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

Optimal. Leaf size=140 \[ \frac{\sqrt{\pi } a^2 \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^3 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{4 b^3 c^{3/2} \log ^{\frac{3}{2}}(f)}+\frac{(a+b x) f^{c (a+b x)^2}}{2 b^3 c \log (f)}-\frac{a f^{c (a+b x)^2}}{b^3 c \log (f)} \]

[Out]

-(Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(4*b^3*c^(3/2)*Log[f]^(3/2)) -
(a*f^(c*(a + b*x)^2))/(b^3*c*Log[f]) + (f^(c*(a + b*x)^2)*(a + b*x))/(2*b^3*c*Lo
g[f]) + (a^2*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(2*b^3*Sqrt[c]*Sqrt[
Log[f]])

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Rubi [A]  time = 0.207403, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\sqrt{\pi } a^2 \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^3 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{4 b^3 c^{3/2} \log ^{\frac{3}{2}}(f)}+\frac{(a+b x) f^{c (a+b x)^2}}{2 b^3 c \log (f)}-\frac{a f^{c (a+b x)^2}}{b^3 c \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(c*(a + b*x)^2)*x^2,x]

[Out]

-(Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(4*b^3*c^(3/2)*Log[f]^(3/2)) -
(a*f^(c*(a + b*x)^2))/(b^3*c*Log[f]) + (f^(c*(a + b*x)^2)*(a + b*x))/(2*b^3*c*Lo
g[f]) + (a^2*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(2*b^3*Sqrt[c]*Sqrt[
Log[f]])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0985372, size = 83, normalized size = 0.59 \[ \frac{\sqrt{\pi } \left (2 a^2 c \log (f)-1\right ) \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )-2 \sqrt{c} \sqrt{\log (f)} (a-b x) f^{c (a+b x)^2}}{4 b^3 c^{3/2} \log ^{\frac{3}{2}}(f)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c]*f^(c*(a + b*x)^2)*(a - b*x)*Sqrt[Log[f]] + Sqrt[Pi]*Erfi[Sqrt[c]*(a
+ b*x)*Sqrt[Log[f]]]*(-1 + 2*a^2*c*Log[f]))/(4*b^3*c^(3/2)*Log[f]^(3/2))

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Maple [A]  time = 0.037, size = 140, normalized size = 1. \[{\frac{{f}^{c \left ( bx+a \right ) ^{2}}x}{2\,c{b}^{2}\ln \left ( f \right ) }}-{\frac{a{f}^{c \left ( bx+a \right ) ^{2}}}{2\,c{b}^{3}\ln \left ( f \right ) }}-{\frac{{a}^{2}\sqrt{\pi }}{2\,{b}^{3}}{\it Erf} \left ( -b\sqrt{-c\ln \left ( f \right ) }x+{ac\ln \left ( f \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{\frac{\sqrt{\pi }}{4\,c{b}^{3}\ln \left ( f \right ) }{\it Erf} \left ( -b\sqrt{-c\ln \left ( f \right ) }x+{ac\ln \left ( f \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c*(b*x+a)^2)*x^2,x)

[Out]

