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

Optimal. Leaf size=68 \[ \frac{f^{c (a+b x)^2}}{2 b^2 c \log (f)}-\frac{\sqrt{\pi } a \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^2 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

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

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Rubi [A]  time = 0.0922748, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{f^{c (a+b x)^2}}{2 b^2 c \log (f)}-\frac{\sqrt{\pi } a \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^2 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]  Int[f^(c*(a + b*x)^2)*x,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a \int f^{\frac{\left (2 a b c + 2 b^{2} c x\right )^{2}}{4 b^{2} c}}\, dx}{b} + \frac{f^{c \left (a + b x\right )^{2}}}{2 b^{2} c \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*Integral(f**((2*a*b*c + 2*b**2*c*x)**2/(4*b**2*c)), x)/b + f**(c*(a + b*x)**2
)/(2*b**2*c*log(f))

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Mathematica [A]  time = 0.0355843, size = 63, normalized size = 0.93 \[ \frac{f^{c (a+b x)^2}-\sqrt{\pi } a \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^2 c \log (f)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.031, size = 66, normalized size = 1. \[{\frac{{f}^{c \left ( bx+a \right ) ^{2}}}{2\,c{b}^{2}\ln \left ( f \right ) }}+{\frac{a\sqrt{\pi }}{2\,{b}^{2}}{\it Erf} \left ( -b\sqrt{-c\ln \left ( f \right ) }x+{ac\ln \left ( f \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c*(b*x+a)^2)*x,x)

[Out]

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

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Maxima [A]  time = 0.857717, size = 211, normalized size = 3.1 \[ -\frac{\frac{\sqrt{\pi }{\left (b^{2} c x \log \left (f\right ) + a b c \log \left (f\right )\right )} a b c{\left (\operatorname{erf}\left (\sqrt{-\frac{{\left (b^{2} c x \log \left (f\right ) + a b c \log \left (f\right )\right )}^{2}}{b^{2} c \log \left (f\right )}}\right ) - 1\right )} \log \left (f\right )}{\left (b^{2} c \log \left (f\right )\right )^{\frac{3}{2}} \sqrt{-\frac{{\left (b^{2} c x \log \left (f\right ) + a b c \log \left (f\right )\right )}^{2}}{b^{2} c \log \left (f\right )}}} - \frac{b^{2} c e^{\left (\frac{{\left (b^{2} c x \log \left (f\right ) + a b c \log \left (f\right )\right )}^{2}}{b^{2} c \log \left (f\right )}\right )} \log \left (f\right )}{\left (b^{2} c \log \left (f\right )\right )^{\frac{3}{2}}}}{2 \, \sqrt{b^{2} c \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(sqrt(pi)*(b^2*c*x*log(f) + a*b*c*log(f))*a*b*c*(erf(sqrt(-(b^2*c*x*log(f)
+ a*b*c*log(f))^2/(b^2*c*log(f)))) - 1)*log(f)/((b^2*c*log(f))^(3/2)*sqrt(-(b^2*
c*x*log(f) + a*b*c*log(f))^2/(b^2*c*log(f)))) - b^2*c*e^((b^2*c*x*log(f) + a*b*c
*log(f))^2/(b^2*c*log(f)))*log(f)/(b^2*c*log(f))^(3/2))/sqrt(b^2*c*log(f))

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Fricas [A]  time = 0.259283, size = 116, normalized size = 1.71 \[ -\frac{\sqrt{\pi } a b c \operatorname{erf}\left (\frac{\sqrt{-b^{2} c \log \left (f\right )}{\left (b x + a\right )}}{b}\right ) \log \left (f\right ) - \sqrt{-b^{2} c \log \left (f\right )} f^{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{2 \, \sqrt{-b^{2} c \log \left (f\right )} b^{2} c \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(f**(c*(a + b*x)**2)*x, x)

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GIAC/XCAS [A]  time = 0.254261, size = 104, normalized size = 1.53 \[ \frac{\frac{\sqrt{\pi } a \operatorname{erf}\left (-\sqrt{-c{\rm ln}\left (f\right )} b{\left (x + \frac{a}{b}\right )}\right )}{\sqrt{-c{\rm ln}\left (f\right )} b} + \frac{e^{\left (b^{2} c x^{2}{\rm ln}\left (f\right ) + 2 \, a b c x{\rm ln}\left (f\right ) + a^{2} c{\rm ln}\left (f\right )\right )}}{b c{\rm ln}\left (f\right )}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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