3.617 \(\int \frac{e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=115 \[ \frac{8}{15} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )-\frac{8 e^{a+b x+c x^2}}{15 \sqrt{a+b x+c x^2}}-\frac{4 e^{a+b x+c x^2}}{15 \left (a+b x+c x^2\right )^{3/2}}-\frac{2 e^{a+b x+c x^2}}{5 \left (a+b x+c x^2\right )^{5/2}} \]

[Out]

(-2*E^(a + b*x + c*x^2))/(5*(a + b*x + c*x^2)^(5/2)) - (4*E^(a + b*x + c*x^2))/(
15*(a + b*x + c*x^2)^(3/2)) - (8*E^(a + b*x + c*x^2))/(15*Sqrt[a + b*x + c*x^2])
+ (8*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]])/15

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Rubi [A]  time = 0.877851, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{8}{15} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )-\frac{8 e^{a+b x+c x^2}}{15 \sqrt{a+b x+c x^2}}-\frac{4 e^{a+b x+c x^2}}{15 \left (a+b x+c x^2\right )^{3/2}}-\frac{2 e^{a+b x+c x^2}}{5 \left (a+b x+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(E^(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^(7/2),x]

[Out]

(-2*E^(a + b*x + c*x^2))/(5*(a + b*x + c*x^2)^(5/2)) - (4*E^(a + b*x + c*x^2))/(
15*(a + b*x + c*x^2)^(3/2)) - (8*E^(a + b*x + c*x^2))/(15*Sqrt[a + b*x + c*x^2])
+ (8*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]])/15

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Rubi in Sympy [A]  time = 99.3292, size = 109, normalized size = 0.95 \[ \frac{8 \sqrt{\pi } \operatorname{erfi}{\left (\sqrt{a + b x + c x^{2}} \right )}}{15} - \frac{8 e^{a + b x + c x^{2}}}{15 \sqrt{a + b x + c x^{2}}} - \frac{4 e^{a + b x + c x^{2}}}{15 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{2 e^{a + b x + c x^{2}}}{5 \left (a + b x + c x^{2}\right )^{\frac{5}{2}}} \]

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)**(7/2),x)

[Out]

8*sqrt(pi)*erfi(sqrt(a + b*x + c*x**2))/15 - 8*exp(a + b*x + c*x**2)/(15*sqrt(a
+ b*x + c*x**2)) - 4*exp(a + b*x + c*x**2)/(15*(a + b*x + c*x**2)**(3/2)) - 2*ex
p(a + b*x + c*x**2)/(5*(a + b*x + c*x**2)**(5/2))

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Mathematica [A]  time = 0.251465, size = 106, normalized size = 0.92 \[ \frac{2}{15} \left (4 \sqrt{\pi } \text{Erfi}\left (\sqrt{a+x (b+c x)}\right )-\frac{e^{a+x (b+c x)} \left (4 a^2+a \left (8 b x+8 c x^2+2\right )+4 b^2 x^2+2 b \left (4 c x^3+x\right )+4 c^2 x^4+2 c x^2+3\right )}{(a+x (b+c x))^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(E^(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*(-((E^(a + x*(b + c*x))*(3 + 4*a^2 + 4*b^2*x^2 + 2*c*x^2 + 4*c^2*x^4 + a*(2 +
8*b*x + 8*c*x^2) + 2*b*(x + 4*c*x^3)))/(a + x*(b + c*x))^(5/2)) + 4*Sqrt[Pi]*Er
fi[Sqrt[a + x*(b + c*x)]]))/15

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Maple [A]  time = 0.01, size = 95, normalized size = 0.8 \[ -{\frac{2\,{{\rm e}^{c{x}^{2}+bx+a}}}{5} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{5}{2}}}}-{\frac{4\,{{\rm e}^{c{x}^{2}+bx+a}}}{15} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,\sqrt{\pi }}{15}{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) }-{\frac{8\,{{\rm e}^{c{x}^{2}+bx+a}}}{15}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \]

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)^(7/2),x)

[Out]

-2/5*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^(5/2)-4/15*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^(3
/2)+8/15*erfi((c*x^2+b*x+a)^(1/2))*Pi^(1/2)-8/15*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. \[ \int \frac{{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{2}}}\,{d x} \]

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)^(7/2),x, algorithm="maxima")

[Out]

integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x +{\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x^{2}\right )} \sqrt{c x^{2} + b x + a}}, x\right ) \]

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)^(7/2),x, algorithm="fricas")

[Out]

integral((2*c*x + b)*e^(c*x^2 + b*x + a)/((c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*
c^2)*x^4 + 3*a^2*b*x + (b^3 + 6*a*b*c)*x^3 + a^3 + 3*(a*b^2 + a^2*c)*x^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)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{2}}}\,{d x} \]

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)^(7/2),x, algorithm="giac")

[Out]

integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^(7/2), x)