3.426 \(\int f^{a+b x+c x^2} x^3 \, dx\)

Optimal. Leaf size=217 \[ \frac{3 \sqrt{\pi } b f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{8 c^{5/2} \log ^{\frac{3}{2}}(f)}+\frac{b^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}-\frac{\sqrt{\pi } b^3 f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{16 c^{7/2} \sqrt{\log (f)}}-\frac{f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}-\frac{b x f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{x^2 f^{a+b x+c x^2}}{2 c \log (f)} \]

[Out]

-f^(a + b*x + c*x^2)/(2*c^2*Log[f]^2) + (3*b*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b
+ 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(8*c^(5/2)*Log[f]^(3/2)) + (b^2*f^(a + b*x
+ c*x^2))/(8*c^3*Log[f]) - (b*f^(a + b*x + c*x^2)*x)/(4*c^2*Log[f]) + (f^(a + b
*x + c*x^2)*x^2)/(2*c*Log[f]) - (b^3*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x
)*Sqrt[Log[f]])/(2*Sqrt[c])])/(16*c^(7/2)*Sqrt[Log[f]])

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Rubi [A]  time = 0.3645, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{3 \sqrt{\pi } b f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{8 c^{5/2} \log ^{\frac{3}{2}}(f)}+\frac{b^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}-\frac{\sqrt{\pi } b^3 f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{16 c^{7/2} \sqrt{\log (f)}}-\frac{f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}-\frac{b x f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{x^2 f^{a+b x+c x^2}}{2 c \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x + c*x^2)*x^3,x]

[Out]

-f^(a + b*x + c*x^2)/(2*c^2*Log[f]^2) + (3*b*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b
+ 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(8*c^(5/2)*Log[f]^(3/2)) + (b^2*f^(a + b*x
+ c*x^2))/(8*c^3*Log[f]) - (b*f^(a + b*x + c*x^2)*x)/(4*c^2*Log[f]) + (f^(a + b
*x + c*x^2)*x^2)/(2*c*Log[f]) - (b^3*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x
)*Sqrt[Log[f]])/(2*Sqrt[c])])/(16*c^(7/2)*Sqrt[Log[f]])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{b^{3} f^{a - \frac{b^{2}}{4 c}} \int f^{\frac{b^{2}}{4 c} + b x + c x^{2}}\, dx}{8 c^{3}} + \frac{b^{2} f^{a + b x + c x^{2}}}{8 c^{3} \log{\left (f \right )}} + \frac{3 b f^{a - \frac{b^{2}}{4 c}} \int f^{\frac{b^{2}}{4 c} + b x + c x^{2}}\, dx}{4 c^{2} \log{\left (f \right )}} - \frac{b f^{a + b x + c x^{2}} x}{4 c^{2} \log{\left (f \right )}} + \frac{f^{a + b x + c x^{2}} x^{2}}{2 c \log{\left (f \right )}} - \frac{f^{a + b x + c x^{2}}}{2 c^{2} \log{\left (f \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(c*x**2+b*x+a)*x**3,x)

[Out]

-b**3*f**(a - b**2/(4*c))*Integral(f**(b**2/(4*c) + b*x + c*x**2), x)/(8*c**3) +
b**2*f**(a + b*x + c*x**2)/(8*c**3*log(f)) + 3*b*f**(a - b**2/(4*c))*Integral(f
**(b**2/(4*c) + b*x + c*x**2), x)/(4*c**2*log(f)) - b*f**(a + b*x + c*x**2)*x/(4
*c**2*log(f)) + f**(a + b*x + c*x**2)*x**2/(2*c*log(f)) - f**(a + b*x + c*x**2)/
(2*c**2*log(f)**2)

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Mathematica [A]  time = 0.238616, size = 122, normalized size = 0.56 \[ \frac{f^{a-\frac{b^2}{4 c}} \left (2 \sqrt{c} f^{\frac{(b+2 c x)^2}{4 c}} \left (\log (f) \left (b^2-2 b c x+4 c^2 x^2\right )-4 c\right )+\sqrt{\pi } b \sqrt{\log (f)} \left (6 c-b^2 \log (f)\right ) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )\right )}{16 c^{7/2} \log ^2(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x + c*x^2)*x^3,x]

