3.612 \(\int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=82 \[ \frac{3}{4} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{3}{2} e^{a+b x+c x^2} \sqrt{a+b x+c x^2} \]

[Out]

(-3*E^(a + b*x + c*x^2)*Sqrt[a + b*x + c*x^2])/2 + E^(a + b*x + c*x^2)*(a + b*x
+ c*x^2)^(3/2) + (3*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]])/4

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Rubi [A]  time = 0.555962, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{3}{4} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{3}{2} e^{a+b x+c x^2} \sqrt{a+b x+c x^2} \]

Antiderivative was successfully verified.

[In]  Int[E^(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2),x]

[Out]

(-3*E^(a + b*x + c*x^2)*Sqrt[a + b*x + c*x^2])/2 + E^(a + b*x + c*x^2)*(a + b*x
+ c*x^2)^(3/2) + (3*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]])/4

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Rubi in Sympy [A]  time = 78.8483, size = 76, normalized size = 0.93 \[ \left (a + b x + c x^{2}\right )^{\frac{3}{2}} e^{a + b x + c x^{2}} - \frac{3 \sqrt{a + b x + c x^{2}} e^{a + b x + c x^{2}}}{2} + \frac{3 \sqrt{\pi } \operatorname{erfi}{\left (\sqrt{a + b x + c x^{2}} \right )}}{4} \]

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)**(3/2),x)

[Out]

(a + b*x + c*x**2)**(3/2)*exp(a + b*x + c*x**2) - 3*sqrt(a + b*x + c*x**2)*exp(a
+ b*x + c*x**2)/2 + 3*sqrt(pi)*erfi(sqrt(a + b*x + c*x**2))/4

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Mathematica [A]  time = 0.121832, size = 67, normalized size = 0.82 \[ \frac{1}{4} \left (3 \sqrt{\pi } \text{Erfi}\left (\sqrt{a+x (b+c x)}\right )+2 e^{a+x (b+c x)} \sqrt{a+x (b+c x)} \left (2 a+2 b x+2 c x^2-3\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2),x]

[Out]

(2*E^(a + x*(b + c*x))*(-3 + 2*a + 2*b*x + 2*c*x^2)*Sqrt[a + x*(b + c*x)] + 3*Sq
rt[Pi]*Erfi[Sqrt[a + x*(b + c*x)]])/4

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Maple [A]  time = 0.01, size = 69, normalized size = 0.8 \[{{\rm e}^{c{x}^{2}+bx+a}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}+{\frac{3\,\sqrt{\pi }}{4}{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) }-{\frac{3\,{{\rm e}^{c{x}^{2}+bx+a}}}{2}\sqrt{c{x}^{2}+bx+a}} \]

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)^(3/2),x)

[Out]

exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^(3/2)+3/4*erfi((c*x^2+b*x+a)^(1/2))*Pi^(1/2)-3/2*
exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)*e^(c*x^2 + b*x + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (2 \, c^{2} x^{3} + 3 \, b c x^{2} + a b +{\left (b^{2} + 2 \, a c\right )} x\right )} \sqrt{c x^{2} + b x + a} e^{\left (c x^{2} + b x + a\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

integral((2*c^2*x^3 + 3*b*c*x^2 + a*b + (b^2 + 2*a*c)*x)*sqrt(c*x^2 + b*x + a)*e
^(c*x^2 + b*x + a), x)

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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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)*e^(c*x^2 + b*x + a), x)