Optimal. Leaf size=67 \[ \frac {\sqrt {\pi } \sqrt {b} F^a \sqrt {\log (F)} \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{d}-\frac {F^{a+b (c+d x)^2}}{d (c+d x)} \]
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Rubi [A] time = 0.08, 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 = {2214, 2204} \[ \frac {\sqrt {\pi } \sqrt {b} F^a \sqrt {\log (F)} \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{d}-\frac {F^{a+b (c+d x)^2}}{d (c+d x)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2214
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 = 0.05, size = 63, normalized size = 0.94 \[ \frac {F^a \left (\sqrt {\pi } \sqrt {b} \sqrt {\log (F)} \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )-\frac {F^{b (c+d x)^2}}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 83, normalized size = 1.24 \[ -\frac {\sqrt {\pi } \sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right ) + F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a} d}{d^{3} x + c d^{2}} \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 62, normalized size = 0.93 \[ \frac {\sqrt {\pi }\, b \,F^{a} \erf \left (\sqrt {-b \ln \relax (F )}\, \left (d x +c \right )\right ) \ln \relax (F )}{\sqrt {-b \ln \relax (F )}\, d}-\frac {F^{a} F^{\left (d x +c \right )^{2} b}}{\left (d x +c \right ) d} \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \]
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 \[ \frac {F^a\,b\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (F)\,d^2+b\,c\,\ln \relax (F)\,d}{\sqrt {b\,d^2\,\ln \relax (F)}}\right )\,\ln \relax (F)}{\sqrt {b\,d^2\,\ln \relax (F)}}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}}{d\,\left (c+d\,x\right )} \]
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 \[ \int \frac {F^{a + b \left (c + d x\right )^{2}}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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