{2} - {\left (d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6} + {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{6 \, d} \]
[In]
[Out]
\[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^5 \, dx=\int F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (c + d x\right )^{5}\, dx \]
[In]
[Out]
\[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^5 \, dx=\int { {\left (d x + c\right )}^{5} F^{a + \frac {b}{{\left (d x + c\right )}^{3}}} \,d x } \]
[In]
[Out]
\[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^5 \, dx=\int { {\left (d x + c\right )}^{5} F^{a + \frac {b}{{\left (d x + c\right )}^{3}}} \,d x } \]
[In]
[Out]
Time = 0.45 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^5 \, dx=\frac {F^a\,b^2\,{\ln \left (F\right )}^2\,\left (\frac {\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )}{2}+F^{\frac {b}{{\left (c+d\,x\right )}^3}}\,\left (\frac {{\left (c+d\,x\right )}^3}{2\,b\,\ln \left (F\right )}+\frac {{\left (c+d\,x\right )}^6}{2\,b^2\,{\ln \left (F\right )}^2}\right )\right )}{3\,d} \]
[In]
[Out]