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]

1/2*F^a*erfi((d*x+c)*b^(1/2)*ln(F)^(1/2))*Pi^(1/2)/d/b^(1/2)/ln(F)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2204} \[ \frac {\sqrt {\pi } F^a \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]

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]])

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin {align*} \int F^{a+b (c+d x)^2} \, dx &=\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d \sqrt {\log (F)}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 44, normalized size = 1.00 \[ \frac {\sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]

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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fricas [A]  time = 0.44, size = 48, normalized size = 1.09 \[ -\frac {\sqrt {\pi } \sqrt {-b d^{2} \log \relax (F)} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right )}{2 \, b d^{2} \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 36, normalized size = 0.82 \[ -\frac {\sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right )}{2 \, \sqrt {-b \log \relax (F)} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 58, normalized size = 1.32 \[ -\frac {\sqrt {\pi }\, F^{-b \,c^{2}} F^{b \,c^{2}+a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+(d*x+c)^2*b),x)

[Out]

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

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maxima [A]  time = 0.96, size = 58, normalized size = 1.32 \[ \frac {\sqrt {\pi } F^{b c^{2} + a} \operatorname {erf}\left (\sqrt {-b \log \relax (F)} d x - \frac {b c \log \relax (F)}{\sqrt {-b \log \relax (F)}}\right )}{2 \, \sqrt {-b \log \relax (F)} F^{b c^{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2),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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mupad [B]  time = 0.04, size = 48, normalized size = 1.09 \[ -\frac {F^a\,\sqrt {\pi }\,\mathrm {erf}\left (\frac {1{}\mathrm {i}\,b\,x\,\ln \relax (F)\,d^2+1{}\mathrm {i}\,b\,c\,\ln \relax (F)\,d}{\sqrt {b\,d^2\,\ln \relax (F)}}\right )\,1{}\mathrm {i}}{2\,\sqrt {b\,d^2\,\ln \relax (F)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + b \left (c + d x\right )^{2}}\, dx \]

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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