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 = 0.197395, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{\sqrt{\pi } F^a (d e-c f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^2 \sqrt{\log (F)}}+\frac{f F^{a+b (c+d x)^2}}{2 b d^2 \log (F)} \]
Antiderivative was successfully verified.
[In] Int[F^(a + b*(c + d*x)^2)*(e + f*x),x]
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Rubi in Sympy [A] time = 15.0032, size = 73, normalized size = 0.9 \[ - \frac{\sqrt{\pi } F^{a} \left (c f - d e\right ) \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{2 \sqrt{b} d^{2} \sqrt{\log{\left (F \right )}}} + \frac{F^{a + b \left (c + d x\right )^{2}} f}{2 b d^{2} \log{\left (F \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(a+b*(d*x+c)**2)*(f*x+e),x)
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Mathematica [A] time = 0.0968409, 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.
[In] Integrate[F^(a + b*(c + d*x)^2)*(e + f*x),x]
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Maple [A] time = 0.034, size = 127, normalized size = 1.6 \[ -{\frac{{F}^{a}\sqrt{\pi }e}{2\,d}{\it Erf} \left ( -d\sqrt{-b\ln \left ( F \right ) }x+{cb\ln \left ( F \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}}+{\frac{{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}f}{2\,\ln \left ( F \right ) b{d}^{2}}}+{\frac{cf\sqrt{\pi }{F}^{a}}{2\,{d}^{2}}{\it Erf} \left ( -d\sqrt{-b\ln \left ( F \right ) }x+{cb\ln \left ( F \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(a+b*(d*x+c)^2)*(f*x+e),x)
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Maxima [A] time = 0.822991, size = 317, normalized size = 3.91 \[ -\frac{{\left (\frac{\sqrt{\pi }{\left (b d^{2} x \log \left (F\right ) + b c d \log \left (F\right )\right )} b c d{\left (\operatorname{erf}\left (\sqrt{-\frac{{\left (b d^{2} x \log \left (F\right ) + b c d \log \left (F\right )\right )}^{2}}{b d^{2} \log \left (F\right )}}\right ) - 1\right )} \log \left (F\right )}{\left (b d^{2} \log \left (F\right )\right )^{\frac{3}{2}} \sqrt{-\frac{{\left (b d^{2} x \log \left (F\right ) + b c d \log \left (F\right )\right )}^{2}}{b d^{2} \log \left (F\right )}}} - \frac{b d^{2} e^{\left (\frac{{\left (b d^{2} x \log \left (F\right ) + b c d \log \left (F\right )\right )}^{2}}{b d^{2} \log \left (F\right )}\right )} \log \left (F\right )}{\left (b d^{2} \log \left (F\right )\right )^{\frac{3}{2}}}\right )} F^{b c^{2} + a} f}{2 \, \sqrt{b d^{2} \log \left (F\right )} F^{b c^{2}}} + \frac{\sqrt{\pi } F^{b c^{2} + a} e \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} d x - \frac{b c \log \left (F\right )}{\sqrt{-b \log \left (F\right )}}\right )}{2 \, \sqrt{-b \log \left (F\right )} F^{b c^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*F^((d*x + c)^2*b + a),x, algorithm="maxima")
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Fricas [A] time = 0.259912, size = 135, normalized size = 1.67 \[ \frac{\sqrt{\pi }{\left (b d^{2} e - b c d f\right )} F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) \log \left (F\right ) + \sqrt{-b d^{2} \log \left (F\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a} f}{2 \, \sqrt{-b d^{2} \log \left (F\right )} b d^{2} \log \left (F\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*F^((d*x + c)^2*b + a),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int F^{a + b \left (c + d x\right )^{2}} \left (e + f x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(a+b*(d*x+c)**2)*(f*x+e),x)
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GIAC/XCAS [A] time = 0.254743, size = 174, normalized size = 2.15 \[ -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a{\rm ln}\left (F\right ) + 1\right )}}{2 \, \sqrt{-b{\rm ln}\left (F\right )} d} + \frac{\frac{\sqrt{\pi } c f \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a{\rm ln}\left (F\right )\right )}}{\sqrt{-b{\rm ln}\left (F\right )} d} + \frac{f e^{\left (b d^{2} x^{2}{\rm ln}\left (F\right ) + 2 \, b c d x{\rm ln}\left (F\right ) + b c^{2}{\rm ln}\left (F\right ) + a{\rm ln}\left (F\right )\right )}}{b d{\rm ln}\left (F\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*F^((d*x + c)^2*b + a),x, algorithm="giac")
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