\(\int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx\) [350]

Optimal result

Integrand size = 19, antiderivative size = 206 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=-\frac {b^2 (b c-a d) e^{\frac {e}{(c+d x)^3}} (c+d x)^3}{d^4}+\frac {b^2 (b c-a d) e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )}{d^4}+\frac {b^3 \left (-\frac {e}{(c+d x)^3}\right )^{4/3} (c+d x)^4 \Gamma \left (-\frac {4}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4}+\frac {b (b c-a d)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{d^4}-\frac {(b c-a d)^3 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.95 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\frac {-3 b^2 (b c-a d) e^{\frac {e}{(c+d x)^3}} (c+d x)^3+3 b^2 (b c-a d) e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )+b^3 \left (-\frac {e}{(c+d x)^3}\right )^{4/3} (c+d x)^4 \Gamma \left (-\frac {4}{3},-\frac {e}{(c+d x)^3}\right )+3 b (b c-a d)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )-(b c-a d)^3 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4} \] Input:

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

Output:

(-3*b^2*(b*c - a*d)*E^(e/(c + d*x)^3)*(c + d*x)^3 + 3*b^2*(b*c - a*d)*e*Ex 
pIntegralEi[e/(c + d*x)^3] + b^3*(-(e/(c + d*x)^3))^(4/3)*(c + d*x)^4*Gamm 
a[-4/3, -(e/(c + d*x)^3)] + 3*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)] - (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)
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2656, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2656

\(\displaystyle \int \left (-\frac {3 b^2 (c+d x)^2 (b c-a d) e^{\frac {e}{(c+d x)^3}}}{d^3}+\frac {(a d-b c)^3 e^{\frac {e}{(c+d x)^3}}}{d^3}+\frac {3 b (c+d x) (b c-a d)^2 e^{\frac {e}{(c+d x)^3}}}{d^3}+\frac {b^3 (c+d x)^3 e^{\frac {e}{(c+d x)^3}}}{d^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b^2 e (b c-a d) \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )}{d^4}-\frac {b^2 (c+d x)^3 (b c-a d) e^{\frac {e}{(c+d x)^3}}}{d^4}+\frac {b (c+d x)^2 (b c-a d)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{d^4}-\frac {(c+d x) (b c-a d)^3 \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4}+\frac {b^3 (c+d x)^4 \left (-\frac {e}{(c+d x)^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4}\)

Input:

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

Output:

-((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*ExpIntegralEi[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)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2656

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(Px_), x_Symbol] :> Int[ 
ExpandLinearProduct[F^(a + b*(c + d*x)^n), Px, c, d, x], x] /; FreeQ[{F, a, 
 b, c, d, n}, x] && PolynomialQ[Px, x]
 

Maple [F]

\[\int {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}} \left (b x +a \right )^{3}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.69 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\frac {4 \, {\left (b^{3} c - a b^{2} d\right )} e {\rm Ei}\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 6 \, {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + {\left (4 \, b^{3} c^{3} d - 12 \, a b^{2} c^{2} d^{2} + 12 \, a^{2} b c d^{3} - 4 \, a^{3} d^{4} - 3 \, b^{3} d e\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + {\left (b^{3} d^{4} x^{4} + 4 \, a b^{2} d^{4} x^{3} + 6 \, a^{2} b d^{4} x^{2} - b^{3} c^{4} + 4 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} + 4 \, a^{3} c d^{3} + 3 \, b^{3} c e + {\left (4 \, a^{3} d^{4} + 3 \, b^{3} d e\right )} x\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{4 \, d^{4}} \] Input:

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

Output:

1/4*(4*(b^3*c - a*b^2*d)*e*Ei(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) 
 - 6*(b^3*c^2*d^2 - 2*a*b^2*c*d^3 + a^2*b*d^4)*(-e/d^3)^(2/3)*gamma(1/3, - 
e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + (4*b^3*c^3*d - 12*a*b^2*c^2 
*d^2 + 12*a^2*b*c*d^3 - 4*a^3*d^4 - 3*b^3*d*e)*(-e/d^3)^(1/3)*gamma(2/3, - 
e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + (b^3*d^4*x^4 + 4*a*b^2*d^4* 
x^3 + 6*a^2*b*d^4*x^2 - b^3*c^4 + 4*a*b^2*c^3*d - 6*a^2*b*c^2*d^2 + 4*a^3* 
c*d^3 + 3*b^3*c*e + (4*a^3*d^4 + 3*b^3*d*e)*x)*e^(e/(d^3*x^3 + 3*c*d^2*x^2 
 + 3*c^2*d*x + c^3)))/d^4
 

Sympy [F]

\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\int \left (a + b x\right )^{3} e^{\frac {e}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\int { {\left (b x + a\right )}^{3} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \] Input:

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

Output:

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)
 

Giac [F]

\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\int { {\left (b x + a\right )}^{3} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^3}}\,{\left (a+b\,x\right )}^3 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\text {too large to display} \] Input:

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

Output:

( - 2296*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a**3*c**10 
*d**3*e - 18424*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a** 
3*c**9*d**4*e*x - 62496*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x* 
*3))*a**3*c**8*d**5*e*x**2 - 112224*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x* 
*2 + d**3*x**3))*a**3*c**7*d**6*e*x**3 + 1176*e**(e/(c**3 + 3*c**2*d*x + 3 
*c*d**2*x**2 + d**3*x**3))*a**3*c**7*d**3*e**2 - 101136*e**(e/(c**3 + 3*c* 
*2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a**3*c**6*d**7*e*x**4 + 3696*e**(e/(c 
**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a**3*c**6*d**4*e**2*x - 705 
6*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a**3*c**5*d**8*e* 
x**5 - 2520*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a**3*c* 
*5*d**5*e**2*x**2 + 89376*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3* 
x**3))*a**3*c**4*d**9*e*x**6 - 26880*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x 
**2 + d**3*x**3))*a**3*c**4*d**6*e**2*x**3 - 252*e**(e/(c**3 + 3*c**2*d*x 
+ 3*c*d**2*x**2 + d**3*x**3))*a**3*c**4*d**3*e**3 + 105504*e**(e/(c**3 + 3 
*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a**3*c**3*d**10*e*x**7 - 49560*e** 
(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a**3*c**3*d**7*e**2*x* 
*4 + 1260*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a**3*c**3 
*d**4*e**3*x + 59976*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3) 
)*a**3*c**2*d**11*e*x**8 - 43344*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 
+ d**3*x**3))*a**3*c**2*d**8*e**2*x**5 + 5292*e**(e/(c**3 + 3*c**2*d*x ...