\(\int f^{c (a+b x)^2} \, dx\) [198]

Optimal result

Integrand size = 11, antiderivative size = 41 \[ \int f^{c (a+b x)^2} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2235} \[ \int f^{c (a+b x)^2} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]

[In]

Int[f^(c*(a + b*x)^2),x]

[Out]

(Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(2*b*Sqrt[c]*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 {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int f^{c (a+b x)^2} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]

[In]

Integrate[f^(c*(a + b*x)^2),x]

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {\sqrt {\pi }\, \operatorname {erf}\left (-b \sqrt {-c \ln \left (f \right )}\, x +\frac {a c \ln \left (f \right )}{\sqrt {-c \ln \left (f \right )}}\right )}{2 b \sqrt {-c \ln \left (f \right )}}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.10 \[ \int f^{c (a+b x)^2} \, dx=-\frac {\sqrt {\pi } \sqrt {-b^{2} c \log \left (f\right )} \operatorname {erf}\left (\frac {\sqrt {-b^{2} c \log \left (f\right )} {\left (b x + a\right )}}{b}\right )}{2 \, b^{2} c \log \left (f\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate(f**(c*(b*x+a)**2),x)

[Out]

Integral(f**(c*(a + b*x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.98 \[ \int f^{c (a+b x)^2} \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} b x - \frac {a c \log \left (f\right )}{\sqrt {-c \log \left (f\right )}}\right )}{2 \, \sqrt {-c \log \left (f\right )} b} \]

[In]

integrate(f^(c*(b*x+a)^2),x, algorithm="maxima")

[Out]

1/2*sqrt(pi)*erf(sqrt(-c*log(f))*b*x - a*c*log(f)/sqrt(-c*log(f)))/(sqrt(-c*log(f))*b)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int f^{c (a+b x)^2} \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} b {\left (x + \frac {a}{b}\right )}\right )}{2 \, \sqrt {-c \log \left (f\right )} b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.10 \[ \int f^{c (a+b x)^2} \, dx=-\frac {\sqrt {\pi }\,\mathrm {erf}\left (\frac {1{}\mathrm {i}\,c\,x\,\ln \left (f\right )\,b^2+1{}\mathrm {i}\,a\,c\,\ln \left (f\right )\,b}{\sqrt {b^2\,c\,\ln \left (f\right )}}\right )\,1{}\mathrm {i}}{2\,\sqrt {b^2\,c\,\ln \left (f\right )}} \]

[In]

int(f^(c*(a + b*x)^2),x)

[Out]

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