Integrand size = 19, antiderivative size = 151 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\frac {b^2 e^{\frac {e}{(c+d x)^3}} (c+d x)^3}{3 d^3}-\frac {b^2 e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )}{3 d^3}-\frac {2 b (b c-a d) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {(b c-a d)^2 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3} \]
1/3*b^2*exp(e/(d*x+c)^3)*(d*x+c)^3/d^3-1/3*b^2*e*Ei(e/(d*x+c)^3)/d^3-2/3*b *(-a*d+b*c)*(-e/(d*x+c)^3)^(2/3)*(d*x+c)^2*GAMMA(-2/3,-e/(d*x+c)^3)/d^3+1/ 3*(-a*d+b*c)^2*(-e/(d*x+c)^3)^(1/3)*(d*x+c)*GAMMA(-1/3,-e/(d*x+c)^3)/d^3
Time = 0.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.90 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\frac {b^2 e^{\frac {e}{(c+d x)^3}} (c+d x)^3-b^2 e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )-2 b (b c-a d) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )+(b c-a d)^2 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3} \]
(b^2*E^(e/(c + d*x)^3)*(c + d*x)^3 - b^2*e*ExpIntegralEi[e/(c + d*x)^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)] + (b*c - a*d)^2*(-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-1/3, -( e/(c + d*x)^3)])/(3*d^3)
Time = 0.33 (sec) , antiderivative size = 151, 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.
\(\displaystyle \int (a+b x)^2 e^{\frac {e}{(c+d x)^3}} \, dx\) |
\(\Big \downarrow \) 2656 |
\(\displaystyle \int \left (\frac {(a d-b c)^2 e^{\frac {e}{(c+d x)^3}}}{d^2}-\frac {2 b (c+d x) (b c-a d) e^{\frac {e}{(c+d x)^3}}}{d^2}+\frac {b^2 (c+d x)^2 e^{\frac {e}{(c+d x)^3}}}{d^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 b (c+d x)^2 (b c-a d) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {(c+d x) (b c-a d)^2 \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}-\frac {b^2 e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {b^2 (c+d x)^3 e^{\frac {e}{(c+d x)^3}}}{3 d^3}\) |
(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*G amma[-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)*Gamma[-1/3, -(e/(c + d*x)^3)])/(3*d^3)
3.5.17.3.1 Defintions of rubi rules used
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]
\[\int {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}} \left (b x +a \right )^{2}d x\]
Time = 0.11 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.72 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=-\frac {b^{2} e {\rm Ei}\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 3 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + 3 \, a^{2} d^{3} x + b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{3 \, d^{3}} \]
-1/3*(b^2*e*Ei(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - 3*(b^2*c*d^2 - a*b*d^3)*(-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)) + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(-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^2*d^3*x^3 + 3*a*b*d^3 *x^2 + 3*a^2*d^3*x + b^2*c^3 - 3*a*b*c^2*d + 3*a^2*c*d^2)*e^(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)))/d^3
\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\int \left (a + b x\right )^{2} e^{\frac {e}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}}\, dx \]
\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\int { {\left (b x + a\right )}^{2} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \]
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)
\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\int { {\left (b x + a\right )}^{2} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \]
Timed out. \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^3}}\,{\left (a+b\,x\right )}^2 \,d x \]