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

Optimal. Leaf size=75 \[ \frac{(b+2 c x) f^{b x+c x^2}}{\log (f)}-\frac{\sqrt{\pi } \sqrt{c} f^{-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{\log ^{\frac{3}{2}}(f)} \]

[Out]

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

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Rubi [A]  time = 0.0575022, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2237, 2234, 2204} \[ \frac{(b+2 c x) f^{b x+c x^2}}{\log (f)}-\frac{\sqrt{\pi } \sqrt{c} f^{-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{\log ^{\frac{3}{2}}(f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(b*x + c*x^2)*(b + 2*c*x)^2,x]

[Out]

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

Rule 2237

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] - Dist[((m - 1)*e^2)/(2*c*Log[F]), Int[(d + e*x)^(m - 2)*F^(a + b*x + c
*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0] && GtQ[m, 1]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2204

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*} \int f^{b x+c x^2} (b+2 c x)^2 \, dx &=\frac{f^{b x+c x^2} (b+2 c x)}{\log (f)}-\frac{(2 c) \int f^{b x+c x^2} \, dx}{\log (f)}\\ &=\frac{f^{b x+c x^2} (b+2 c x)}{\log (f)}-\frac{\left (2 c f^{-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{\log (f)}\\ &=-\frac{\sqrt{c} f^{-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{\log ^{\frac{3}{2}}(f)}+\frac{f^{b x+c x^2} (b+2 c x)}{\log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0619255, size = 84, normalized size = 1.12 \[ \frac{f^{-\frac{b^2}{4 c}} \left (\sqrt{\log (f)} (b+2 c x) f^{\frac{(b+2 c x)^2}{4 c}}-\sqrt{\pi } \sqrt{c} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )\right )}{\log ^{\frac{3}{2}}(f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.036, size = 90, normalized size = 1.2 \begin{align*} 2\,{\frac{cx{f}^{c{x}^{2}}{f}^{bx}}{\ln \left ( f \right ) }}+{\frac{b{f}^{c{x}^{2}}{f}^{bx}}{\ln \left ( f \right ) }}+{\frac{c\sqrt{\pi }}{\ln \left ( f \right ) }{f}^{-{\frac{{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 ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*x^2+b*x)*(2*c*x+b)^2,x)

[Out]

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

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Maxima [B]  time = 1.24178, size = 444, normalized size = 5.92 \begin{align*} \frac{\sqrt{\pi } b^{2} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x - \frac{b \log \left (f\right )}{2 \, \sqrt{-c \log \left (f\right )}}\right )}{2 \, \sqrt{-c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}}} - \frac{{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b{\left (\operatorname{erf}\left (\frac{1}{2} \, \sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{2}}{\sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac{3}{2}}} - \frac{2 \, c f^{\frac{{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )}{\left (c \log \left (f\right )\right )^{\frac{3}{2}}}\right )} b c}{\sqrt{c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}}} + \frac{{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b^{2}{\left (\operatorname{erf}\left (\frac{1}{2} \, \sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{3}}{\sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac{5}{2}}} - \frac{4 \,{\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac{3}{2}, -\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{4 \, c}\right ) \log \left (f\right )^{3}}{\left (-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}\right )^{\frac{3}{2}} \left (c \log \left (f\right )\right )^{\frac{5}{2}}} - \frac{4 \, b c f^{\frac{{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac{5}{2}}}\right )} c^{2}}{2 \, \sqrt{c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(pi)*b^2*erf(sqrt(-c*log(f))*x - 1/2*b*log(f)/sqrt(-c*log(f)))/(sqrt(-c*log(f))*f^(1/4*b^2/c)) - (sqrt
(pi)*(2*c*x + b)*b*(erf(1/2*sqrt(-(2*c*x + b)^2*log(f)/c)) - 1)*log(f)^2/(sqrt(-(2*c*x + b)^2*log(f)/c)*(c*log
(f))^(3/2)) - 2*c*f^(1/4*(2*c*x + b)^2/c)*log(f)/(c*log(f))^(3/2))*b*c/(sqrt(c*log(f))*f^(1/4*b^2/c)) + 1/2*(s
qrt(pi)*(2*c*x + b)*b^2*(erf(1/2*sqrt(-(2*c*x + b)^2*log(f)/c)) - 1)*log(f)^3/(sqrt(-(2*c*x + b)^2*log(f)/c)*(
c*log(f))^(5/2)) - 4*(2*c*x + b)^3*gamma(3/2, -1/4*(2*c*x + b)^2*log(f)/c)*log(f)^3/((-(2*c*x + b)^2*log(f)/c)
^(3/2)*(c*log(f))^(5/2)) - 4*b*c*f^(1/4*(2*c*x + b)^2/c)*log(f)^2/(c*log(f))^(5/2))*c^2/(sqrt(c*log(f))*f^(1/4
*b^2/c))

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Fricas [A]  time = 1.54022, size = 171, normalized size = 2.28 \begin{align*} \frac{{\left (2 \, c x + b\right )} f^{c x^{2} + b x} \log \left (f\right ) + \frac{\sqrt{\pi } \sqrt{-c \log \left (f\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 \, c}}}}{\log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23602, size = 104, normalized size = 1.39 \begin{align*} \frac{c{\left (2 \, x + \frac{b}{c}\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right )\right )}}{\log \left (f\right )} + \frac{\sqrt{\pi } c \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right )}{\sqrt{-c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}} \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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