[Out]

(f^(a - b^2/(4*c))*(b*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt
[Log[f]]*(6*c - b^2*Log[f]) + 2*Sqrt[c]*f^((b + 2*c*x)^2/(4*c))*(-4*c + (b^2 - 2
*b*c*x + 4*c^2*x^2)*Log[f])))/(16*c^(7/2)*Log[f]^2)

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Maple [A]  time = 0.103, size = 214, normalized size = 1. \[{\frac{{f}^{c{x}^{2}+bx+a}{x}^{2}}{2\,c\ln \left ( f \right ) }}-{\frac{b{f}^{c{x}^{2}+bx+a}x}{4\,\ln \left ( f \right ){c}^{2}}}+{\frac{{b}^{2}{f}^{c{x}^{2}+bx+a}}{8\,{c}^{3}\ln \left ( f \right ) }}+{\frac{{b}^{3}\sqrt{\pi }}{16\,{c}^{3}}{f}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{3\,b\sqrt{\pi }}{8\,\ln \left ( f \right ){c}^{2}}{f}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{{f}^{c{x}^{2}+bx+a}}{2\,{c}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c*x^2+b*x+a)*x^3,x)

[Out]

1/2*f^(c*x^2+b*x+a)*x^2/c/ln(f)-1/4*b*f^(c*x^2+b*x+a)*x/c^2/ln(f)+1/8*b^2*f^(c*x
^2+b*x+a)/c^3/ln(f)+1/16*b^3/c^3*Pi^(1/2)*f^(1/4*(4*a*c-b^2)/c)/(-c*ln(f))^(1/2)
*erf(-(-c*ln(f))^(1/2)*x+1/2*ln(f)*b/(-c*ln(f))^(1/2))-3/8*b/c^2/ln(f)*Pi^(1/2)*
f^(1/4*(4*a*c-b^2)/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*ln(f)*b/(-c*l
n(f))^(1/2))-1/2*f^(c*x^2+b*x+a)/c^2/ln(f)^2

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Maxima [A]  time = 0.880951, size = 344, normalized size = 1.59 \[ -\frac{{\left (\frac{\sqrt{\pi }{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )} b^{3}{\left (\operatorname{erf}\left (\frac{1}{2} \, \sqrt{-\frac{{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )}^{2}}{c \log \left (f\right )}}\right ) - 1\right )} \log \left (f\right )^{3}}{\left (c \log \left (f\right )\right )^{\frac{7}{2}} \sqrt{-\frac{{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )}^{2}}{c \log \left (f\right )}}} - \frac{6 \, b^{2} c e^{\left (\frac{{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )}^{2}}{4 \, c \log \left (f\right )}\right )} \log \left (f\right )^{3}}{\left (c \log \left (f\right )\right )^{\frac{7}{2}}} + \frac{8 \, c^{2} \Gamma \left (2, -\frac{{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )}^{2}}{4 \, c \log \left (f\right )}\right ) \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac{7}{2}}} - \frac{12 \,{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )}^{3} b \Gamma \left (\frac{3}{2}, -\frac{{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )}^{2}}{4 \, c \log \left (f\right )}\right ) \log \left (f\right )}{\left (c \log \left (f\right )\right )^{\frac{7}{2}} \left (-\frac{{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )}^{2}}{c \log \left (f\right )}\right )^{\frac{3}{2}}}\right )} f^{a}}{16 \, \sqrt{c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c*x^2 + b*x + a)*x^3,x, algorithm="maxima")

[Out]

