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

Optimal. Leaf size=142 \[ \frac{105}{16} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}-\frac{7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+\frac{35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{105}{8} e^{a+b x+c x^2} \sqrt{a+b x+c x^2} \]

[Out]

(-105*E^(a + b*x + c*x^2)*Sqrt[a + b*x + c*x^2])/8 + (35*E^(a + b*x + c*x^2)*(a
+ b*x + c*x^2)^(3/2))/4 - (7*E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(5/2))/2 + E^
(a + b*x + c*x^2)*(a + b*x + c*x^2)^(7/2) + (105*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*
x^2]])/16

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Rubi [A]  time = 1.01358, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{105}{16} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}-\frac{7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+\frac{35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{105}{8} e^{a+b x+c x^2} \sqrt{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

(-105*E^(a + b*x + c*x^2)*Sqrt[a + b*x + c*x^2])/8 + (35*E^(a + b*x + c*x^2)*(a
+ b*x + c*x^2)^(3/2))/4 - (7*E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(5/2))/2 + E^
(a + b*x + c*x^2)*(a + b*x + c*x^2)^(7/2) + (105*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*
x^2]])/16

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Rubi in Sympy [A]  time = 86.1223, size = 134, normalized size = 0.94 \[ \left (a + b x + c x^{2}\right )^{\frac{7}{2}} e^{a + b x + c x^{2}} - \frac{7 \left (a + b x + c x^{2}\right )^{\frac{5}{2}} e^{a + b x + c x^{2}}}{2} + \frac{35 \left (a + b x + c x^{2}\right )^{\frac{3}{2}} e^{a + b x + c x^{2}}}{4} - \frac{105 \sqrt{a + b x + c x^{2}} e^{a + b x + c x^{2}}}{8} + \frac{105 \sqrt{\pi } \operatorname{erfi}{\left (\sqrt{a + b x + c x^{2}} \right )}}{16} \]

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]

(a + b*x + c*x**2)**(7/2)*exp(a + b*x + c*x**2) - 7*(a + b*x + c*x**2)**(5/2)*ex
p(a + b*x + c*x**2)/2 + 35*(a + b*x + c*x**2)**(3/2)*exp(a + b*x + c*x**2)/4 - 1
05*sqrt(a + b*x + c*x**2)*exp(a + b*x + c*x**2)/8 + 105*sqrt(pi)*erfi(sqrt(a + b
*x + c*x**2))/16

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Mathematica [A]  time = 0.177632, size = 91, normalized size = 0.64 \[ \frac{105}{16} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+x (b+c x)}\right )+\frac{1}{8} e^{a+x (b+c x)} \sqrt{a+x (b+c x)} \left (8 (a+x (b+c x))^3-28 (a+x (b+c x))^2+70 (a+x (b+c x))-105\right ) \]

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]

(E^(a + x*(b + c*x))*Sqrt[a + x*(b + c*x)]*(-105 + 70*(a + x*(b + c*x)) - 28*(a
+ x*(b + c*x))^2 + 8*(a + x*(b + c*x))^3))/8 + (105*Sqrt[Pi]*Erfi[Sqrt[a + x*(b
+ c*x)]])/16

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Maple [A]  time = 0.011, size = 119, normalized size = 0.8 \[{\frac{35\,{{\rm e}^{c{x}^{2}+bx+a}}}{4} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{{\rm e}^{c{x}^{2}+bx+a}}}{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}+{{\rm e}^{c{x}^{2}+bx+a}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{2}}}+{\frac{105\,\sqrt{\pi }}{16}{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) }-{\frac{105\,{{\rm e}^{c{x}^{2}+bx+a}}}{8}\sqrt{c{x}^{2}+bx+a}} \]

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]

35/4*exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^(3/2)-7/2*exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^(5/
2)+exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^(7/2)+105/16*erfi((c*x^2+b*x+a)^(1/2))*Pi^(1/2
)-105/8*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. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{7}{2}}{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(7/2)*(2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(7/2)*(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. \[{\rm integral}\left ({\left (2 \, c^{4} x^{7} + 7 \, b c^{3} x^{6} + 3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{5} + 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} x^{4} + a^{3} b +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{3} + 3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} x^{2} +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x\right )} \sqrt{c x^{2} + b x + a} e^{\left (c x^{2} + b x + a\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(7/2)*(2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

integral((2*c^4*x^7 + 7*b*c^3*x^6 + 3*(3*b^2*c^2 + 2*a*c^3)*x^5 + 5*(b^3*c + 3*a
*b*c^2)*x^4 + a^3*b + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*x^3 + 3*(a*b^3 + 3*a^2*b*c)
*x^2 + (3*a^2*b^2 + 2*a^3*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)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(7/2)*(2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

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