Integrand size = 25, antiderivative size = 100 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-2 n} \, dx=-\frac {F^{a+b (c+d x)^n} (c+d x)^{-2 n}}{2 d n}-\frac {b F^{a+b (c+d x)^n} (c+d x)^{-n} \log (F)}{2 d n}+\frac {b^2 F^a \operatorname {ExpIntegralEi}\left (b (c+d x)^n \log (F)\right ) \log ^2(F)}{2 d n} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2246, 2241} \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-2 n} \, dx=\frac {b^2 F^a \log ^2(F) \operatorname {ExpIntegralEi}\left (b (c+d x)^n \log (F)\right )}{2 d n}-\frac {(c+d x)^{-2 n} F^{a+b (c+d x)^n}}{2 d n}-\frac {b \log (F) (c+d x)^{-n} F^{a+b (c+d x)^n}}{2 d n} \]
[In]
[Out]
Rule 2241
Rule 2246
Rubi steps \begin{align*} \text {integral}& = -\frac {F^{a+b (c+d x)^n} (c+d x)^{-2 n}}{2 d n}+\frac {1}{2} (b \log (F)) \int F^{a+b (c+d x)^n} (c+d x)^{-1-n} \, dx \\ & = -\frac {F^{a+b (c+d x)^n} (c+d x)^{-2 n}}{2 d n}-\frac {b F^{a+b (c+d x)^n} (c+d x)^{-n} \log (F)}{2 d n}+\frac {1}{2} \left (b^2 \log ^2(F)\right ) \int \frac {F^{a+b (c+d x)^n}}{c+d x} \, dx \\ & = -\frac {F^{a+b (c+d x)^n} (c+d x)^{-2 n}}{2 d n}-\frac {b F^{a+b (c+d x)^n} (c+d x)^{-n} \log (F)}{2 d n}+\frac {b^2 F^a \text {Ei}\left (b (c+d x)^n \log (F)\right ) \log ^2(F)}{2 d n} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.32 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-2 n} \, dx=-\frac {b^2 F^a \Gamma \left (-2,-b (c+d x)^n \log (F)\right ) \log ^2(F)}{d n} \]
[In]
[Out]
Time = 0.63 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.99
method | result | size |
risch | \(-\frac {F^{b \left (d x +c \right )^{n}} F^{a} \left (d x +c \right )^{-2 n}}{2 n d}-\frac {\ln \left (F \right ) b \,F^{b \left (d x +c \right )^{n}} F^{a} \left (d x +c \right )^{-n}}{2 n d}-\frac {\ln \left (F \right )^{2} b^{2} F^{a} \operatorname {Ei}_{1}\left (-b \left (d x +c \right )^{n} \ln \left (F \right )\right )}{2 n d}\) | \(99\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.84 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-2 n} \, dx=\frac {{\left (d x + c\right )}^{2 \, n} F^{a} b^{2} {\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \left (F\right )\right ) \log \left (F\right )^{2} - {\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + 1\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{2 \, {\left (d x + c\right )}^{2 \, n} d n} \]
[In]
[Out]
\[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-2 n} \, dx=\int F^{a + b \left (c + d x\right )^{n}} \left (c + d x\right )^{- 2 n - 1}\, dx \]
[In]
[Out]
\[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-2 n} \, dx=\int { {\left (d x + c\right )}^{-2 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a} \,d x } \]
[In]
[Out]
\[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-2 n} \, dx=\int { {\left (d x + c\right )}^{-2 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-2 n} \, dx=\int \frac {F^{a+b\,{\left (c+d\,x\right )}^n}}{{\left (c+d\,x\right )}^{2\,n+1}} \,d x \]
[In]
[Out]