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

Optimal. Leaf size=206 \[ \frac{b (c+d x)^2 (b c-a d)^2 \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{e}{(c+d x)^3}\right )}{d^4}-\frac{(c+d x) (b c-a d)^3 \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^4}+\frac{b^3 (c+d x)^4 \left (-\frac{e}{(c+d x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^4}+\frac{b^2 e (b c-a d) \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^3}\right )}{d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{(c+d x)^3}}}{d^4} \]

[Out]

-((b^2*(b*c - a*d)*E^(e/(c + d*x)^3)*(c + d*x)^3)/d^4) + (b^2*(b*c - a*d)*e*ExpI
ntegralEi[e/(c + d*x)^3])/d^4 + (b^3*(-(e/(c + d*x)^3))^(4/3)*(c + d*x)^4*Gamma[
-4/3, -(e/(c + d*x)^3)])/(3*d^4) + (b*(b*c - a*d)^2*(-(e/(c + d*x)^3))^(2/3)*(c
+ d*x)^2*Gamma[-2/3, -(e/(c + d*x)^3)])/d^4 - ((b*c - a*d)^3*(-(e/(c + d*x)^3))^
(1/3)*(c + d*x)*Gamma[-1/3, -(e/(c + d*x)^3)])/(3*d^4)

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Rubi [A]  time = 0.323752, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{b (c+d x)^2 (b c-a d)^2 \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{e}{(c+d x)^3}\right )}{d^4}-\frac{(c+d x) (b c-a d)^3 \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^4}+\frac{b^3 (c+d x)^4 \left (-\frac{e}{(c+d x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^4}+\frac{b^2 e (b c-a d) \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^3}\right )}{d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{(c+d x)^3}}}{d^4} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2*(b*c - a*d)*E^(e/(c + d*x)^3)*(c + d*x)^3)/d^4) + (b^2*(b*c - a*d)*e*ExpI
ntegralEi[e/(c + d*x)^3])/d^4 + (b^3*(-(e/(c + d*x)^3))^(4/3)*(c + d*x)^4*Gamma[
-4/3, -(e/(c + d*x)^3)])/(3*d^4) + (b*(b*c - a*d)^2*(-(e/(c + d*x)^3))^(2/3)*(c
+ d*x)^2*Gamma[-2/3, -(e/(c + d*x)^3)])/d^4 - ((b*c - a*d)^3*(-(e/(c + d*x)^3))^
(1/3)*(c + d*x)*Gamma[-1/3, -(e/(c + d*x)^3)])/(3*d^4)

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Rubi in Sympy [A]  time = 42.635, size = 196, normalized size = 0.95 \[ \frac{b^{3} \left (- \frac{e}{\left (c + d x\right )^{3}}\right )^{\frac{4}{3}} \left (c + d x\right )^{4} \Gamma{\left (- \frac{4}{3},- \frac{e}{\left (c + d x\right )^{3}} \right )}}{3 d^{4}} - \frac{b^{2} e \left (a d - b c\right ) \operatorname{Ei}{\left (\frac{e}{\left (c + d x\right )^{3}} \right )}}{d^{4}} + \frac{b^{2} \left (c + d x\right )^{3} \left (a d - b c\right ) e^{\frac{e}{\left (c + d x\right )^{3}}}}{d^{4}} + \frac{b \left (- \frac{e}{\left (c + d x\right )^{3}}\right )^{\frac{2}{3}} \left (c + d x\right )^{2} \left (a d - b c\right )^{2} \Gamma{\left (- \frac{2}{3},- \frac{e}{\left (c + d x\right )^{3}} \right )}}{d^{4}} + \frac{\sqrt [3]{- \frac{e}{\left (c + d x\right )^{3}}} \left (c + d x\right ) \left (a d - b c\right )^{3} \Gamma{\left (- \frac{1}{3},- \frac{e}{\left (c + d x\right )^{3}} \right )}}{3 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**3*(-e/(c + d*x)**3)**(4/3)*(c + d*x)**4*Gamma(-4/3, -e/(c + d*x)**3)/(3*d**4)
- b**2*e*(a*d - b*c)*Ei(e/(c + d*x)**3)/d**4 + b**2*(c + d*x)**3*(a*d - b*c)*ex
p(e/(c + d*x)**3)/d**4 + b*(-e/(c + d*x)**3)**(2/3)*(c + d*x)**2*(a*d - b*c)**2*
Gamma(-2/3, -e/(c + d*x)**3)/d**4 + (-e/(c + d*x)**3)**(1/3)*(c + d*x)*(a*d - b*
c)**3*Gamma(-1/3, -e/(c + d*x)**3)/(3*d**4)

