Optimal. Leaf size=81 \[ \frac {\sqrt {\pi } F^a (d e-c f) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^2 \sqrt {\log (F)}}+\frac {f F^{a+b (c+d x)^2}}{2 b d^2 \log (F)} \]
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Rubi [A] time = 0.15, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2226, 2204, 2209} \[ \frac {\sqrt {\pi } F^a (d e-c f) \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^2 \sqrt {\log (F)}}+\frac {f F^{a+b (c+d x)^2}}{2 b d^2 \log (F)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2209
Rule 2226
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^2} (e+f x) \, dx &=\int \left (\frac {(d e-c f) F^{a+b (c+d x)^2}}{d}+\frac {f F^{a+b (c+d x)^2} (c+d x)}{d}\right ) \, dx\\ &=\frac {f \int F^{a+b (c+d x)^2} (c+d x) \, dx}{d}+\frac {(d e-c f) \int F^{a+b (c+d x)^2} \, dx}{d}\\ &=\frac {f F^{a+b (c+d x)^2}}{2 b d^2 \log (F)}+\frac {(d e-c f) F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d^2 \sqrt {\log (F)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 74, normalized size = 0.91 \[ \frac {F^a \left (\sqrt {\pi } \sqrt {b} \sqrt {\log (F)} (d e-c f) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )+f F^{b (c+d x)^2}\right )}{2 b d^2 \log (F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 85, normalized size = 1.05 \[ -\frac {\sqrt {\pi } \sqrt {-b d^{2} \log \relax (F)} {\left (d e - c f\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right ) - F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a} d f}{2 \, b d^{3} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 127, normalized size = 1.57 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right ) e^{\left (a \log \relax (F) + 1\right )}}{2 \, \sqrt {-b \log \relax (F)} d} + \frac {\frac {\sqrt {\pi } F^{a} c f \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right )}{\sqrt {-b \log \relax (F)} d} + \frac {f e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F)\right )}}{b d \log \relax (F)}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 132, normalized size = 1.63 \[ \frac {\sqrt {\pi }\, c f \,F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d^{2}}-\frac {\sqrt {\pi }\, e \,F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d}+\frac {f \,F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{2} \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.76, size = 195, normalized size = 2.41 \[ -\frac {{\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b c {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}\right ) - 1\right )} \log \relax (F)^{2}}{\left (b \log \relax (F)\right )^{\frac {3}{2}} d^{2} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}} - \frac {F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b \log \relax (F)}{\left (b \log \relax (F)\right )^{\frac {3}{2}} d}\right )} F^{a} f}{2 \, \sqrt {b \log \relax (F)} d} + \frac {\sqrt {\pi } F^{b c^{2} + a} e \operatorname {erf}\left (\sqrt {-b \log \relax (F)} d x - \frac {b c \log \relax (F)}{\sqrt {-b \log \relax (F)}}\right )}{2 \, \sqrt {-b \log \relax (F)} F^{b c^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 96, normalized size = 1.19 \[ \frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,f}{2\,b\,d^2\,\ln \relax (F)}-\frac {F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (F)\,d^2+b\,c\,\ln \relax (F)\,d}{\sqrt {b\,d^2\,\ln \relax (F)}}\right )\,\left (c\,f-d\,e\right )}{2\,d\,\sqrt {b\,d^2\,\ln \relax (F)}} \]
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 \[ \int F^{a + b \left (c + d x\right )^{2}} \left (e + f x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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