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