3.417 \(\int e^{\frac{e}{(c+d x)^3}} (a+b x)^2 \, dx\)

Optimal. Leaf size=151 \[ -\frac{2 b (c+d x)^2 (b c-a d) \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^3}+\frac{(c+d x) (b c-a d)^2 \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^3}-\frac{b^2 e \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^3}\right )}{3 d^3}+\frac{b^2 (c+d x)^3 e^{\frac{e}{(c+d x)^3}}}{3 d^3} \]

[Out]

(b^2*E^(e/(c + d*x)^3)*(c + d*x)^3)/(3*d^3) - (b^2*e*ExpIntegralEi[e/(c + d*x)^3
])/(3*d^3) - (2*b*(b*c - a*d)*(-(e/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-2/3, -
(e/(c + d*x)^3)])/(3*d^3) + ((b*c - a*d)^2*(-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Ga
mma[-1/3, -(e/(c + d*x)^3)])/(3*d^3)

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Rubi [A]  time = 0.234966, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{2 b (c+d x)^2 (b c-a d) \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^3}+\frac{(c+d x) (b c-a d)^2 \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^3}-\frac{b^2 e \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^3}\right )}{3 d^3}+\frac{b^2 (c+d x)^3 e^{\frac{e}{(c+d x)^3}}}{3 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(b^2*E^(e/(c + d*x)^3)*(c + d*x)^3)/(3*d^3) - (b^2*e*ExpIntegralEi[e/(c + d*x)^3
])/(3*d^3) - (2*b*(b*c - a*d)*(-(e/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-2/3, -
(e/(c + d*x)^3)])/(3*d^3) + ((b*c - a*d)^2*(-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Ga
mma[-1/3, -(e/(c + d*x)^3)])/(3*d^3)

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Rubi in Sympy [A]  time = 29.9945, size = 143, normalized size = 0.95 \[ - \frac{b^{2} e \operatorname{Ei}{\left (\frac{e}{\left (c + d x\right )^{3}} \right )}}{3 d^{3}} + \frac{b^{2} \left (c + d x\right )^{3} e^{\frac{e}{\left (c + d x\right )^{3}}}}{3 d^{3}} + \frac{2 b \left (- \frac{e}{\left (c + d x\right )^{3}}\right )^{\frac{2}{3}} \left (c + d x\right )^{2} \left (a d - b c\right ) \Gamma{\left (- \frac{2}{3},- \frac{e}{\left (c + d x\right )^{3}} \right )}}{3 d^{3}} + \frac{\sqrt [3]{- \frac{e}{\left (c + d x\right )^{3}}} \left (c + d x\right ) \left (a d - b c\right )^{2} \Gamma{\left (- \frac{1}{3},- \frac{e}{\left (c + d x\right )^{3}} \right )}}{3 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(e/(d*x+c)**3)*(b*x+a)**2,x)

[Out]

-b**2*e*Ei(e/(c + d*x)**3)/(3*d**3) + b**2*(c + d*x)**3*exp(e/(c + d*x)**3)/(3*d
**3) + 2*b*(-e/(c + d*x)**3)**(2/3)*(c + d*x)**2*(a*d - b*c)*Gamma(-2/3, -e/(c +
d*x)**3)/(3*d**3) + (-e/(c + d*x)**3)**(1/3)*(c + d*x)*(a*d - b*c)**2*Gamma(-1/
3, -e/(c + d*x)**3)/(3*d**3)

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Mathematica [A]  time = 1.44339, size = 184, normalized size = 1.22 \[ \frac{3 b (c+d x)^2 (b c-a d) \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (\frac{1}{3},-\frac{e}{(c+d x)^3}\right )-3 (c+d x) (b c-a d)^2 \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (\frac{2}{3},-\frac{e}{(c+d x)^3}\right )+c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right ) e^{\frac{e}{(c+d x)^3}}+d^3 x \left (3 a^2+3 a b x+b^2 x^2\right ) e^{\frac{e}{(c+d x)^3}}-b^2 e \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^3}\right )}{3 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*E^(e/(c + d*x)^3) + d^3*E^(e/(c + d*x)^3)*x
*(3*a^2 + 3*a*b*x + b^2*x^2) - b^2*e*ExpIntegralEi[e/(c + d*x)^3] + 3*b*(b*c - a
*d)*(-(e/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[1/3, -(e/(c + d*x)^3)] - 3*(b*c -
a*d)^2*(-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[2/3, -(e/(c + d*x)^3)])/(3*d^3)

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{3}}}}} \left ( bx+a \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(e/(d*x+c)^3)*(b*x+a)^2,x)

[Out]

int(exp(e/(d*x+c)^3)*(b*x+a)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{3} \,{\left (b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )} + \int \frac{{\left (b^{2} d e x^{3} + 3 \, a b d e x^{2} + 3 \, a^{2} d e x\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b^2*x^3 + 3*a*b*x^2 + 3*a^2*x)*e^(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^
3)) + integrate((b^2*d*e*x^3 + 3*a*b*d*e*x^2 + 3*a^2*d*e*x)*e^(e/(d^3*x^3 + 3*c*
d^2*x^2 + 3*c^2*d*x + c^3))/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x +
c^4), x)

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Fricas [A]  time = 0.260902, size = 370, normalized size = 2.45 \[ -\frac{3 \,{\left (b^{2} c d - a b d^{2}\right )} e \left (-\frac{e}{d^{3}}\right )^{\frac{1}{3}} \Gamma \left (\frac{1}{3}, -\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e \Gamma \left (\frac{2}{3}, -\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) +{\left (b^{2} d^{2} e{\rm Ei}\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) -{\left (b^{2} d^{5} x^{3} + 3 \, a b d^{5} x^{2} + 3 \, a^{2} d^{5} x + b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + 3 \, a^{2} c d^{4}\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}\right )} \left (-\frac{e}{d^{3}}\right )^{\frac{2}{3}}}{3 \, d^{5} \left (-\frac{e}{d^{3}}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*(b^2*c*d - a*b*d^2)*e*(-e/d^3)^(1/3)*gamma(1/3, -e/(d^3*x^3 + 3*c*d^2*x^
2 + 3*c^2*d*x + c^3)) - 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e*gamma(2/3, -e/(d^3*x
^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + (b^2*d^2*e*Ei(e/(d^3*x^3 + 3*c*d^2*x^2 +
3*c^2*d*x + c^3)) - (b^2*d^5*x^3 + 3*a*b*d^5*x^2 + 3*a^2*d^5*x + b^2*c^3*d^2 - 3
*a*b*c^2*d^3 + 3*a^2*c*d^4)*e^(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)))*(-e
/d^3)^(2/3))/(d^5*(-e/d^3)^(2/3))

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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