\(\int F^{a+b (c+d x)^2} \, dx\) [273]

Optimal result

Integrand size = 13, antiderivative size = 44 \[ \int F^{a+b (c+d x)^2} \, dx=\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2235} \[ \int F^{a+b (c+d x)^2} \, dx=\frac {\sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]

[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 2235

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int F^{a+b (c+d x)^2} \, dx=\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]

[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]])

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.32

method	result	size
risch	\(-\frac {\sqrt {\pi }\, F^{b \,c^{2}+a} F^{-b \,c^{2}} \operatorname {erf}\left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{2 d \sqrt {-b \ln \left (F \right )}}\)	\(58\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09 \[ \int F^{a+b (c+d x)^2} \, dx=-\frac {\sqrt {\pi } \sqrt {-b d^{2} \log \left (F\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )}}{d}\right )}{2 \, b d^{2} \log \left (F\right )} \]

[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))

Sympy [F]

\[ \int F^{a+b (c+d x)^2} \, dx=\int F^{a + b \left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

Integral(F**(a + b*(c + d*x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.32 \[ \int F^{a+b (c+d x)^2} \, dx=\frac {\sqrt {\pi } F^{b c^{2} + a} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} d x - \frac {b c \log \left (F\right )}{\sqrt {-b \log \left (F\right )}}\right )}{2 \, \sqrt {-b \log \left (F\right )} F^{b c^{2}} d} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int F^{a+b (c+d x)^2} \, dx=-\frac {\sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} d {\left (x + \frac {c}{d}\right )}\right )}{2 \, \sqrt {-b \log \left (F\right )} d} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09 \[ \int F^{a+b (c+d x)^2} \, dx=-\frac {F^a\,\sqrt {\pi }\,\mathrm {erf}\left (\frac {1{}\mathrm {i}\,b\,x\,\ln \left (F\right )\,d^2+1{}\mathrm {i}\,b\,c\,\ln \left (F\right )\,d}{\sqrt {b\,d^2\,\ln \left (F\right )}}\right )\,1{}\mathrm {i}}{2\,\sqrt {b\,d^2\,\ln \left (F\right )}} \]

[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))