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

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Maxima [A]  time = 1.02154, size = 350, normalized size = 2.5 \[ \frac{\frac{\sqrt{\pi }{\left (b^{2} c x \log \left (f\right ) + a b c \log \left (f\right )\right )} a^{2} b^{2} c^{2}{\left (\operatorname{erf}\left (\sqrt{-\frac{{\left (b^{2} c x \log \left (f\right ) + a b c \log \left (f\right )\right )}^{2}}{b^{2} c \log \left (f\right )}}\right ) - 1\right )} \log \left (f\right )^{2}}{\left (b^{2} c \log \left (f\right )\right )^{\frac{5}{2}} \sqrt{-\frac{{\left (b^{2} c x \log \left (f\right ) + a b c \log \left (f\right )\right )}^{2}}{b^{2} c \log \left (f\right )}}} - \frac{2 \, a b^{3} c^{2} e^{\left (\frac{{\left (b^{2} c x \log \left (f\right ) + a b c \log \left (f\right )\right )}^{2}}{b^{2} c \log \left (f\right )}\right )} \log \left (f\right )^{2}}{\left (b^{2} c \log \left (f\right )\right )^{\frac{5}{2}}} - \frac{{\left (b^{2} c x \log \left (f\right ) + a b c \log \left (f\right )\right )}^{3} \Gamma \left (\frac{3}{2}, -\frac{{\left (b^{2} c x \log \left (f\right ) + a b c \log \left (f\right )\right )}^{2}}{b^{2} c \log \left (f\right )}\right )}{\left (b^{2} c \log \left (f\right )\right )^{\frac{5}{2}} \left (-\frac{{\left (b^{2} c x \log \left (f\right ) + a b c \log \left (f\right )\right )}^{2}}{b^{2} c \log \left (f\right )}\right )^{\frac{3}{2}}}}{2 \, \sqrt{b^{2} c \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(sqrt(pi)*(b^2*c*x*log(f) + a*b*c*log(f))*a^2*b^2*c^2*(erf(sqrt(-(b^2*c*x*lo
g(f) + a*b*c*log(f))^2/(b^2*c*log(f)))) - 1)*log(f)^2/((b^2*c*log(f))^(5/2)*sqrt
(-(b^2*c*x*log(f) + a*b*c*log(f))^2/(b^2*c*log(f)))) - 2*a*b^3*c^2*e^((b^2*c*x*l
og(f) + a*b*c*log(f))^2/(b^2*c*log(f)))*log(f)^2/(b^2*c*log(f))^(5/2) - (b^2*c*x
*log(f) + a*b*c*log(f))^3*gamma(3/2, -(b^2*c*x*log(f) + a*b*c*log(f))^2/(b^2*c*l
og(f)))/((b^2*c*log(f))^(5/2)*(-(b^2*c*x*log(f) + a*b*c*log(f))^2/(b^2*c*log(f))
)^(3/2)))/sqrt(b^2*c*log(f))

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Fricas [A]  time = 0.26551, size = 136, normalized size = 0.97 \[ \frac{2 \, \sqrt{-b^{2} c \log \left (f\right )}{\left (b x - a\right )} f^{b^{2} c x^{2} + 2 \, a b c x + a^{2} c} + \sqrt{\pi }{\left (2 \, a^{2} b c \log \left (f\right ) - b\right )} \operatorname{erf}\left (\frac{\sqrt{-b^{2} c \log \left (f\right )}{\left (b x + a\right )}}{b}\right )}{4 \, \sqrt{-b^{2} c \log \left (f\right )} b^{3} c \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(f**(c*(a + b*x)**2)*x**2, x)

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GIAC/XCAS [A]  time = 0.238722, size = 144, normalized size = 1.03 \[ -\frac{\frac{\sqrt{\pi }{\left (2 \, a^{2} c{\rm ln}\left (f\right ) - 1\right )} \operatorname{erf}\left (-\sqrt{-c{\rm ln}\left (f\right )} b{\left (x + \frac{a}{b}\right )}\right )}{\sqrt{-c{\rm ln}\left (f\right )} b c{\rm ln}\left (f\right )} - \frac{2 \,{\left (b{\left (x + \frac{a}{b}\right )} - 2 \, a\right )} e^{\left (b^{2} c x^{2}{\rm ln}\left (f\right ) + 2 \, a b c x{\rm ln}\left (f\right ) + a^{2} c{\rm ln}\left (f\right )\right )}}{b c{\rm ln}\left (f\right )}}{4 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(sqrt(pi)*(2*a^2*c*ln(f) - 1)*erf(-sqrt(-c*ln(f))*b*(x + a/b))/(sqrt(-c*ln(
f))*b*c*ln(f)) - 2*(b*(x + a/b) - 2*a)*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^2