Integrand size = 31, antiderivative size = 90 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^3 \, dx=-6 e^{a+b x+c x^2}+6 e^{a+b x+c x^2} \left (a+b x+c x^2\right )-3 e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3 \] Output:
-6*exp(c*x^2+b*x+a)+6*exp(c*x^2+b*x+a)*(c*x^2+b*x+a)-3*exp(c*x^2+b*x+a)*(c *x^2+b*x+a)^2+exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^3
Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.54 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^3 \, dx=e^{a+x (b+c x)} \left (-6+6 (a+x (b+c x))-3 (a+x (b+c x))^2+(a+x (b+c x))^3\right ) \] 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))*(-6 + 6*(a + x*(b + c*x)) - 3*(a + x*(b + c*x))^2 + (a + x*(b + c*x))^3)
Time = 0.69 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {7258, 2607, 2607, 2607, 2624}
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.
\(\displaystyle \int (b+2 c x) e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3 \, dx\) |
\(\Big \downarrow \) 7258 |
\(\displaystyle \int e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3d\left (a+b x+c x^2\right )\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3-3 \int e^{c x^2+b x+a} \left (c x^2+b x+a\right )^2d\left (c x^2+b x+a\right )\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3-3 \left (e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2-2 \int e^{c x^2+b x+a} \left (c x^2+b x+a\right )d\left (c x^2+b x+a\right )\right )\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3-3 \left (e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2-2 \left (e^{a+b x+c x^2} \left (a+b x+c x^2\right )-\int e^{c x^2+b x+a}d\left (c x^2+b x+a\right )\right )\right )\) |
\(\Big \downarrow \) 2624 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3-3 \left (e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2-2 \left (e^{a+b x+c x^2} \left (a+b x+c x^2\right )-e^{a+b x+c x^2}\right )\right )\) |
Input:
Int[E^(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^3,x]
Output:
E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^3 - 3*(E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^2 - 2*(-E^(a + b*x + c*x^2) + E^(a + b*x + c*x^2)*(a + b*x + c*x^ 2)))
Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m _.), x_Symbol] :> Simp[(c + d*x)^m*((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Simp[d*(m/(f*g*n*Log[F])) 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] && GtQ[m, 0] && IntegerQ[2* m] && !TrueQ[$UseGamma]
Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; FreeQ[{F, n}, x] && LinearQ[v, x]
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 = 0.62 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(-6 \,{\mathrm e}^{c \,x^{2}+b x +a}+6 \,{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )-3 \,{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )^{2}+{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )^{3}\) | \(87\) |
default | \(-6 \,{\mathrm e}^{c \,x^{2}+b x +a}+6 \,{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )-3 \,{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )^{2}+{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )^{3}\) | \(87\) |
