Integrand size = 17, antiderivative size = 92 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\frac {b \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2}-\frac {(b c-a d) \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2} \] Output:
1/3*b*(-e/(d*x+c)^3)^(2/3)*(d*x+c)^2*GAMMA(-2/3,-e/(d*x+c)^3)/d^2-1/3*(-a* d+b*c)*(-e/(d*x+c)^3)^(1/3)*(d*x+c)*GAMMA(-1/3,-e/(d*x+c)^3)/d^2
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.92 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\frac {(c+d x) \left (b \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )+(-b c+a d) \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )\right )}{3 d^2} \] Input:
Integrate[E^(e/(c + d*x)^3)*(a + b*x),x]
Output:
((c + d*x)*(b*(-(e/(c + d*x)^3))^(2/3)*(c + d*x)*Gamma[-2/3, -(e/(c + d*x) ^3)] + (-(b*c) + a*d)*(-(e/(c + d*x)^3))^(1/3)*Gamma[-1/3, -(e/(c + d*x)^3 )]))/(3*d^2)
Time = 0.39 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
\(\displaystyle \int (a+b x) e^{\frac {e}{(c+d x)^3}} \, dx\) |
\(\Big \downarrow \) 2656 |
\(\displaystyle \int \left (\frac {(a d-b c) e^{\frac {e}{(c+d x)^3}}}{d}+\frac {b (c+d x) e^{\frac {e}{(c+d x)^3}}}{d}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b (c+d x)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2}-\frac {(c+d x) (b c-a d) \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2}\) |
Input:
Int[E^(e/(c + d*x)^3)*(a + b*x),x]
Output:
(b*(-(e/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-2/3, -(e/(c + d*x)^3)])/(3* d^2) - ((b*c - a*d)*(-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-1/3, -(e/(c + d*x)^3)])/(3*d^2)
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]
\[\int {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}} \left (b x +a \right )d x\]
Input:
int(exp(e/(d*x+c)^3)*(b*x+a),x)
Output:
int(exp(e/(d*x+c)^3)*(b*x+a),x)
Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (80) = 160\).
Time = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.84 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=-\frac {b d^{2} \left (-\frac {e}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 2 \, {\left (b c d - a d^{2}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (b d^{2} x^{2} + 2 \, a d^{2} x - b c^{2} + 2 \, a c d\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{2 \, d^{2}} \] Input:
integrate(exp(e/(d*x+c)^3)*(b*x+a),x, algorithm="fricas")
Output:
-1/2*(b*d^2*(-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)) - 2*(b*c*d - a*d^2)*(-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*d^2*x^2 + 2*a*d^2*x - b*c^2 + 2*a*c*d)*e^ (e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)))/d^2
\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\int \left (a + b x\right ) e^{\frac {e}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}}\, dx \] Input:
integrate(exp(e/(d*x+c)**3)*(b*x+a),x)
Output:
Integral((a + b*x)*exp(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3)), x)
\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\int { {\left (b x + a\right )} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \] Input:
integrate(exp(e/(d*x+c)^3)*(b*x+a),x, algorithm="maxima")
Output:
1/2*(b*x^2 + 2*a*x)*e^(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + inte grate(3/2*(b*d*e*x^2 + 2*a*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)
\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\int { {\left (b x + a\right )} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \] Input:
integrate(exp(e/(d*x+c)^3)*(b*x+a),x, algorithm="giac")
Output:
integrate((b*x + a)*e^(e/(d*x + c)^3), x)
Timed out. \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^3}}\,\left (a+b\,x\right ) \,d x \] Input:
int(exp(e/(c + d*x)^3)*(a + b*x),x)
Output:
int(exp(e/(c + d*x)^3)*(a + b*x), x)
\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\text {too large to display} \] Input:
int(exp(e/(d*x+c)^3)*(b*x+a),x)
Output:
( - 7448*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a*c**13*d - 83216*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a*c**12*d** 2*x - 417648*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a*c**1 1*d**3*x**2 - 1232000*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3 ))*a*c**10*d**4*x**3 + 4368*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d** 3*x**3))*a*c**10*d*e - 2331560*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a*c**9*d**5*x**4 + 30072*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x **2 + d**3*x**3))*a*c**9*d**2*e*x - 2849616*e**(e/(c**3 + 3*c**2*d*x + 3*c *d**2*x**2 + d**3*x**3))*a*c**8*d**6*x**5 + 74088*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a*c**8*d**3*e*x**2 - 2003232*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a*c**7*d**7*x**6 + 34272*e**(e/(c **3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a*c**7*d**4*e*x**3 - 1260*e **(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a*c**7*d*e**2 - 2069 76*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a*c**6*d**8*x**7 - 225792*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a*c**6*d* *5*e*x**4 - 2016*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a* c**6*d**2*e**2*x + 1191960*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3 *x**3))*a*c**5*d**9*x**8 - 613872*e**(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*a*c**5*d**6*e*x**5 + 14364*e**(e/(c**3 + 3*c**2*d*x + 3*c*d **2*x**2 + d**3*x**3))*a*c**5*d**3*e**2*x**2 + 1410640*e**(e/(c**3 + 3*...