3.613 \(\int e^{a+b x+c x^2} (b+2 c x) \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=52 \[ e^{a+b x+c x^2} \sqrt{a+b x+c x^2}-\frac{1}{2} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right ) \]

[Out]

E^(a + b*x + c*x^2)*Sqrt[a + b*x + c*x^2] - (Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]
])/2

_______________________________________________________________________________________

Rubi [A]  time = 0.346253, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ e^{a+b x+c x^2} \sqrt{a+b x+c x^2}-\frac{1}{2} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^(a + b*x + c*x^2)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2],x]

[Out]

E^(a + b*x + c*x^2)*Sqrt[a + b*x + c*x^2] - (Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]
])/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 76.8862, size = 46, normalized size = 0.88 \[ \sqrt{a + b x + c x^{2}} e^{a + b x + c x^{2}} - \frac{\sqrt{\pi } \operatorname{erfi}{\left (\sqrt{a + b x + c x^{2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(c*x**2+b*x+a)*(2*c*x+b)*(c*x**2+b*x+a)**(1/2),x)

[Out]

sqrt(a + b*x + c*x**2)*exp(a + b*x + c*x**2) - sqrt(pi)*erfi(sqrt(a + b*x + c*x*
*2))/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.0388514, size = 49, normalized size = 0.94 \[ e^{a+x (b+c x)} \sqrt{a+x (b+c x)}-\frac{1}{2} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+x (b+c x)}\right ) \]

Antiderivative was successfully verified.

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

[Out]

E^(a + x*(b + c*x))*Sqrt[a + x*(b + c*x)] - (Sqrt[Pi]*Erfi[Sqrt[a + x*(b + c*x)]
])/2

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 44, normalized size = 0.9 \[ -{\frac{\sqrt{\pi }}{2}{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) }+{{\rm e}^{c{x}^{2}+bx+a}}\sqrt{c{x}^{2}+bx+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2),x)

[Out]

-1/2*erfi((c*x^2+b*x+a)^(1/2))*Pi^(1/2)+exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^2 + b*x + a)*(2*c*x + b)*e^(c*x^2 + b*x + a), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(c*x^2 + b*x + a)*(2*c*x + b)*e^(c*x^2 + b*x + a), x)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(c*x**2+b*x+a)*(2*c*x+b)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.243511, size = 63, normalized size = 1.21 \[ -\frac{1}{2} \, \sqrt{\pi } i \operatorname{erf}\left (-\sqrt{c x^{2} + b x + a} i\right ) + \sqrt{c x^{2} + b x + a} e^{\left (c x^{2} + b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(pi)*i*erf(-sqrt(c*x^2 + b*x + a)*i) + sqrt(c*x^2 + b*x + a)*e^(c*x^2 +
b*x + a)