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

Optimal. Leaf size=41 \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b \sqrt{c} \sqrt{\log (f)}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0213947, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[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]])

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a^{2} c + 2 a b c x + b^{2} c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00597152, size = 41, normalized size = 1. \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[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]  time = 0.028, size = 41, normalized size = 1. \[ -{\frac{\sqrt{\pi }}{2\,b}{\it Erf} \left ( -b\sqrt{-c\ln \left ( f \right ) }x+{ac\ln \left ( f \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c*(b*x+a)^2),x)

[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))

_______________________________________________________________________________________

Maxima [A]  time = 0.833927, size = 54, normalized size = 1.32 \[ \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} b x - \frac{a c \log \left (f\right )}{\sqrt{-c \log \left (f\right )}}\right )}{2 \, \sqrt{-c \log \left (f\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^((b*x + a)^2*c),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)

_______________________________________________________________________________________

Fricas [A]  time = 0.271681, size = 47, normalized size = 1.15 \[ \frac{\sqrt{\pi } \operatorname{erf}\left (\frac{\sqrt{-b^{2} c \log \left (f\right )}{\left (b x + a\right )}}{b}\right )}{2 \, \sqrt{-b^{2} c \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(c*(b*x+a)**2),x)

[Out]

Integral(f**(c*(a + b*x)**2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.253157, size = 45, normalized size = 1.1 \[ -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c{\rm ln}\left (f\right )} b{\left (x + \frac{a}{b}\right )}\right )}{2 \, \sqrt{-c{\rm ln}\left (f\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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