Optimal. Leaf size=47 \[ \frac {\sqrt {\pi } \text {erf}\left (\sqrt {c} (a+b x)^{n/2} \sqrt {\log (F)}\right )}{b \sqrt {c} n \sqrt {\log (F)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2242, 2236}
\begin {gather*} \frac {\sqrt {\pi } \text {Erf}\left (\sqrt {c} \sqrt {\log (F)} (a+b x)^{n/2}\right )}{b \sqrt {c} n \sqrt {\log (F)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2236
Rule 2242
Rubi steps
\begin {align*} \int F^{-c (a+b x)^n} (a+b x)^{-1+\frac {n}{2}} \, dx &=\frac {2 \text {Subst}\left (\int F^{-c x^2} \, dx,x,(a+b x)^{n/2}\right )}{b n}\\ &=\frac {\sqrt {\pi } \text {erf}\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 \begin {gather*} \frac {\sqrt {\pi } \text {erf}\left (\sqrt {c} (a+b x)^{n/2} \sqrt {\log (F)}\right )}{b \sqrt {c} n \sqrt {\log (F)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 34, normalized size = 0.72
method | result | size |
risch | \(\frac {\sqrt {\pi }\, \erf \left (\sqrt {c \ln \left (F \right )}\, \left (b x +a \right )^{\frac {n}{2}}\right )}{n b \sqrt {c \ln \left (F \right )}}\) | \(34\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 47, normalized size = 1.00 \begin {gather*} \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 )} \end {gather*}
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 \begin {gather*} \int F^{- c \left (a + b x\right )^{n}} \left (a + b x\right )^{\frac {n}{2} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.92, size = 35, normalized size = 0.74 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.04, size = 35, normalized size = 0.74 \begin {gather*} \frac {\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {c}\,\sqrt {\ln \left (F\right )}\,{\left (a+b\,x\right )}^{n/2}\right )}{b\,\sqrt {c}\,n\,\sqrt {\ln \left (F\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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