Integrand size = 33, antiderivative size = 82 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {3}{2} e^{a+b x+c x^2} \sqrt {a+b x+c x^2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+\frac {3}{4} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \] Output:
-3/2*exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^(1/2)+exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^( 3/2)+3/4*Pi^(1/2)*erfi((c*x^2+b*x+a)^(1/2))
Time = 2.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.56 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {-a-x (b+c x)} \Gamma \left (\frac {5}{2},-a-x (b+c x)\right )}{\sqrt {a+x (b+c x)}} \] Input:
Integrate[E^(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2),x]
Output:
(Sqrt[-a - x*(b + c*x)]*Gamma[5/2, -a - x*(b + c*x)])/Sqrt[a + x*(b + c*x) ]
Time = 0.86 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {7258, 2607, 2607, 2611, 2633}
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.
\(\displaystyle \int (b+2 c x) e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 7258 |
\(\displaystyle \int e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}d\left (a+b x+c x^2\right )\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac {3}{2} \int e^{c x^2+b x+a} \sqrt {c x^2+b x+a}d\left (c x^2+b x+a\right )\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac {3}{2} \left (e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {1}{2} \int \frac {e^{c x^2+b x+a}}{\sqrt {c x^2+b x+a}}d\left (c x^2+b x+a\right )\right )\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac {3}{2} \left (e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\int e^{c x^2+b x+a}d\sqrt {c x^2+b x+a}\right )\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac {3}{2} \left (e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )\right )\) |
Input:
Int[E^(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2),x]
Output:
E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(3/2) - (3*(E^(a + b*x + c*x^2)*Sqrt [a + b*x + c*x^2] - (Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]])/2))/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[$UseGamma]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
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 = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(-\frac {3 \,{\mathrm e}^{c \,x^{2}+b x +a} \sqrt {c \,x^{2}+b x +a}}{2}+{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}+\frac {3 \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right )}{4}\) | \(69\) |
default | \(-\frac {3 \,{\mathrm e}^{c \,x^{2}+b x +a} \sqrt {c \,x^{2}+b x +a}}{2}+{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}+\frac {3 \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right )}{4}\) | \(69\) |
Input:
int(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE )
Output:
-3/2*exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^(1/2)+exp(c*x^2+b*x+a)*(c*x^2+b*x+a)^( 3/2)+3/4*Pi^(1/2)*erfi((c*x^2+b*x+a)^(1/2))
\[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )} \,d x } \] Input:
integrate(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2),x, algorithm="fri cas")
Output:
integral((2*c^2*x^3 + 3*b*c*x^2 + a*b + (b^2 + 2*a*c)*x)*sqrt(c*x^2 + b*x + a)*e^(c*x^2 + b*x + a), x)
Time = 43.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.15 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\left (- \sqrt {- a - b x - c x^{2}} \left (a + b x + c x^{2} - \frac {3}{2}\right ) e^{a + b x + c x^{2}} + \frac {3 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- a - b x - c x^{2}} \right )}}{4}\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (- a - b x - c x^{2}\right )^{\frac {3}{2}}} \] Input:
integrate(exp(c*x**2+b*x+a)*(2*c*x+b)*(c*x**2+b*x+a)**(3/2),x)
Output:
(-sqrt(-a - b*x - c*x**2)*(a + b*x + c*x**2 - 3/2)*exp(a + b*x + c*x**2) + 3*sqrt(pi)*erfc(sqrt(-a - b*x - c*x**2))/4)*(a + b*x + c*x**2)**(3/2)/(-a - b*x - c*x**2)**(3/2)
\[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )} \,d x } \] Input:
integrate(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2),x, algorithm="max ima")
Output:
integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)*e^(c*x^2 + b*x + a), x)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{2} \, {\left (2 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {c x^{2} + b x + a}\right )} e^{\left (c x^{2} + b x + a\right )} + \frac {3}{4} i \, \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {c x^{2} + b x + a}\right ) \] Input:
integrate(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2),x, algorithm="gia c")
Output:
1/2*(2*(c*x^2 + b*x + a)^(3/2) - 3*sqrt(c*x^2 + b*x + a))*e^(c*x^2 + b*x + a) + 3/4*I*sqrt(pi)*erf(-I*sqrt(c*x^2 + b*x + a))
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {3\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,{\left (-c\,x^2-b\,x-a\right )}^{3/2}}-\frac {3\,{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{c\,x^2}\,\sqrt {c\,x^2+b\,x+a}}{2}+{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{c\,x^2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \] Input:
int(exp(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2),x)
Output:
(3*pi^(1/2)*erfc((- a - b*x - c*x^2)^(1/2))*(a + b*x + c*x^2)^(3/2))/(4*(- a - b*x - c*x^2)^(3/2)) - (3*exp(b*x)*exp(a)*exp(c*x^2)*(a + b*x + c*x^2) ^(1/2))/2 + exp(b*x)*exp(a)*exp(c*x^2)*(a + b*x + c*x^2)^(3/2)
\[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \, dx=e^{a} \left (2 \left (\int e^{c \,x^{2}+b x} \sqrt {c \,x^{2}+b x +a}\, x^{3}d x \right ) c^{2}+3 \left (\int e^{c \,x^{2}+b x} \sqrt {c \,x^{2}+b x +a}\, x^{2}d x \right ) b c +2 \left (\int e^{c \,x^{2}+b x} \sqrt {c \,x^{2}+b x +a}\, x d x \right ) a c +\left (\int e^{c \,x^{2}+b x} \sqrt {c \,x^{2}+b x +a}\, x d x \right ) b^{2}+\left (\int e^{c \,x^{2}+b x} \sqrt {c \,x^{2}+b x +a}d x \right ) a b \right ) \] Input:
int(exp(c*x^2+b*x+a)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2),x)
Output:
e**a*(2*int(e**(b*x + c*x**2)*sqrt(a + b*x + c*x**2)*x**3,x)*c**2 + 3*int( e**(b*x + c*x**2)*sqrt(a + b*x + c*x**2)*x**2,x)*b*c + 2*int(e**(b*x + c*x **2)*sqrt(a + b*x + c*x**2)*x,x)*a*c + int(e**(b*x + c*x**2)*sqrt(a + b*x + c*x**2)*x,x)*b**2 + int(e**(b*x + c*x**2)*sqrt(a + b*x + c*x**2),x)*a*b)