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

Optimal. Leaf size=189 \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} (2 c d-b e)^2 \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{8 c^{5/2} \sqrt{\log (f)}}-\frac{\sqrt{\pi } e^2 f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \log ^{\frac{3}{2}}(f)}+\frac{e (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.108072, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2241, 2240, 2234, 2204} \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} (2 c d-b e)^2 \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{8 c^{5/2} \sqrt{\log (f)}}-\frac{\sqrt{\pi } e^2 f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \log ^{\frac{3}{2}}(f)}+\frac{e (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x + c*x^2)*(d + e*x)^2,x]

[Out]

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

Rule 2241

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[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)
, x], 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] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]

Rule 2240

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

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^{a+b x+c x^2} (d+e x)^2 \, dx &=\frac{e f^{a+b x+c x^2} (d+e x)}{2 c \log (f)}-\frac{(-2 c d+b e) \int f^{a+b x+c x^2} (d+e x) \, dx}{2 c}-\frac{e^2 \int f^{a+b x+c x^2} \, dx}{2 c \log (f)}\\ &=\frac{e (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{e f^{a+b x+c x^2} (d+e x)}{2 c \log (f)}+\frac{(2 c d-b e)^2 \int f^{a+b x+c x^2} \, dx}{4 c^2}-\frac{\left (e^2 f^{a-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{2 c \log (f)}\\ &=-\frac{e^2 f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{4 c^{3/2} \log ^{\frac{3}{2}}(f)}+\frac{e (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{e f^{a+b x+c x^2} (d+e x)}{2 c \log (f)}+\frac{\left ((2 c d-b e)^2 f^{a-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{4 c^2}\\ &=-\frac{e^2 f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{4 c^{3/2} \log ^{\frac{3}{2}}(f)}+\frac{e (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{e f^{a+b x+c x^2} (d+e x)}{2 c \log (f)}+\frac{(2 c d-b e)^2 f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{8 c^{5/2} \sqrt{\log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.17507, size = 123, normalized size = 0.65 \[ \frac{f^{a-\frac{b^2}{4 c}} \left (\sqrt{\pi } \left (\log (f) (b e-2 c d)^2-2 c e^2\right ) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )+2 \sqrt{c} e \sqrt{\log (f)} f^{\frac{(b+2 c x)^2}{4 c}} (-b e+4 c d+2 c e x)\right )}{8 c^{5/2} \log ^{\frac{3}{2}}(f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 307, normalized size = 1.6 \begin{align*} -{\frac{{d}^{2}\sqrt{\pi }{f}^{a}}{2}{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 ) }}}}+{\frac{{e}^{2}x{f}^{c{x}^{2}}{f}^{bx}{f}^{a}}{2\,c\ln \left ( f \right ) }}-{\frac{b{e}^{2}{f}^{c{x}^{2}}{f}^{bx}{f}^{a}}{4\,\ln \left ( f \right ){c}^{2}}}-{\frac{{b}^{2}{e}^{2}\sqrt{\pi }{f}^{a}}{8\,{c}^{2}}{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 ) }}}}+{\frac{{e}^{2}\sqrt{\pi }{f}^{a}}{4\,c\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 ) }}}}+{\frac{de{f}^{c{x}^{2}}{f}^{bx}{f}^{a}}{c\ln \left ( f \right ) }}+{\frac{bde\sqrt{\pi }{f}^{a}}{2\,c}{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+a)*(e*x+d)^2,x)

[Out]

-1/2*d^2*Pi^(1/2)*f^a*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))+1/
2*e^2/c/ln(f)*x*f^(c*x^2)*f^(b*x)*f^a-1/4*e^2/c^2*b/ln(f)*f^(c*x^2)*f^(b*x)*f^a-1/8*e^2/c^2*b^2*Pi^(1/2)*f^a*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))+1/4*e^2/c/ln(f)*Pi^(1/2)*
f^a*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))+d*e/c/ln(f)*f^(c*x^2
)*f^(b*x)*f^a+1/2*d*e*b/c*Pi^(1/2)*f^a*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 = 1.24732, size = 448, normalized size = 2.37 \begin{align*} -\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 )} d e f^{a - \frac{b^{2}}{4 \, c}}}{2 \, \sqrt{c \log \left (f\right )}} + \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 )} e^{2} f^{a - \frac{b^{2}}{4 \, c}}}{8 \, \sqrt{c \log \left (f\right )}} + \frac{\sqrt{\pi } d^{2} f^{a} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(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))*d*e*f^(a - 1/4*b^2/c)/sqrt(c*log(
f)) + 1/8*(sqrt(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))*e^2*f^(a - 1/4
*b^2/c)/sqrt(c*log(f)) + 1/2*sqrt(pi)*d^2*f^a*erf(sqrt(-c*log(f))*x - 1/2*b*log(f)/sqrt(-c*log(f)))/(sqrt(-c*l
og(f))*f^(1/4*b^2/c))

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Fricas [A]  time = 1.59018, size = 311, normalized size = 1.65 \begin{align*} \frac{2 \,{\left (2 \, c^{2} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} f^{c x^{2} + b x + a} \log \left (f\right ) + \frac{\sqrt{\pi }{\left (2 \, c e^{2} -{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \log \left (f\right )\right )} \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 \, a c}{4 \, c}}}}{8 \, c^{3} \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+a)*(e*x+d)^2,x, algorithm="fricas")

[Out]

1/8*(2*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*f^(c*x^2 + b*x + a)*log(f) + sqrt(pi)*(2*c*e^2 - (4*c^2*d^2 - 4*b*c
*d*e + b^2*e^2)*log(f))*sqrt(-c*log(f))*erf(1/2*(2*c*x + b)*sqrt(-c*log(f))/c)/f^(1/4*(b^2 - 4*a*c)/c))/(c^3*l
og(f)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*x**2+b*x+a)*(e*x+d)**2,x)

[Out]

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

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Giac [A]  time = 1.2856, size = 340, normalized size = 1.8 \begin{align*} -\frac{\sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{2 \, \sqrt{-c \log \left (f\right )}} + \frac{\frac{\sqrt{\pi } b d \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right ) - 4 \, c}{4 \, c}\right )}}{\sqrt{-c \log \left (f\right )}} + \frac{2 \, d e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right ) + 1\right )}}{\log \left (f\right )}}{2 \, c} - \frac{\frac{\sqrt{\pi }{\left (b^{2} \log \left (f\right ) - 2 \, c\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right ) - 8 \, c}{4 \, c}\right )}}{\sqrt{-c \log \left (f\right )} \log \left (f\right )} - \frac{2 \,{\left (c{\left (2 \, x + \frac{b}{c}\right )} - 2 \, b\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right ) + 2\right )}}{\log \left (f\right )}}{8 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(pi)*d^2*erf(-1/2*sqrt(-c*log(f))*(2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*log(f))/c)/sqrt(-c*log(f))
+ 1/2*(sqrt(pi)*b*d*erf(-1/2*sqrt(-c*log(f))*(2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*log(f) - 4*c)/c)/sqrt(-
c*log(f)) + 2*d*e^(c*x^2*log(f) + b*x*log(f) + a*log(f) + 1)/log(f))/c - 1/8*(sqrt(pi)*(b^2*log(f) - 2*c)*erf(
-1/2*sqrt(-c*log(f))*(2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*log(f) - 8*c)/c)/(sqrt(-c*log(f))*log(f)) - 2*(c
*(2*x + b/c) - 2*b)*e^(c*x^2*log(f) + b*x*log(f) + a*log(f) + 2)/log(f))/c^2