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

Optimal. Leaf size=21 \[ \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right ) \]

[Out]

Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]

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Rubi [A]  time = 0.400767, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right ) \]

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]

Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]

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Rubi in Sympy [A]  time = 77.2556, size = 19, normalized size = 0.9 \[ \sqrt{\pi } \operatorname{erfi}{\left (\sqrt{a + b x + c x^{2}} \right )} \]

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(pi)*erfi(sqrt(a + b*x + c*x**2))

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Mathematica [A]  time = 0.0258226, size = 20, normalized size = 0.95 \[ \sqrt{\pi } \text{Erfi}\left (\sqrt{a+x (b+c x)}\right ) \]

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]

Sqrt[Pi]*Erfi[Sqrt[a + x*(b + c*x)]]

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Maple [A]  time = 0.016, size = 18, normalized size = 0.9 \[{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) \sqrt{\pi } \]

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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]

(Integral(b*exp(b*x)*exp(c*x**2)/sqrt(a + b*x + c*x**2), x) + Integral(2*c*x*exp
(b*x)*exp(c*x**2)/sqrt(a + b*x + c*x**2), x))*exp(a)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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