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

Optimal. Leaf size=90 \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \sqrt {\log (f)}}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2240, 2234, 2204} \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \sqrt {\log (f)}}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x + c*x^2)*(d + e*x),x]

[Out]

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

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]

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 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]

Rubi steps

\begin {align*} \int f^{a+b x+c x^2} (d+e x) \, dx &=\frac {e f^{a+b x+c x^2}}{2 c \log (f)}-\frac {(-2 c d+b e) \int f^{a+b x+c x^2} \, dx}{2 c}\\ &=\frac {e f^{a+b x+c x^2}}{2 c \log (f)}+\frac {\left ((2 c d-b e) f^{a-\frac {b^2}{4 c}}\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{2 c}\\ &=\frac {e f^{a+b x+c x^2}}{2 c \log (f)}+\frac {(2 c d-b e) 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} \sqrt {\log (f)}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 96, normalized size = 1.07 \[ \frac {f^{a-\frac {b^2}{4 c}} \left (\sqrt {\pi } \sqrt {\log (f)} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )+2 \sqrt {c} e f^{\frac {(b+2 c x)^2}{4 c}}\right )}{4 c^{3/2} \log (f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 83, normalized size = 0.92 \[ \frac {2 \, c e f^{c x^{2} + b x + a} - \frac {\sqrt {\pi } {\left (2 \, c d - b e\right )} \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{4 \, c^{2} \log \relax (f)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.47, size = 136, normalized size = 1.51 \[ -\frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f)}{4 \, c}\right )}}{2 \, \sqrt {-c \log \relax (f)}} + \frac {\frac {\sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f) - 4 \, c}{4 \, c}\right )}}{\sqrt {-c \log \relax (f)}} + \frac {2 \, e^{\left (c x^{2} \log \relax (f) + b x \log \relax (f) + a \log \relax (f) + 1\right )}}{\log \relax (f)}}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 131, normalized size = 1.46 \[ \frac {\sqrt {\pi }\, b e \,f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}\, c}-\frac {\sqrt {\pi }\, d \,f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{2 \sqrt {-c \ln \relax (f )}}+\frac {e \,f^{a} f^{b x} f^{c \,x^{2}}}{2 c \ln \relax (f )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.42, size = 160, normalized size = 1.78 \[ -\frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{2}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {3}{2}}} - \frac {2 \, c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)}{\left (c \log \relax (f)\right )^{\frac {3}{2}}}\right )} e f^{a - \frac {b^{2}}{4 \, c}}}{4 \, \sqrt {c \log \relax (f)}} + \frac {\sqrt {\pi } d f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f)}}\right )}{2 \, \sqrt {-c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.71, size = 80, normalized size = 0.89 \[ \frac {e\,f^a\,f^{c\,x^2}\,f^{b\,x}}{2\,c\,\ln \relax (f)}-\frac {f^{a-\frac {b^2}{4\,c}}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \relax (f)}{2}+c\,x\,\ln \relax (f)}{\sqrt {c\,\ln \relax (f)}}\right )\,\left (\frac {b\,e}{4}-\frac {c\,d}{2}\right )}{c\,\sqrt {c\,\ln \relax (f)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x + c*x^2)*(d + e*x),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + b x + c x^{2}} \left (d + e x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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