gosper | \(\left (c^{3} x^{6}+3 b \,c^{2} x^{5}+3 a \,c^{2} x^{4}+3 b^{2} c \,x^{4}+6 a b c \,x^{3}+b^{3} x^{3}-3 c^{2} x^{4}+3 a^{2} c \,x^{2}+3 a \,b^{2} x^{2}-6 b c \,x^{3}+3 a^{2} b x -6 a c \,x^{2}-3 b^{2} x^{2}+a^{3}-6 a b x +6 c \,x^{2}-3 a^{2}+6 b x +6 a -6\right ) {\mathrm e}^{c \,x^{2}+b x +a}\) | \(145\) |
risch | \(\left (c^{3} x^{6}+3 b \,c^{2} x^{5}+3 a \,c^{2} x^{4}+3 b^{2} c \,x^{4}+6 a b c \,x^{3}+b^{3} x^{3}-3 c^{2} x^{4}+3 a^{2} c \,x^{2}+3 a \,b^{2} x^{2}-6 b c \,x^{3}+3 a^{2} b x -6 a c \,x^{2}-3 b^{2} x^{2}+a^{3}-6 a b x +6 c \,x^{2}-3 a^{2}+6 b x +6 a -6\right ) {\mathrm e}^{c \,x^{2}+b x +a}\) | \(145\) |
orering | \(\left (c^{3} x^{6}+3 b \,c^{2} x^{5}+3 a \,c^{2} x^{4}+3 b^{2} c \,x^{4}+6 a b c \,x^{3}+b^{3} x^{3}-3 c^{2} x^{4}+3 a^{2} c \,x^{2}+3 a \,b^{2} x^{2}-6 b c \,x^{3}+3 a^{2} b x -6 a c \,x^{2}-3 b^{2} x^{2}+a^{3}-6 a b x +6 c \,x^{2}-3 a^{2}+6 b x +6 a -6\right ) {\mathrm e}^{c \,x^{2}+b x +a}\) | \(145\) |
norman | \(\left (a^{3}-3 a^{2}+6 a -6\right ) {\mathrm e}^{c \,x^{2}+b x +a}+c^{3} x^{6} {\mathrm e}^{c \,x^{2}+b x +a}+\left (3 c^{2} a +3 b^{2} c -3 c^{2}\right ) x^{4} {\mathrm e}^{c \,x^{2}+b x +a}+\left (3 a^{2} c +3 a \,b^{2}-6 a c -3 b^{2}+6 c \right ) x^{2} {\mathrm e}^{c \,x^{2}+b x +a}+b \left (6 a c +b^{2}-6 c \right ) x^{3} {\mathrm e}^{c \,x^{2}+b x +a}+3 b \,c^{2} x^{5} {\mathrm e}^{c \,x^{2}+b x +a}+3 b \left (a^{2}-2 a +2\right ) x \,{\mathrm e}^{c \,x^{2}+b x +a}\) | \(188\) |
parallelrisch | \(c^{3} x^{6} {\mathrm e}^{c \,x^{2}+b x +a}+3 b \,c^{2} x^{5} {\mathrm e}^{c \,x^{2}+b x +a}+3 x^{4} {\mathrm e}^{c \,x^{2}+b x +a} a \,c^{2}+3 x^{4} {\mathrm e}^{c \,x^{2}+b x +a} b^{2} c -3 x^{4} {\mathrm e}^{c \,x^{2}+b x +a} c^{2}+6 x^{3} {\mathrm e}^{c \,x^{2}+b x +a} a b c +x^{3} {\mathrm e}^{c \,x^{2}+b x +a} b^{3}-6 x^{3} {\mathrm e}^{c \,x^{2}+b x +a} b c +3 x^{2} {\mathrm e}^{c \,x^{2}+b x +a} a^{2} c +3 x^{2} {\mathrm e}^{c \,x^{2}+b x +a} a \,b^{2}-6 x^{2} {\mathrm e}^{c \,x^{2}+b x +a} a c -3 x^{2} {\mathrm e}^{c \,x^{2}+b x +a} b^{2}+3 x \,{\mathrm e}^{c \,x^{2}+b x +a} a^{2} b +6 x^{2} {\mathrm e}^{c \,x^{2}+b x +a} c -6 x \,{\mathrm e}^{c \,x^{2}+b x +a} a b +{\mathrm e}^{c \,x^{2}+b x +a} a^{3}+6 x \,{\mathrm e}^{c \,x^{2}+b x +a} b -3 \,{\mathrm e}^{c \,x^{2}+b x +a} a^{2}+6 \,{\mathrm e}^{c \,x^{2}+b x +a} a -6 \,{\mathrm e}^{c \,x^{2}+b x +a}\) | \(355\) |
parts | \(\text {Expression too large to display}\) | \(1562\) |
Input:
int(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
-6*exp(c*x^2+b*x+a)+6*exp(c*x^2+b*x+a)*(c*x^2+b*x+a)-3*exp(c*x^2+b*x+a)*(c *x^2+b*x+a)^2+exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^3
Time = 0.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.21 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^3 \, dx={\left (c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + {\left (a - 1\right )} c^{2}\right )} x^{4} + {\left (b^{3} + 6 \, {\left (a - 1\right )} b c\right )} x^{3} + a^{3} + 3 \, {\left (a^{2} - 2 \, a + 2\right )} b x + 3 \, {\left ({\left (a - 1\right )} b^{2} + {\left (a^{2} - 2 \, a + 2\right )} c\right )} x^{2} - 3 \, a^{2} + 6 \, a - 6\right )} e^{\left (c x^{2} + b x + a\right )} \] Input:
