3.386 \(\int F^{a+b (c+d x)^2} (e+f x) \, dx\)

Optimal. Leaf size=81 \[ \frac{\sqrt{\pi } F^a (d e-c f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^2 \sqrt{\log (F)}}+\frac{f F^{a+b (c+d x)^2}}{2 b d^2 \log (F)} \]

[Out]

(f*F^(a + b*(c + d*x)^2))/(2*b*d^2*Log[F]) + ((d*e - c*f)*F^a*Sqrt[Pi]*Erfi[Sqrt
[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^2*Sqrt[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]

[Out]

(f*F^(a + b*(c + d*x)^2))/(2*b*d^2*Log[F]) + ((d*e - c*f)*F^a*Sqrt[Pi]*Erfi[Sqrt
[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^2*Sqrt[Log[F]])

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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)

[Out]

-sqrt(pi)*F**a*(c*f - d*e)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/(2*sqrt(b)*d**2*
sqrt(log(F))) + F**(a + b*(c + d*x)**2)*f/(2*b*d**2*log(F))

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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]

[Out]

(F^a*(f*F^(b*(c + d*x)^2) + Sqrt[b]*(d*e - c*f)*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*
Sqrt[Log[F]]]*Sqrt[Log[F]]))/(2*b*d^2*Log[F])

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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)

[Out]

-1/2*e*Pi^(1/2)*F^a/d/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*l
n(F))^(1/2))+1/2*f/ln(F)/b/d^2*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+1/2*f*c/d^2*Pi^(1
/2)*F^a/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))

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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")

[Out]

-1/2*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b*c*d*(erf(sqrt(-(b*d^2*x*log(F)
+ b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)/((b*d^2*log(F))^(3/2)*sqrt(-(b*d^
2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - b*d^2*e^((b*d^2*x*log(F) + b*c*d
*log(F))^2/(b*d^2*log(F)))*log(F)/(b*d^2*log(F))^(3/2))*F^(b*c^2 + a)*f/(sqrt(b*
d^2*log(F))*F^(b*c^2)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*e*erf(sqrt(-b*log(F))*d*x -
b*c*log(F)/sqrt(-b*log(F)))/(sqrt(-b*log(F))*F^(b*c^2)*d)

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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")

[Out]

1/2*(sqrt(pi)*(b*d^2*e - b*c*d*f)*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d)*log(F
) + sqrt(-b*d^2*log(F))*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)*f)/(sqrt(-b*d^2*lo
g(F))*b*d^2*log(F))

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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)

[Out]

Integral(F**(a + b*(c + d*x)**2)*(e + f*x), 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")

[Out]

-1/2*sqrt(pi)*erf(-sqrt(-b*ln(F))*d*(x + c/d))*e^(a*ln(F) + 1)/(sqrt(-b*ln(F))*d
) + 1/2*(sqrt(pi)*c*f*erf(-sqrt(-b*ln(F))*d*(x + c/d))*e^(a*ln(F))/(sqrt(-b*ln(F
))*d) + f*e^(b*d^2*x^2*ln(F) + 2*b*c*d*x*ln(F) + b*c^2*ln(F) + a*ln(F))/(b*d*ln(
F)))/d