Optimal. Leaf size=47 \[ \frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} \sqrt {\log (F)} (a+b x)^{n/2}\right )}{b \sqrt {c} n \sqrt {\log (F)}} \]
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Rubi [A] time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2211, 2204} \[ \frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {c} \sqrt {\log (F)} (a+b x)^{n/2}\right )}{b \sqrt {c} n \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2211
Rubi steps
\begin {align*} \int F^{c (a+b x)^n} (a+b x)^{-1+\frac {n}{2}} \, dx &=\frac {2 \operatorname {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*}
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Mathematica [A] time = 0.01, size = 47, normalized size = 1.00 \[ \frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} \sqrt {\log (F)} (a+b x)^{n/2}\right )}{b \sqrt {c} n \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 50, normalized size = 1.06 \[ -\frac {\sqrt {\pi } \sqrt {-c \log \relax (F)} \operatorname {erf}\left ({\left (b x + a\right )} \sqrt {-c \log \relax (F)} {\left (b x + a\right )}^{\frac {1}{2} \, n - 1}\right )}{b c n \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 37, normalized size = 0.79 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (F)} \sqrt {{\left (b x + a\right )}^{n}}\right )}{\sqrt {-c \log \relax (F)} b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 36, normalized size = 0.77 \[ \frac {\sqrt {\pi }\, \erf \left (\sqrt {-c \ln \relax (F )}\, \left (b x +a \right )^{\frac {n}{2}}\right )}{\sqrt {-c \ln \relax (F )}\, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{\frac {1}{2} \, n - 1} F^{{\left (b x + a\right )}^{n} c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.95, size = 39, normalized size = 0.83 \[ -\frac {\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {c}\,\sqrt {\ln \relax (F)}\,{\left (a+b\,x\right )}^{n/2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b\,\sqrt {c}\,n\,\sqrt {\ln \relax (F)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{c \left (a + b x\right )^{n}} \left (a + b x\right )^{\frac {n}{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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