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

Optimal. Leaf size=112 \[ -\frac{15}{8} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}-\frac{5}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+\frac{15}{4} e^{a+b x+c x^2} \sqrt{a+b x+c x^2} \]

[Out]

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

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Rubi [A]  time = 0.722838, antiderivative size = 112, 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 \[ -\frac{15}{8} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}-\frac{5}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+\frac{15}{4} e^{a+b x+c x^2} \sqrt{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 82.0956, size = 105, normalized size = 0.94 \[ \left (a + b x + c x^{2}\right )^{\frac{5}{2}} e^{a + b x + c x^{2}} - \frac{5 \left (a + b x + c x^{2}\right )^{\frac{3}{2}} e^{a + b x + c x^{2}}}{2} + \frac{15 \sqrt{a + b x + c x^{2}} e^{a + b x + c x^{2}}}{4} - \frac{15 \sqrt{\pi } \operatorname{erfi}{\left (\sqrt{a + b x + c x^{2}} \right )}}{8} \]

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]

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

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Mathematica [A]  time = 0.165651, size = 78, normalized size = 0.7 \[ \frac{1}{4} e^{a+x (b+c x)} \sqrt{a+x (b+c x)} \left (4 (a+x (b+c x))^2-10 (a+x (b+c x))+15\right )-\frac{15}{8} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+x (b+c x)}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(E^(a + x*(b + c*x))*Sqrt[a + x*(b + c*x)]*(15 - 10*(a + x*(b + c*x)) + 4*(a + x
*(b + c*x))^2))/4 - (15*Sqrt[Pi]*Erfi[Sqrt[a + x*(b + c*x)]])/8

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Maple [A]  time = 0.01, size = 94, normalized size = 0.8 \[ -{\frac{5\,{{\rm e}^{c{x}^{2}+bx+a}}}{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{{\rm e}^{c{x}^{2}+bx+a}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}-{\frac{15\,\sqrt{\pi }}{8}{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) }+{\frac{15\,{{\rm e}^{c{x}^{2}+bx+a}}}{4}\sqrt{c{x}^{2}+bx+a}} \]

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]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x + a)^(5/2)*(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. \[{\rm integral}\left ({\left (2 \, c^{3} x^{5} + 5 \, b c^{2} x^{4} + 4 \,{\left (b^{2} c + a c^{2}\right )} x^{3} + a^{2} b +{\left (b^{3} + 6 \, a b c\right )} x^{2} + 2 \,{\left (a b^{2} + a^{2} c\right )} x\right )} \sqrt{c x^{2} + b x + a} e^{\left (c x^{2} + b x + a\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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