3.616 \(\int \frac{e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac{4}{3} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )-\frac{4 e^{a+b x+c x^2}}{3 \sqrt{a+b x+c x^2}}-\frac{2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.775344, antiderivative size = 85, 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 \[ \frac{4}{3} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )-\frac{4 e^{a+b x+c x^2}}{3 \sqrt{a+b x+c x^2}}-\frac{2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(E^(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 88.9101, size = 80, normalized size = 0.94 \[ \frac{4 \sqrt{\pi } \operatorname{erfi}{\left (\sqrt{a + b x + c x^{2}} \right )}}{3} - \frac{4 e^{a + b x + c x^{2}}}{3 \sqrt{a + b x + c x^{2}}} - \frac{2 e^{a + b x + c x^{2}}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.157838, size = 67, normalized size = 0.79 \[ \frac{4}{3} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+x (b+c x)}\right )-\frac{2 e^{a+x (b+c x)} \left (2 a+2 b x+2 c x^2+1\right )}{3 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(E^(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*E^(a + x*(b + c*x))*(1 + 2*a + 2*b*x + 2*c*x^2))/(3*(a + x*(b + c*x))^(3/2))
+ (4*Sqrt[Pi]*Erfi[Sqrt[a + x*(b + c*x)]])/3

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Maple [A]  time = 0.01, size = 70, normalized size = 0.8 \[ -{\frac{2\,{{\rm e}^{c{x}^{2}+bx+a}}}{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,\sqrt{\pi }}{3}{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) }-{\frac{4\,{{\rm e}^{c{x}^{2}+bx+a}}}{3}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \]

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)^(5/2),x)

[Out]

-2/3*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^(3/2)+4/3*erfi((c*x^2+b*x+a)^(1/2))*Pi^(1/2)
-4/3*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. \[ \int \frac{{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

integral((2*c*x + b)*e^(c*x^2 + b*x + a)/((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2
+ 2*a*c)*x^2 + a^2)*sqrt(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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^(5/2), x)