Integrand size = 31, antiderivative size = 72 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )^2}-\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \operatorname {ExpIntegralEi}\left (a+b x+c x^2\right ) \] Output:
-1/2*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^2-exp(c*x^2+b*x+a)/(2*c*x^2+2*b*x+2*a) +1/2*Ei(c*x^2+b*x+a)
Time = 1.45 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (-\frac {e^{a+x (b+c x)} \left (1+a+b x+c x^2\right )}{(a+x (b+c x))^2}+\operatorname {ExpIntegralEi}(a+x (b+c x))\right ) \] Input:
Integrate[(E^(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^3,x]
Output:
(-((E^(a + x*(b + c*x))*(1 + a + b*x + c*x^2))/(a + x*(b + c*x))^2) + ExpI ntegralEi[a + x*(b + c*x)])/2
Time = 0.70 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {7258, 2608, 2608, 2609}
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.
\(\displaystyle \int \frac {(b+2 c x) e^{a+b x+c x^2}}{\left (a+b x+c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 7258 |
\(\displaystyle \int \frac {e^{a+b x+c x^2}}{\left (a+b x+c x^2\right )^3}d\left (a+b x+c x^2\right )\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle \frac {1}{2} \int \frac {e^{c x^2+b x+a}}{\left (c x^2+b x+a\right )^2}d\left (c x^2+b x+a\right )-\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle \frac {1}{2} \left (\int \frac {e^{c x^2+b x+a}}{c x^2+b x+a}d\left (c x^2+b x+a\right )-\frac {e^{a+b x+c x^2}}{a+b x+c x^2}\right )-\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {1}{2} \left (\operatorname {ExpIntegralEi}\left (c x^2+b x+a\right )-\frac {e^{a+b x+c x^2}}{a+b x+c x^2}\right )-\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )^2}\) |
Input:
Int[(E^(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^3,x]
Output:
-1/2*E^(a + b*x + c*x^2)/(a + b*x + c*x^2)^2 + (-(E^(a + b*x + c*x^2)/(a + b*x + c*x^2)) + ExpIntegralEi[a + b*x + c*x^2])/2
Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m _), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))) , x] - Simp[f*g*n*(Log[F]/(d*(m + 1))) Int[(c + d*x)^(m + 1)*(b*F^(g*(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && In tegerQ[2*m] && !TrueQ[$UseGamma]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^(v_)*(u_)*(w_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q Subst[Int[x^m*F^x, x], x, v], x] /; !FalseQ[q]] /; FreeQ[ {F, m}, x] && EqQ[w, v]
Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )^{2}}-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )}-\frac {\operatorname {expIntegral}_{1}\left (-c \,x^{2}-b x -a \right )}{2}\) | \(70\) |
default | \(-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )^{2}}-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )}-\frac {\operatorname {expIntegral}_{1}\left (-c \,x^{2}-b x -a \right )}{2}\) | \(70\) |
risch | \(-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )^{2}}-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )}-\frac {\operatorname {expIntegral}_{1}\left (-c \,x^{2}-b x -a \right )}{2}\) | \(70\) |
Input:
int(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^2-1/2*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)-1 /2*Ei(1,-c*x^2-b*x-a)
Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.54 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} {\rm Ei}\left (c x^{2} + b x + a\right ) - {\left (c x^{2} + b x + a + 1\right )} e^{\left (c x^{2} + b x + a\right )}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \] Input:
integrate(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^3,x, algorithm="fricas" )
Output:
1/2*((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*Ei(c*x^2 + b*x + a) - (c*x^2 + b*x + a + 1)*e^(c*x^2 + b*x + a))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)
Timed out. \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(exp(c*x**2+b*x+a)*(2*c*x+b)/(c*x**2+b*x+a)**3,x)
Output:
Timed out
\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \] Input:
integrate(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^3,x, algorithm="maxima" )
Output:
integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^3, x)
\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \] Input:
integrate(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^3, x)
Time = 0.43 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.86 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {\mathrm {expint}\left (-c\,x^2-b\,x-a\right )}{2}-{\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (\frac {1}{2\,\left (c\,x^2+b\,x+a\right )}+\frac {1}{2\,{\left (c\,x^2+b\,x+a\right )}^2}\right ) \] Input:
int((exp(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^3,x)
Output:
- expint(- a - b*x - c*x^2)/2 - exp(a + b*x + c*x^2)*(1/(2*(a + b*x + c*x^ 2)) + 1/(2*(a + b*x + c*x^2)^2))
\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=e^{a} \left (\left (\int \frac {e^{c \,x^{2}+b x}}{c^{3} x^{6}+3 b \,c^{2} x^{5}+3 a \,c^{2} x^{4}+3 b^{2} c \,x^{4}+6 a b c \,x^{3}+b^{3} x^{3}+3 a^{2} c \,x^{2}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) b +2 \left (\int \frac {e^{c \,x^{2}+b x} x}{c^{3} x^{6}+3 b \,c^{2} x^{5}+3 a \,c^{2} x^{4}+3 b^{2} c \,x^{4}+6 a b c \,x^{3}+b^{3} x^{3}+3 a^{2} c \,x^{2}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) c \right ) \] Input:
int(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^3,x)
Output:
e**a*(int(e**(b*x + c*x**2)/(a**3 + 3*a**2*b*x + 3*a**2*c*x**2 + 3*a*b**2* x**2 + 6*a*b*c*x**3 + 3*a*c**2*x**4 + b**3*x**3 + 3*b**2*c*x**4 + 3*b*c**2 *x**5 + c**3*x**6),x)*b + 2*int((e**(b*x + c*x**2)*x)/(a**3 + 3*a**2*b*x + 3*a**2*c*x**2 + 3*a*b**2*x**2 + 6*a*b*c*x**3 + 3*a*c**2*x**4 + b**3*x**3 + 3*b**2*c*x**4 + 3*b*c**2*x**5 + c**3*x**6),x)*c)