Optimal. Leaf size=67 \[ -\frac {F^{a+b (c+d x)^2}}{d (c+d x)}+\frac {\sqrt {b} F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \sqrt {\log (F)}}{d} \]
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Rubi [A]
time = 0.06, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2245, 2235}
\begin {gather*} \frac {\sqrt {\pi } \sqrt {b} F^a \sqrt {\log (F)} \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{d}-\frac {F^{a+b (c+d x)^2}}{d (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2245
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^2} \, dx &=-\frac {F^{a+b (c+d x)^2}}{d (c+d x)}+(2 b \log (F)) \int F^{a+b (c+d x)^2} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{d (c+d x)}+\frac {\sqrt {b} F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \sqrt {\log (F)}}{d}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 63, normalized size = 0.94 \begin {gather*} \frac {F^a \left (-\frac {F^{b (c+d x)^2}}{c+d x}+\sqrt {b} \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \sqrt {\log (F)}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 62, normalized size = 0.93
method | result | size |
risch | \(-\frac {F^{b \left (d x +c \right )^{2}} F^{a}}{d \left (d x +c \right )}+\frac {b \ln \left (F \right ) \sqrt {\pi }\, F^{a} \erf \left (\sqrt {-b \ln \left (F \right )}\, \left (d x +c \right )\right )}{d \sqrt {-b \ln \left (F \right )}}\) | \(62\) |
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 = 0.37, size = 83, normalized size = 1.24 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )}}{d}\right ) + F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a} d}{d^{3} x + c d^{2}} \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 \frac {F^{a + b \left (c + d x\right )^{2}}}{\left (c + d x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.06, size = 86, normalized size = 1.28 \begin {gather*} \frac {F^a\,b\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \left (F\right )\,d^2+b\,c\,\ln \left (F\right )\,d}{\sqrt {b\,d^2\,\ln \left (F\right )}}\right )\,\ln \left (F\right )}{\sqrt {b\,d^2\,\ln \left (F\right )}}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}}{d\,\left (c+d\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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