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Mathematica [A]  time = 1.05482, size = 279, normalized size = 1.35 \[ \frac{(c+d x) \sqrt [3]{-\frac{e}{(c+d x)^3}} \left (-4 a^3 d^3+12 a^2 b c d^2-12 a b^2 c^2 d+b^3 \left (4 c^3-3 e\right )\right ) \text{Gamma}\left (\frac{2}{3},-\frac{e}{(c+d x)^3}\right )-6 b (c+d x)^2 (b c-a d)^2 \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (\frac{1}{3},-\frac{e}{(c+d x)^3}\right )-c e^{\frac{e}{(c+d x)^3}} \left (-4 a^3 d^3+6 a^2 b c d^2-4 a b^2 c^2 d+b^3 \left (c^3-3 e\right )\right )}{4 d^4}+\frac{d x e^{\frac{e}{(c+d x)^3}} \left (4 a^3 d^3+6 a^2 b d^3 x+4 a b^2 d^3 x^2+b^3 \left (d^3 x^3+3 e\right )\right )+4 b^2 e (b c-a d) \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^3}\right )}{4 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(d*E^(e/(c + d*x)^3)*x*(4*a^3*d^3 + 6*a^2*b*d^3*x + 4*a*b^2*d^3*x^2 + b^3*(3*e +
d^3*x^3)) + 4*b^2*(b*c - a*d)*e*ExpIntegralEi[e/(c + d*x)^3])/(4*d^4) + (-(c*(-
4*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 4*a^3*d^3 + b^3*(c^3 - 3*e))*E^(e/(c + d*x)^3))
- 6*b*(b*c - a*d)^2*(-(e/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[1/3, -(e/(c + d*x
)^3)] + (-12*a*b^2*c^2*d + 12*a^2*b*c*d^2 - 4*a^3*d^3 + b^3*(4*c^3 - 3*e))*(-(e/
(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[2/3, -(e/(c + d*x)^3)])/(4*d^4)

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{3}}}}} \left ( bx+a \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (b^{3} d^{3} x^{4} + 4 \, a b^{2} d^{3} x^{3} + 6 \, a^{2} b d^{3} x^{2} +{\left (4 \, a^{3} d^{3} + 3 \, b^{3} e\right )} x\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{4 \, d^{3}} + \int -\frac{3 \,{\left (b^{3} c^{4} e + 4 \,{\left (b^{3} c d^{3} e - a b^{2} d^{4} e\right )} x^{3} + 6 \,{\left (b^{3} c^{2} d^{2} e - a^{2} b d^{4} e\right )} x^{2} -{\left (4 \, a^{3} d^{4} e -{\left (4 \, c^{3} d e - 3 \, d e^{2}\right )} b^{3}\right )} x\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{4 \,{\left (d^{7} x^{4} + 4 \, c d^{6} x^{3} + 6 \, c^{2} d^{5} x^{2} + 4 \, c^{3} d^{4} x + c^{4} d^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b^3*d^3*x^4 + 4*a*b^2*d^3*x^3 + 6*a^2*b*d^3*x^2 + (4*a^3*d^3 + 3*b^3*e)*x)*
e^(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/d^3 + integrate(-3/4*(b^3*c^4*e
+ 4*(b^3*c*d^3*e - a*b^2*d^4*e)*x^3 + 6*(b^3*c^2*d^2*e - a^2*b*d^4*e)*x^2 - (4*a
^3*d^4*e - (4*c^3*d*e - 3*d*e^2)*b^3)*x)*e^(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x
+ c^3))/(d^7*x^4 + 4*c*d^6*x^3 + 6*c^2*d^5*x^2 + 4*c^3*d^4*x + c^4*d^3), x)

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Fricas [A]  time = 0.270777, size = 505, normalized size = 2.45 \[ \frac{6 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e \left (-\frac{e}{d^{3}}\right )^{\frac{1}{3}} \Gamma \left (\frac{1}{3}, -\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) +{\left (3 \, b^{3} e^{2} - 4 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e\right )} \Gamma \left (\frac{2}{3}, -\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) +{\left (4 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e{\rm Ei}\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) +{\left (b^{3} d^{6} x^{4} + 4 \, a b^{2} d^{6} x^{3} + 6 \, a^{2} b d^{6} x^{2} - b^{3} c^{4} d^{2} + 4 \, a b^{2} c^{3} d^{3} - 6 \, a^{2} b c^{2} d^{4} + 4 \, a^{3} c d^{5} + 3 \, b^{3} c d^{2} e +{\left (4 \, a^{3} d^{6} + 3 \, b^{3} d^{3} e\right )} x\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}\right )} \left (-\frac{e}{d^{3}}\right )^{\frac{2}{3}}}{4 \, d^{6} \left (-\frac{e}{d^{3}}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(6*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*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^3*e^2 - 4*(b^3*c^3 - 3*a*b^2*c^2
*d + 3*a^2*b*c*d^2 - a^3*d^3)*e)*gamma(2/3, -e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*
x + c^3)) + (4*(b^3*c*d^2 - a*b^2*d^3)*e*Ei(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x
+ c^3)) + (b^3*d^6*x^4 + 4*a*b^2*d^6*x^3 + 6*a^2*b*d^6*x^2 - b^3*c^4*d^2 + 4*a*
b^2*c^3*d^3 - 6*a^2*b*c^2*d^4 + 4*a^3*c*d^5 + 3*b^3*c*d^2*e + (4*a^3*d^6 + 3*b^3
*d^3*e)*x)*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^6
*(-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)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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