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

Optimal. Leaf size=44 \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]

[Out]

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

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Rubi [A]  time = 0.02603, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.61782, size = 41, normalized size = 0.93 \[ \frac{\sqrt{\pi } F^{a} \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{2 \sqrt{b} d \sqrt{\log{\left (F \right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**2),x)

[Out]

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

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Mathematica [A]  time = 0.00764215, size = 44, normalized size = 1. \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^2),x]

[Out]

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

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Maple [A]  time = 0.027, size = 44, normalized size = 1. \[ -{\frac{{F}^{a}\sqrt{\pi }}{2\,d}{\it Erf} \left ( -d\sqrt{-b\ln \left ( F \right ) }x+{cb\ln \left ( F \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^2),x)

[Out]

-1/2*Pi^(1/2)*F^a/d/(-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 = 0.925755, size = 78, normalized size = 1.77 \[ \frac{\sqrt{\pi } F^{b c^{2} + a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} d x - \frac{b c \log \left (F\right )}{\sqrt{-b \log \left (F\right )}}\right )}{2 \, \sqrt{-b \log \left (F\right )} F^{b c^{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^2*b + a),x, algorithm="maxima")

[Out]

1/2*sqrt(pi)*F^(b*c^2 + a)*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 = 0.262297, size = 51, normalized size = 1.16 \[ \frac{\sqrt{\pi } F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right )}{2 \, \sqrt{-b d^{2} \log \left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^2*b + a),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int F^{a + b \left (c + d x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b*(d*x+c)**2),x)

[Out]

Integral(F**(a + b*(c + d*x)**2), x)

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GIAC/XCAS [A]  time = 0.236733, size = 51, normalized size = 1.16 \[ -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a{\rm ln}\left (F\right )\right )}}{2 \, \sqrt{-b{\rm ln}\left (F\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(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))/(sqrt(-b*ln(F))*d)