integrate(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^3,x, algorithm="fricas" )
Output:
(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + (a - 1)*c^2)*x^4 + (b^3 + 6*(a - 1)*b* c)*x^3 + a^3 + 3*(a^2 - 2*a + 2)*b*x + 3*((a - 1)*b^2 + (a^2 - 2*a + 2)*c) *x^2 - 3*a^2 + 6*a - 6)*e^(c*x^2 + b*x + a)
Time = 0.12 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.78 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^3 \, dx=\left (a^{3} + 3 a^{2} b x + 3 a^{2} c x^{2} - 3 a^{2} + 3 a b^{2} x^{2} + 6 a b c x^{3} - 6 a b x + 3 a c^{2} x^{4} - 6 a c x^{2} + 6 a + b^{3} x^{3} + 3 b^{2} c x^{4} - 3 b^{2} x^{2} + 3 b c^{2} x^{5} - 6 b c x^{3} + 6 b x + c^{3} x^{6} - 3 c^{2} x^{4} + 6 c x^{2} - 6\right ) e^{a + b x + c x^{2}} \] Input:
integrate(exp(c*x**2+b*x+a)*(2*c*x+b)*(c*x**2+b*x+a)**3,x)
Output:
(a**3 + 3*a**2*b*x + 3*a**2*c*x**2 - 3*a**2 + 3*a*b**2*x**2 + 6*a*b*c*x**3 - 6*a*b*x + 3*a*c**2*x**4 - 6*a*c*x**2 + 6*a + b**3*x**3 + 3*b**2*c*x**4 - 3*b**2*x**2 + 3*b*c**2*x**5 - 6*b*c*x**3 + 6*b*x + c**3*x**6 - 3*c**2*x* *4 + 6*c*x**2 - 6)*exp(a + b*x + c*x**2)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.78 (sec) , antiderivative size = 2381, normalized size of antiderivative = 26.46 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^3,x, algorithm="maxima" )
Output:
1/2*sqrt(pi)*a^3*b*erf(sqrt(-c)*x - 1/2*b/sqrt(-c))*e^(a - 1/4*b^2/c)/sqrt (-c) - 3/4*(sqrt(pi)*(2*c*x + b)*b*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/( sqrt(-(2*c*x + b)^2/c)*c^(3/2)) - 2*e^(1/4*(2*c*x + b)^2/c)/sqrt(c))*a^2*b ^2*e^(a - 1/4*b^2/c)/sqrt(c) + 3/8*(sqrt(pi)*(2*c*x + b)*b^2*(erf(1/2*sqrt (-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(5/2)) - 4*b*e^(1/4*(2* c*x + b)^2/c)/c^(3/2) - 4*(2*c*x + b)^3*gamma(3/2, -1/4*(2*c*x + b)^2/c)/( (-(2*c*x + b)^2/c)^(3/2)*c^(5/2)))*a*b^3*e^(a - 1/4*b^2/c)/sqrt(c) - 1/16* (sqrt(pi)*(2*c*x + b)*b^3*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2* c*x + b)^2/c)*c^(7/2)) - 6*b^2*e^(1/4*(2*c*x + b)^2/c)/c^(5/2) - 12*(2*c*x + b)^3*b*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(3/2)*c^(7/ 2)) + 8*gamma(2, -1/4*(2*c*x + b)^2/c)/c^(3/2))*b^4*e^(a - 1/4*b^2/c)/sqrt (c) - 1/2*(sqrt(pi)*(2*c*x + b)*b*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(s qrt(-(2*c*x + b)^2/c)*c^(3/2)) - 2*e^(1/4*(2*c*x + b)^2/c)/sqrt(c))*a^3*sq rt(c)*e^(a - 1/4*b^2/c) + 9/8*(sqrt(pi)*(2*c*x + b)*b^2*(erf(1/2*sqrt(-(2* c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(5/2)) - 4*b*e^(1/4*(2*c*x + b)^2/c)/c^(3/2) - 4*(2*c*x + b)^3*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2* c*x + b)^2/c)^(3/2)*c^(5/2)))*a^2*b*sqrt(c)*e^(a - 1/4*b^2/c) - 3/4*(sqrt( pi)*(2*c*x + b)*b^3*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(7/2)) - 6*b^2*e^(1/4*(2*c*x + b)^2/c)/c^(5/2) - 12*(2*c*x + b)^ 3*b*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(3/2)*c^(7/2))...
Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.59 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^3 \, dx={\left ({\left (c x^{2} + b x + a\right )}^{3} + 6 \, c x^{2} - 3 \, {\left (c x^{2} + b x + a\right )}^{2} + 6 \, b x + 6 \, a - 6\right )} e^{\left (c x^{2} + b x + a\right )} \] Input:
integrate(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
((c*x^2 + b*x + a)^3 + 6*c*x^2 - 3*(c*x^2 + b*x + a)^2 + 6*b*x + 6*a - 6)* e^(c*x^2 + b*x + a)
Time = 0.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.61 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^3 \, dx={\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{c\,x^2}\,\left (a^3+3\,a^2\,b\,x+3\,a^2\,c\,x^2-3\,a^2+3\,a\,b^2\,x^2+6\,a\,b\,c\,x^3-6\,a\,b\,x+3\,a\,c^2\,x^4-6\,a\,c\,x^2+6\,a+b^3\,x^3+3\,b^2\,c\,x^4-3\,b^2\,x^2+3\,b\,c^2\,x^5-6\,b\,c\,x^3+6\,b\,x+c^3\,x^6-3\,c^2\,x^4+6\,c\,x^2-6\right ) \] Input:
int(exp(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^3,x)
Output:
exp(b*x)*exp(a)*exp(c*x^2)*(6*a + 6*b*x + 6*c*x^2 - 3*a^2 + a^3 - 3*b^2*x^ 2 + b^3*x^3 - 3*c^2*x^4 + c^3*x^6 + 3*a*b^2*x^2 + 3*a^2*c*x^2 + 3*a*c^2*x^ 4 + 3*b^2*c*x^4 + 3*b*c^2*x^5 - 6*a*b*x + 3*a^2*b*x - 6*a*c*x^2 - 6*b*c*x^ 3 + 6*a*b*c*x^3 - 6)
Time = 0.17 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.61 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^3 \, dx=e^{c \,x^{2}+b x +a} \left (c^{3} x^{6}+3 b \,c^{2} x^{5}+3 a \,c^{2} x^{4}+3 b^{2} c \,x^{4}+6 a b c \,x^{3}+b^{3} x^{3}-3 c^{2} x^{4}+3 a^{2} c \,x^{2}+3 a \,b^{2} x^{2}-6 b c \,x^{3}+3 a^{2} b x -6 a c \,x^{2}-3 b^{2} x^{2}+a^{3}-6 a b x +6 c \,x^{2}-3 a^{2}+6 b x +6 a -6\right ) \] Input:
int(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^3,x)
Output:
e**(a + b*x + c*x**2)*(a**3 + 3*a**2*b*x + 3*a**2*c*x**2 - 3*a**2 + 3*a*b* *2*x**2 + 6*a*b*c*x**3 - 6*a*b*x + 3*a*c**2*x**4 - 6*a*c*x**2 + 6*a + b**3 *x**3 + 3*b**2*c*x**4 - 3*b**2*x**2 + 3*b*c**2*x**5 - 6*b*c*x**3 + 6*b*x + c**3*x**6 - 3*c**2*x**4 + 6*c*x**2 - 6)