-1/16*(sqrt(pi)*(2*c*x*log(f) + b*log(f))*b^3*(erf(1/2*sqrt(-(2*c*x*log(f) + b*l
og(f))^2/(c*log(f)))) - 1)*log(f)^3/((c*log(f))^(7/2)*sqrt(-(2*c*x*log(f) + b*lo
g(f))^2/(c*log(f)))) - 6*b^2*c*e^(1/4*(2*c*x*log(f) + b*log(f))^2/(c*log(f)))*lo
g(f)^3/(c*log(f))^(7/2) + 8*c^2*gamma(2, -1/4*(2*c*x*log(f) + b*log(f))^2/(c*log
(f)))*log(f)^2/(c*log(f))^(7/2) - 12*(2*c*x*log(f) + b*log(f))^3*b*gamma(3/2, -1
/4*(2*c*x*log(f) + b*log(f))^2/(c*log(f)))*log(f)/((c*log(f))^(7/2)*(-(2*c*x*log
(f) + b*log(f))^2/(c*log(f)))^(3/2)))*f^a/(sqrt(c*log(f))*f^(1/4*b^2/c))

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Fricas [A]  time = 0.248888, size = 159, normalized size = 0.73 \[ \frac{2 \,{\left ({\left (4 \, c^{2} x^{2} - 2 \, b c x + b^{2}\right )} \log \left (f\right ) - 4 \, c\right )} \sqrt{-c \log \left (f\right )} f^{c x^{2} + b x + a} - \frac{\sqrt{\pi }{\left (b^{3} \log \left (f\right )^{2} - 6 \, b c \log \left (f\right )\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac{b^{2} - 4 \, a c}{4 \, c}}}}{16 \, \sqrt{-c \log \left (f\right )} c^{3} \log \left (f\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c*x^2 + b*x + a)*x^3,x, algorithm="fricas")

[Out]

1/16*(2*((4*c^2*x^2 - 2*b*c*x + b^2)*log(f) - 4*c)*sqrt(-c*log(f))*f^(c*x^2 + b*
x + a) - sqrt(pi)*(b^3*log(f)^2 - 6*b*c*log(f))*erf(1/2*(2*c*x + b)*sqrt(-c*log(
f))/c)/f^(1/4*(b^2 - 4*a*c)/c))/(sqrt(-c*log(f))*c^3*log(f)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + b x + c x^{2}} x^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(c*x**2+b*x+a)*x**3,x)

[Out]

Integral(f**(a + b*x + c*x**2)*x**3, x)

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GIAC/XCAS [A]  time = 0.299912, size = 185, normalized size = 0.85 \[ \frac{\frac{\sqrt{\pi }{\left (b^{3}{\rm ln}\left (f\right ) - 6 \, b c\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c{\rm ln}\left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2}{\rm ln}\left (f\right ) - 4 \, a c{\rm ln}\left (f\right )}{4 \, c}\right )}}{\sqrt{-c{\rm ln}\left (f\right )}{\rm ln}\left (f\right )} + \frac{2 \,{\left (c^{2}{\left (2 \, x + \frac{b}{c}\right )}^{2}{\rm ln}\left (f\right ) - 3 \, b c{\left (2 \, x + \frac{b}{c}\right )}{\rm ln}\left (f\right ) + 3 \, b^{2}{\rm ln}\left (f\right ) - 4 \, c\right )} e^{\left (c x^{2}{\rm ln}\left (f\right ) + b x{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}}{{\rm ln}\left (f\right )^{2}}}{16 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c*x^2 + b*x + a)*x^3,x, algorithm="giac")

[Out]

1/16*(sqrt(pi)*(b^3*ln(f) - 6*b*c)*erf(-1/2*sqrt(-c*ln(f))*(2*x + b/c))*e^(-1/4*
(b^2*ln(f) - 4*a*c*ln(f))/c)/(sqrt(-c*ln(f))*ln(f)) + 2*(c^2*(2*x + b/c)^2*ln(f)
- 3*b*c*(2*x + b/c)*ln(f) + 3*b^2*ln(f) - 4*c)*e^(c*x^2*ln(f) + b*x*ln(f) + a*l
n(f))/ln(f)^2)/c^3