Integrand size = 25, antiderivative size = 47 \[ \int F^{c (a+b x)^n} (a+b x)^{-1+\frac {n}{2}} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x)^{n/2} \sqrt {\log (F)}\right )}{b \sqrt {c} n \sqrt {\log (F)}} \]
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Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2242, 2235} \[ \int F^{c (a+b x)^n} (a+b x)^{-1+\frac {n}{2}} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} \sqrt {\log (F)} (a+b x)^{n/2}\right )}{b \sqrt {c} n \sqrt {\log (F)}} \]
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Rule 2235
Rule 2242
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int F^{c x^2} \, dx,x,(a+b x)^{n/2}\right )}{b n} \\ & = \frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x)^{n/2} \sqrt {\log (F)}\right )}{b \sqrt {c} n \sqrt {\log (F)}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int F^{c (a+b x)^n} (a+b x)^{-1+\frac {n}{2}} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x)^{n/2} \sqrt {\log (F)}\right )}{b \sqrt {c} n \sqrt {\log (F)}} \]
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Time = 0.73 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-c \ln \left (F \right )}\, \left (b x +a \right )^{\frac {n}{2}}\right )}{n b \sqrt {-c \ln \left (F \right )}}\) | \(36\) |
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Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.06 \[ \int F^{c (a+b x)^n} (a+b x)^{-1+\frac {n}{2}} \, dx=-\frac {\sqrt {\pi } \sqrt {-c \log \left (F\right )} \operatorname {erf}\left ({\left (b x + a\right )} \sqrt {-c \log \left (F\right )} {\left (b x + a\right )}^{\frac {1}{2} \, n - 1}\right )}{b c n \log \left (F\right )} \]
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\[ \int F^{c (a+b x)^n} (a+b x)^{-1+\frac {n}{2}} \, dx=\int F^{c \left (a + b x\right )^{n}} \left (a + b x\right )^{\frac {n}{2} - 1}\, dx \]
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\[ \int F^{c (a+b x)^n} (a+b x)^{-1+\frac {n}{2}} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{2} \, n - 1} F^{{\left (b x + a\right )}^{n} c} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79 \[ \int F^{c (a+b x)^n} (a+b x)^{-1+\frac {n}{2}} \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (F\right )} \sqrt {{\left (b x + a\right )}^{n}}\right )}{\sqrt {-c \log \left (F\right )} b n} \]
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Time = 0.87 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83 \[ \int F^{c (a+b x)^n} (a+b x)^{-1+\frac {n}{2}} \, dx=-\frac {\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {c}\,\sqrt {\ln \left (F\right )}\,{\left (a+b\,x\right )}^{n/2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b\,\sqrt {c}\,n\,\sqrt {\ln \left (F\right )}} \]
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